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+SoftTreeMax: Exponential Variance Reduction in Policy Gradient
+via Tree Search
+Gal Dalal * Assaf Hallak * Gugan Thoppe Shie Mannor Gal Chechik
+Abstract
+Despite the popularity of policy gradient meth-
+ods, they are known to suffer from large vari-
+ance and high sample complexity. To mitigate
+this, we introduce SoftTreeMax – a generaliza-
+tion of softmax that takes planning into account.
+In SoftTreeMax, we extend the traditional logits
+with the multi-step discounted cumulative reward,
+topped with the logits of future states. We con-
+sider two variants of SoftTreeMax, one for cumu-
+lative reward and one for exponentiated reward.
+For both, we analyze the gradient variance and
+reveal for the first time the role of a tree expan-
+sion policy in mitigating this variance. We prove
+that the resulting variance decays exponentially
+with the planning horizon as a function of the
+expansion policy. Specifically, we show that the
+closer the resulting state transitions are to uni-
+form, the faster the decay. In a practical imple-
+mentation, we utilize a parallelized GPU-based
+simulator for fast and efficient tree search. Our
+differentiable tree-based policy leverages all gra-
+dients at the tree leaves in each environment step
+instead of the traditional single-sample-based gra-
+dient. We then show in simulation how the vari-
+ance of the gradient is reduced by three orders
+of magnitude, leading to better sample complex-
+ity compared to the standard policy gradient. On
+Atari, SoftTreeMax demonstrates up to 5x better
+performance in a faster run time compared to dis-
+tributed PPO. Lastly, we demonstrate that high
+reward correlates with lower variance.
+1. Introduction
+Policy Gradient (PG; Sutton et al. 1999) methods for Re-
+inforcement Learning (RL) are often the first choice for
+environments that allow numerous interactions at a fast pace
+(Schulman et al., 2017). Their success is attributed to several
+*Equal contribution .
+Correspondence to:
+Gal Dalal
+<gdalal@nvidia.com>, Assaf Hallak <ahallak@nvidia.com>.
+Preperint.
+factors, including that they are easy-to-distribute to multiple
+workers, require no assumptions on an underlying value
+function, and have both on-policy and off-policy variants.
+Despite their popularity, PG algorithms are also notoriously
+unstable since they compute gradients over entire trajec-
+tories (Liu et al., 2020; Xu et al., 2020). As a result, PG
+algorithms tend to be highly inefficient in terms of sample
+complexity. Several solutions were proposed to mitigate the
+instability of PG methods, including baseline subtraction
+(Greensmith et al., 2004; Weaver & Tao, 2001; Thomas &
+Brunskill, 2017; Wu et al., 2018), anchor-point averaging
+(Papini et al., 2018), and other variance reduction techniques
+(Zhang et al., 2021; Shen et al., 2019; Pham et al., 2020).
+A second family of algorithms that achieved state-of-the-art
+results in several domains is based on planning (Silver et al.,
+2016; Ye et al., 2021). Planning is exercised primarily in the
+context of value-based RL and is usually implemented using
+a Tree Search (TS; Coulom 2006; Silver 2009). In this work,
+we combine PG with TS by introducing a parameterized dif-
+ferentiable policy that incorporates tree expansion. Namely,
+our SoftTreeMax policy replaces the standard policy logits
+of a state and action, with the expected value of trajectories
+that originate from these state and action.
+Combining TS into PG suite should be done with care given
+the biggest hurdle of PG – its high gradient variance. This
+raises prominent actionable questions that were ignored un-
+til this work: How does the tree-expansion policy affect
+the PG variance? And, can we design tree-expansion that
+is guaranteed to strongly reduces that variance? Here, we
+analyze the gradient variance of SoftTreeMax, and provide
+a practical methodology to choose the expansion policy to
+minimize the resulting variance. Our main result shows that
+a desirable expansion policy is one that induces transitions
+as close to uniform as possible. More generally, we show
+that the gradient variance of SoftTreeMax decays at an expo-
+nential rate of |λ2|d, where d is the tree depth and λ2 is the
+second eigenvalue of the transition matrix induced by the
+tree expansion policy. This paper is the first to prove such a
+relation between PG variance and TS expansion policy.
+Common practices for expanding the tree rely on a value
+estimate, using UCT (Kocsis & Szepesv´ari, 2006; Browne
+et al., 2012), or based on some prior knowledge such as
+arXiv:2301.13236v1  [cs.LG]  30 Jan 2023
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+human-collected trajectories (Silver et al., 2018). Our work
+raises the question of whether optimal variance reduction
+corresponds to the appealing regret properties of UCT.
+To verify our results, we implemented a practical version of
+SoftTreeMax that exhaustively searches the entire tree and
+applies a neural network on its leaves. We test our algorithm
+in the Atari domain, where it is possible to span the whole
+tree of the (nearly) deterministic Atari environment. Hence,
+the gradient variance has no sampling component, and our
+variance calculations indeed match the empirical gradient
+variance. Our search mechanism uses a GPU simulator that
+allows multiple copies of the environment to be run in par-
+allel (Dalton et al., 2020). To enable a tractable deep search,
+up to depth eight, we also introduce a pruning technique
+that limits the width of the tree. We do so by sampling only
+the most promising nodes at each level.
+We integrate our SoftTreeMax GPU implementation into
+the popular PPO (Schulman et al., 2017) and compare it to
+the flat distributed variant of PPO. For a fair comparison,
+we also run the distributed PPO baseline with the parallel
+GPU emulator by Dalton et al. (2020). In all tested Atari
+games, our results outperform the baseline and obtain up
+to 5x more reward. We further show in Section 6 that the
+associated gradient variance is smaller by three orders of
+magnitude in all games, demonstrating the relation between
+low gradient variance and high reward.
+We summarize our key contributions:
+1. We explore the relation between two seemingly unre-
+lated families of SoTA approaches: PG and TS, and
+show how they can be combined.
+2. We introduce SoftTreeMax: A novel parametric policy
+that generalizes softmax to planning. We propose both
+cumulative and exponentiated reward variants.
+3. We prove that the gradient variance of SoftTreeMax in
+its two variants decays exponentially with its TS depth.
+Our analysis sheds new light on the choice of tree
+expansion policy. It raises the question of optimality in
+terms of variance versus the previously studied regret.
+4. We implement a differentiable deep version of
+SoftTreeMax that employs a parallelized GPU TS. We
+demonstrate how its gradient variance is reduced by
+three orders of magnitude over PPO while obtaining
+up to 5x reward.
+2. Preliminaries
+We follow the standard notation by (Puterman, 2014). Con-
+sider a discounted Markov Decision Process (MDP) M =
+(S, A, P, r, γ), where S is a finite state space of size S, A
+is a finite action space of size A, r : S × A → [0, 1] is
+the reward function, P : S × A → ∆S is the transition
+function, and γ ∈ (0, 1) is the discount factor. In vector
+form, denote the transition matrix starting from state s by
+[Ps]a,s′ = Pr(s′|a, s) ∈ [0, 1]A×S, and the corresponding
+reward vector by Rs = r(s, ·) ∈ RA.
+Let π : S → ∆A be a stationary policy. We define the in-
+duced transition matrix P π(s′|s) = �
+a π(a|s) Pr(s′|s, a)
+and reward function Rπ(s) = �
+a π(a|s)r(s, a).
+De-
+note by µπ ∈ RS the stationary distribution of P π, s.t.
+µ⊤
+π P π = P π. Also, let V π ∈ RS be the value function of
+π defined by V π(s) = Eπ [�∞
+t=0 γtr (st, π(st)) | s0 = s],
+and let Qπ ∈ RS×A be the Q-function such that Qπ(s, a) =
+Eπ [r(s, a) + γV π(s′)].
+Our goal is to find an optimal policy π⋆ such that
+V ⋆(s) ≡ V π⋆(s) = max
+π
+V π(s),
+∀s ∈ S.
+Lastly, for the analysis in Section 4, we introduce the follow-
+ing vector notation. Denote by Θ ∈ RS the vector represen-
+tation of θ(s) ∀s ∈ S. For a vector u, denote by exp(u) the
+coordinate-wise exponent of u and by D(u) the diagonal
+square matrix with u in its diagonal. For matrix A, denote
+its i-th eigenvalue by λi(A). Denote the k-dimensional iden-
+tity matrix and all-ones vector by Ik and 1k, respectively.
+We denote the trace operator by Tr . Finally, We treat all
+vectors as column vectors.
+2.1. Policy Gradient
+PG schemes seek to maximize the cumulative reward as a
+function of the parameterized policy πθ(a|s) by perform-
+ing gradient steps on θ. The celebrated Policy Gradient
+Theorem (Sutton et al., 1999) states that
+∂
+∂θ
+�
+µ⊤
+πθV πθ�
+= Es∼µπθ ,a∼πθ(·|s) [∇θ log πθ(a|s)Qπθ(s, a)] .
+The variance of the gradient is thus
+Vars∼µπθ ,a∼πθ(·|s) (∇θ log πθ(a|s)Qπθ(s, a)) .
+(1)
+In the notation above, we denote the variance of a vector
+random variable X by:
+Varx (X) = Tr
+�
+Ex
+�
+(X − ExX)⊤ (X − ExX]
+��
+,
+similarly as in (Greensmith et al., 2004). From here on, we
+drop the subscript from Var in (1) for brevity.
+When the action space is discrete, a commonly used param-
+eterized policy is softmax:
+πθ(a|s) ∝ exp (θ(s, a)) ,
+where θ : S × A → R is a state-action parametrization.
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+3. SoftTreeMax: Exponent of trajectories
+We introduce a new family of policies called SoftTreeMax,
+which are a model-based generalization of the popu-
+lar softmax.
+We propose two variants:
+Cumulative
+(C-SoftTreeMax) and Exponentiated (E-SoftTreeMax). In
+both variants, we replace the generic softmax logits θ(s, a)
+with the score of a trajectory of horizon d starting from
+s, a, generated by applying a behavior policy πb.
+In
+C-SoftTreeMax, we exponentiate the expectation of the log-
+its. In E-SoftTreeMax, we first exponentiate the logits, and
+only then compute their expectation.
+Logits. Let the SoftTreeMax logit ℓs,a(d; θ) be a random
+variable depicting the score of a trajectory of horizon d
+starting from s, a and following the policy πb :
+ℓs,a(d; θ) =
+d−1
+�
+t=0
+γtrt + γdθ(sd).
+(2)
+Namely, s0 = s, a0 = a, at ∼ πb(·|st) ∀t ≥ 1, and
+rt ≡ r (st, at) . For brevity of the analysis, we let the para-
+metric score θ in (2) be state-based, similarly to a value
+function. Instead, one could use a state-action input analo-
+gous to a Q-function. This freedom allows easy integration
+of SoftTreeMax to the two types of RL algorithm imple-
+mentations in standard packages.
+C-SoftTreeMax. Given an inverse temperature parameter
+β, let C-SoftTreeMax be
+πC
+d,θ(a|s) ∝ exp [βEπbℓs,a(d; θ)] .
+(3)
+C-SoftTreeMax gives higher weight for actions that result
+in higher expected returns. While standard softmax relies
+entirely on parametrization θ, C-SoftTreeMax also interpo-
+lates a Monte-Carlo portion of the reward.
+Using the monotone convergence theorem (since rewards
+are non-negative), it follows that when d → ∞,
+πC
+d→∞,θ(a|s) ∝ exp [βQπb(s, a)] ,
+corresponding to Boltzmann exploration (Sutton et al., 1999)
+using the behavior policy πb.
+E-SoftTreeMax. A second natural operator to consider is
+E-SoftTreeMax, in which the expectation is taken outside
+the exponent:
+πE
+d,θ(a|s) ∝ Eπb exp [(βℓs,a(d; θ))] .
+(4)
+This objective corresponds to the exponentiated reward ob-
+jective which is often used for risk-sensitive RL (Howard
+& Matheson, 1972; Fei et al., 2021; Noorani & Baras,
+2021). The common risk-sensitive objective is of the form
+log E[exp(δR)], where δ is the risk parameter and R is the
+cumulative reward. Similarly to that literature, the exponent
+in (4) emphasizes the most promising trajectories.
+SoftTreeMax properties. SoftTreeMax is a natural model-
+based generalization of softmax. For d = 0, both variants
+above coincide, since (2) becomes deterministic. In that
+case and for a state-action parametrization, they reduce
+to standard softmax. When β → 0, both variants again
+coincide and sample actions uniformly (exploration). When
+β → ∞, the policies become deterministic and greedily
+optimize for the best trajectory (exploitation). The best
+trajectory is in expectation in the case of C-SoftTreeMax,
+and in terms of best sample-path for E-SoftTreeMax.
+SoftTreeMax convergence. Under regularity conditions,
+for any parametric policy, PG converges to local optima
+(Bhatnagar et al., 2009), and thus also SoftTreeMax. Specif-
+ically for softmax PG, asymptotic (Agarwal et al., 2021) and
+rate results (Mei et al., 2020b) were recently obtained. A fu-
+ture direction would be to extend those for the convergence
+properties of SoftTreeMax.
+SoftTreeMax gradient. The two variants of SoftTreeMax
+involve an expectation. This expectation is taken over Sd
+many trajectories from the root state s and are weighted
+according to their probability. Thus, during the PG train-
+ing process, the gradient ∇θ log πθ is calculated using a
+weighted sum of gradients over all reachable states starting
+from s. Our method exploits the exponential number of tra-
+jectories to reduce the variance. Indeed, in the next section
+we prove that the gradient variance of SoftTreeMax decays
+exponentially fast as a function of the behavior policy πb and
+trajectory length d. In the experiments in Section 6, we also
+show how the practical version of SoftTreeMax achieves
+a significant reduction in the noise of the PG process and
+leads to faster convergence and higher reward.
+4. Theoretical Analysis
+In this section, we bound the variance of PG when using
+SoftTreeMax policy. Specifically, we show that the variance
+decreases exponentially with the tree depth, where the rate
+is determined by the second eigenvalue of the transition
+kernel induced by πb. We analyze the gradient variance w.r.t.
+state-action frequencies, as a function of problem param-
+eters. Other types of analyses could have instead focused
+on the estimation aspect in the context of sampling. Indeed,
+in our implementation in Section 5, we manage to avoid
+sampling and directly compute the expectations in Eqs. (3)
+and (4). As we show later, we do so by leveraging efficient
+parallel simulation on the GPU in feasible run-time. In our
+application, due to the nature of the finite action space and
+quasi-deterministic Atari dynamics (Bellemare et al., 2013),
+our expectation estimator is noiseless. We encourage future
+work to account for the finite-sample variance component.
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+We begin with a general variance bound that holds for any
+parametric policy. We defer all the proofs in this section to
+Appendix A.1.
+Lemma 4.1 (Bound on the policy gradient variance). For
+any parametric policy πθ and function Qπθ : S × A → R,
+Var (∇θ log πθ(a|s)Qπθ(s, a))
+��� max
+s,a [Qπθ(s, a)]2 max
+s
+||∇θ log πθ(·|s)||2
+F ,
+where ∇θ log πθ(·|s) ∈ RA×dim(θ) is a matrix whose a-th
+row is ∇θ log πθ(a|s)⊤.
+Hence, to bound (1), it is sufficient to bound the Frobenius
+norm of the policy gradient ∇θ log πθ(·|s) for any s.
+A common assumption in the RL literature (Szepesv´ari,
+2010) that we adopt for the remainder of the section is that
+the transition matrix P πb, induced by the behavior policy
+πb, is irreducible and aperiodic. Subsequently, its second
+highest eigenvalue holds: |λ2(P πb)| < 1.
+From here on, we split the variance results for the
+two variants of SoftTreeMax to two subsections.
+For
+C-SoftTreeMax, the analysis is simpler and we provide an
+exact bound. The case of E-SoftTreeMax is more involved
+and we provide for it a more general result. In both cases,
+we show that the variance decays exponentially with the
+planning horizon.
+4.1. Variance of C-SoftTreeMax
+We express C-SoftTreeMax in vector form as follows.
+Lemma 4.2 (Vector form of C-SoftTreeMax). For d ≥ 1,
+(3) is given by
+πC
+d,θ(·|s) =
+exp
+�
+β
+�
+Cs,d + γdPs (P πb)d−1 Θ
+��
+1⊤
+A exp
+�
+β
+�
+Cs,d + γdPs (P πb)d−1 Θ
+��,
+(5)
+where
+Cs,d = Rs + Ps
+�d−1
+�
+h=1
+γh (P πb)h−1
+�
+Rπb.
+(6)
+The matrix Cs,d ∈ RA×S represents the cumulative dis-
+counted reward in expectation along the trajectory of hori-
+zon d. Starting from the state s, the reward Rs is collected
+and a transition occurs according to Ps. Then, the policy
+πb is applied to obtain the reward Rπb and transition, and
+the process repeats. When depth d is reached, we apply the
+score function on the last state as depicted in (5).
+Next, we express the policy gradient of C-SoftTreeMax
+Lemma
+4.3
+(Gradient
+of
+C-SoftTreeMax).
+The
+C-SoftTreeMax gradient of dimension A × S is given by
+∇θ log πC
+d,θ = βγd �
+IA − 1A(πC
+d,θ)⊤�
+Ps (P πb)d−1 ,
+where for brevity, we drop the s index in the policy above,
+i.e., πC
+d,θ ≡ πC
+d,θ(·|s).
+We are now ready to present our first main result:
+Theorem
+4.4
+(Exponential
+variance
+decay
+of
+C-SoftTreeMax). For every Q
+:
+S × A
+→
+R, the
+C-SoftTreeMax policy gradient is bounded by
+Var
+�
+∇θ log πC
+d,θ(a|s)Q(s, a)
+�
+≤ 2 A2S2β2
+(1 − γ)2 γ2d|λ2(P πb)|2(d−1).
+Although we provide a rigorous proof in Appendix A.1.4,
+since the proof relatively accessible, we briefly outline its
+essence here.
+Proof outline. Lemma 4.1 allows us to bound the variance
+using a direct bound on the gradient norm. The gradient is
+given in Lemma 4.3 as a product of three matrices, which
+we now study from right to left. The matrix P πb is a row-
+stochastic matrix. Because the associated Markov chain is
+irreducible and aperiodic, it has a unique stationary distribu-
+tion. This implies that P πb has one and only one eigenvalue
+equal to 1; all others have magnitude strictly less than 1. Let
+us suppose that all these other eigenvalues have multiplicity
+1 (the general case with repeated eigenvalues can be handled
+via Jordan decompositions as in (Pelletier, 1998, Lemma1)).
+Then, P πb has the spectral decomposition
+P πb = 1Sµ⊤
+πb +
+S
+�
+i=2
+λiviu⊤
+i ,
+where λi is the i-th eigenvalue of P πb (ordered in descend-
+ing order according to their magnitude) and ui and vi are
+the corresponding left and right eigenvectors, respectively.
+Therefore,
+(P πb)d−1 = 1Sµ⊤
+πb +
+S
+�
+i=2
+λd−1
+i
+viu⊤
+i .
+(7)
+The second matrix in the gradient relation in Lemma 4.3, Ps,
+is a rectangular transition matrix that translates the vector
+of all ones from dimension S to A : Ps1S = 1A.
+Lastly,
+the first matrix
+�
+IA − 1A(πC
+d,θ)⊤�
+is a pro-
+jection whose null-space includes the vector 1A, i.e.,
+�
+IA − 1A(πC
+d,θ)⊤�
+1A = 0.
+Combining the three properties above when multiplying
+the three matrices of the gradient, it is easy to see that
+the first term in (7) gets canceled, and we are left with
+bounded summands scaled by λi(P πb)d−1. Recalling that
+|λi(P πb)| < 1 and that |λ2| > |λ3| > . . . for i = 2, . . . , S,
+we obtain the desired result.
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+2
+4
+6
+8
+10
+Depth d
+10
+31
+10
+26
+10
+21
+10
+16
+10
+11
+10
+6
+10
+1
+SoftTreeMax 
+Gradient variance 
+Permutation: True variance
+Permutation: Variance bound
+Random: True variance
+Random: Variance bound
+Uniform: True variance
+Uniform: Variance bound
+Figure 1. A comparison of the analytical PG variance and our
+bound for E-SoftTreeMax on randomly drawn MDPs. We present
+three cases for P πb : (i) close to uniform, (ii) drawn randomly, and
+(iii) close to a permutation matrix. This experiment verifies the
+optimal and worse-case rate decay cases. The variance bounds here
+are taken from Theorem 4.7 where we substitute α = |λ2(P πb)|.
+Theorem 4.4 guarantees that the variance of the gradient
+decays exponentially with d, regardless of γ. It also pro-
+vides a novel insight that drives us to choose the behavior
+policy πb as the policy that minimizes the absolute second
+eigenvalue of the P πb. Indeed, the second eigenvalue of a
+Markov chain has known connections to its connectivity and
+its rate of convergence to the stationary distribution (Levin
+& Peres, 2017).
+Optimal variance decay. To achieve the best reduction
+in variance, the behavior policy πb should be chosen to
+achieve uniformity. That is, that transitions induced by
+the interaction of πb with the environment are uniform. In
+that case, P πb is a rank one matrix of the form 1Sµ⊤
+πb, and
+λ2(P πb) = 0. Then, Var (∇θ log πθ(a|s)Q(s, a)) = 0. As
+we show in Section 5, we choose our tree expansion policy
+accordingly.
+Worst-case variance decay. In contrast, and somewhat
+surprisingly, when πb is chosen so that the dynamics is
+deterministic, there is no guarantee that it will decay expo-
+nentially fast. For example, if P πb is a permutation matrix,
+then λ2(P πb) = 1, and advancing the tree amounts to only
+updating the gradient of one state for every action, as in the
+basic softmax.
+4.2. Variance of E-SoftTreeMax
+The proof of the variance bound for E-SoftTreeMax is sim-
+ilar to that of C-SoftTreeMax, but more involved. It also
+requires the assumption that the reward depends only on the
+state, i.e. r(s, a) ≡ r(s). This is indeed the case in most
+standard RL environments such as Atari and Mujoco.
+We begin with expressing E-SoftTreeMax in vector form.
+Lemma 4.5 (Vector form of E-SoftTreeMax). For d ≥ 1,
+(4) is given by
+πE
+d,θ(·|s) =
+Es,d exp(βγdΘ)
+1⊤
+AEs,d exp(βγdΘ),
+(8)
+where
+Es,d = Ps
+d−1
+�
+h=1
+�
+D
+�
+exp(βγhR)
+�
+P πb�
+.
+(9)
+The vector R above is the S-dimensional vector whose s-th
+coordinate is r(s).
+The matrix Es,d ∈ RA×S has a similar role to Cs,d from (6),
+but it represents the exponentiated cumulative discounted
+reward. Accordingly, it is a product of d matrices as opposed
+to a sum. It captures the expected reward sequence starting
+from s and then iteratively following P πb. After d steps, we
+apply the score function on the last state as in (8).
+Lemma
+4.6
+(Gradient
+of
+E-SoftTreeMax).
+The
+E-SoftTreeMax gradient of dimension A × S is given by
+∇θ log πE
+d,θ =
+βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1
+Es,dD(exp(βγdΘ))
+1⊤
+AEs,d exp(βγdΘ)
+,
+where for brevity, we drop the s index in the policy above,
+i.e., πE
+d,θ ≡ πE
+d,θ(·|s).
+This gradient structure is harder to handle than that of
+C-SoftTreeMax in Lemma 4.3, but here we also prove an
+exponential variance decay nonetheless.
+Theorem
+4.7
+(Exponential
+variance
+decay
+of
+E-SoftTreeMax). There exists α
+∈
+(0, 1) such that,
+for any function Q : S × A → R,
+Var
+�
+∇θ log πE
+d,θ(a|s)Q(s, a)
+�
+∈ O
+�
+β2γ2dα2d�
+.
+If all rewards are equal (r ≡ const), then α = |λ2(P πb)|.
+The proof structure is similar in spirit to that of Theorem 4.4,
+but several new technical arguments are needed. We give it
+in full in Appendix A.1.4, but briefly outline it here.
+Proof outline. Recall that thanks to Lemma 4.1, we can
+bound the PG variance using a direct bound on the gradient
+norm. The definition of the induced norm is
+∥∇θ log πE
+d,θ∥ = max
+z:∥z∥=1 ∥∇θ log πE
+d,θz∥,
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+with ∇θ log πE
+d,θ given in Lemma 4.6. Let z ∈ RS be an
+arbitrary vector such that ∥z∥ = 1. Then, z = �S
+i=1 cizi,
+where ci are scalar coefficients and zi are vectors spanning
+the S-dimensional space. In the full proof, we show our
+specific choice of zi and prove they are linearly independent
+given that choice. We do note that z1 = 1S.
+The first part of the proof relies on the fact that
+(∇θ log πE
+d,θ)z1 = 0. This is easy to verify using Lemma 4.6
+together with (8), and because
+�
+IA − 1A(πE
+d,θ)⊤�
+is a pro-
+jection matrix whose null-space is spanned by 1S. Thus,
+∇θ log πE
+d,θz = ∇θ log πE
+d,θ
+S
+�
+i=2
+cizi.
+In the second part of the proof, we focus on Es,d from (9),
+which appears within ∇θ log πE
+d,θ. Notice that Es,d consists
+of the product �d−1
+h=1
+�
+D
+�
+exp(βγhR
+�
+P πb�
+. Even though
+the elements in this product are not stochastic matrices, in
+the full proof we show how to normalize each of them to a
+stochastic matrix Bh. We thus obtain that
+Es,d = PsD(M1)
+d−1
+�
+h=1
+Bh,
+where M1 ∈ RS is some strictly positive vector. Then,
+we can apply a result by Mathkar & Borkar (2016), which
+itself builds on (Chatterjee & Seneta, 1977). The result
+states that the product of stochastic matrices �d−1
+h=1 Bh of
+our particular form converges exponentially fast to a matrix
+of the form 1Sµ⊤ s.t. ∥1Sµ⊤ − �d−1
+h=1 Bh∥ ≤ Cαd for
+some constant C.
+Lastly, 1Sµ⊤
+πb gets canceled due to our choice of zi, i =
+2, . . . , S. This observation along with the above fact that the
+remainder decays then shows that ∇θ log πE
+d,θ
+�S
+i=2 zi =
+O(αd), which gives the desired result.
+Although our proof guarantees that α = |λ2(P πb)| only
+in the constant-reward case, we conjecture that this is also
+true in the general case. To demonstrate this, we run the
+following simulation. We drew a random finite MDP, pa-
+rameter vector Θ ∈ RS
++, and behavior policy πb. We then
+analytically computed the PG variance of E-SoftTreeMax
+as given in (1) and compared it to |λ2(P πb)|d. As seen, the
+true variance and our bound matched almost identically.
+This suggests that indeed α = |λ2(P πb)|. We repeat this
+experiment three times for different P πb : (i) close to uni-
+form, (ii) drawn randomly, and (iii) close to a permutation
+matrix. The three cases match our takeaways on the opti-
+mal and worst-case rate decay cases. We ran multiple such
+experiments and in all of them the lines match closely; we
+give here one such instance. To account the for constants,
+we match the values for the first point in d = 1.
+𝑊(𝑆!"#
+$,$ )
+logits 
+for 
+logits 
+for
+logits 
+for
+𝑎!"$
+($)
+Policy 
+network
+𝑎!"$
+(()
+𝑎!"$
+($)
+𝑎!"$
+(()
+𝑎!"$
+($)
+𝑎!"$
+(()
+𝑎!")
+($)
+𝑎!")
+(#)
+𝑎!")
+(()
+𝑎!"#
+(%)
+𝑎!"#
+(')
+𝑎!"#
+(()
+𝑆!")
+𝑆!"$
+($)
+𝑆!"$
+(#)
+𝑆!"$
+(()
+𝑆!"#
+($,$)
+𝑆!"#
+(#,$)
+𝑆!"#
+($,()
+𝑆!"#
+(#,()
+𝑆!"#
+((,$)
+𝑆!"#
+((,()
+𝑊(𝑆!"#
+$,( )
+𝑊(𝑆!"#
+#,$ )
+𝑊(𝑆!"#
+#,( )
+𝑊(𝑆!"#
+(,$ )
+𝑊(𝑆!"#
+(,( )
+Figure 2. SoftTreeMax policy. Our exhaustive parallel TS ex-
+pands all actions at each state up to depth d (= 2 here). The leaf
+state of every trajectory is used as input to the policy network.
+The output is then added to the trajectory’s cumulative reward as
+described in (2). I.e., instead of the standard softmax logits, we
+add the cumulative discounted reward to the policy network output.
+This policy is differentiable and can be easily integrated into any
+PG algorithm. In this work, we build on PPO and use its loss
+function to train the policy network.
+5. SoftTreeMax: Deep Parallel
+Implementation
+Following the success of deep RL (Mnih et al., 2015), deep
+neural networks are used nowadays almost exclusively in
+practice. Depending on the RL algorithm, a loss function
+is defined and gradients on the network weights can be
+calculated. In PG methods, the scoring function used in the
+softmax is commonly replaced by a neural network Wθ:
+πθ(a|s) ∝ exp (Wθ(s, a)) .
+Similarly, we implement SoftTreeMax by replacing θ(s) in
+(2) with a neural network Wθ(s). Although both variants of
+SoftTreeMax from Section 3 involve computing an expecta-
+tion, this can be hard in general. One approach to handle it is
+with sampling, though these introduce estimation variance
+into the process. We leave the question of sample-based
+theory and algorithmic implementations for future work.
+Instead, in finite action space environments such as Atari,
+we compute the exact expectation in SoftTreeMax with an
+exhaustive TS of depth d. Despite the exponential computa-
+tional cost of spanning the entire tree, recent advancements
+in parallel GPU-based simulation allow efficient expansion
+of all nodes at the same depth simultaneously (Dalal et al.,
+2021; Rosenberg et al., 2022). This is possible when a simu-
+lator is implemented on GPU (Dalton et al., 2020; Makoviy-
+chuk et al., 2021; Freeman et al., 2021), or when a forward
+model is learned (Kim et al., 2020; Ha & Schmidhuber,
+2018). To reduce the complexity to be linear in depth, we
+apply tree pruning to a limited width in all levels. We do so
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+by sampling only the most promising actions at each level.
+To summarize, in the practical SoftTreeMax algorithm we
+perform an exhaustive TS to obtain all trajectories up to
+depth d. We expand the tree by exhaustively expanding
+all actions, which corresponds to a uniform tree expansion
+policy πb. We apply a neural network on the leaf states,
+and accumulate the result with the rewards along each tra-
+jectory to obtain the logits in (2). Finally, we aggregate
+the results using C-SoftTreeMax. We leave experiments
+E-SoftTreeMax for future work on risk-averse RL. During
+training, the gradient propagates to the NN weights of Wθ.
+When the gradient ∇θ log πd,θ is calculated at each time
+step, it updates Wθ for all leaf states, similarly to Siamese
+networks (Bertinetto et al., 2016). An illustration of the
+policy is given in Figure 2.
+6. Experiments
+We conduct our experiments on multiple games from the
+Atari simulation suite (Bellemare et al., 2013). As a baseline,
+we train a PPO (Schulman et al., 2017) agent with 256
+workers in parallel. In a hyperparameter search, we found
+this number of workers to be the best in terms of run-time.
+The environment engine is the highly efficient Atari-CuLE
+(Dalton et al., 2020), a CUDA-based version of Atari that
+runs on GPU. Similarly, we use Atari-CuLE for the GPU-
+based breadth-first TS as done in (Dalal et al., 2021). We
+then train SoftTreeMax for depths d = 1 . . . 8, with a single
+worker. We use five seeds for each experiment.
+For the implementation, we extend Stable-Baselines3 (Raf-
+fin et al., 2019) with all parameters taken as default from the
+original PPO paper (Schulman et al., 2017). We will release
+the code upon publication. For depths d ≥ 3, we limited
+the tree to a maximum width of 1024 nodes and pruned
+non-promising trajectories in terms of estimated weights.
+Since the distributed PPO baseline advances significantly
+faster in terms of environment steps, for a fair comparison,
+we ran all experiments for one week on the same machine
+and use the wall-clock time as the x-axis. We use Intel(R)
+Xeon(R) CPU E5-2698 v4 @ 2.20GHz equipped with one
+NVIDIA Tesla V100 32GB.
+In Figure 3, we plot the reward and variance of SoftTreeMax
+for each game, as a function of depth. The dashed lines are
+the results for PPO. Each value is taken after convergence,
+i.e., the average over the last 20% of the run. The numbers
+represent the average over five seeds per game. We choose
+to exclude the standard deviation to avoid excessive clutter
+in the plot. The plot conveys three intriguing conclusions.
+First, in all cases, SoftTreeMax achieves significantly higher
+reward than PPO. Its gradient variance is also orders of
+magnitude lower than that of PPO. Second, the reward and
+variance are negatively correlated – they mirror each other
+in almost all of the games. This phenomenon demonstrates
+how crucial it is to lower the variance of PG for improving
+performance. And specifically, it highlights the benefits of
+SoftTreeMax over “flat” PG. The third conclusion is that
+each game has a different sweet-spot in terms of optimal TS
+depth. Recall that we limit the run-time in all experiments
+to one week. The deeper the TS, the slower each step and
+less steps are finished by the end of the run. This type of
+comparison also explains why there is no reason to expect
+monotone variance reduction as a function of depth.
+We also provide the training curves in Figure 4. For brevity,
+we exclude a few of the depths from the plots. As seen, there
+is a clear benefit for SoftTreeMax over distributed PPO with
+the standard softmax policy. In most games, PPO with the
+SoftTreeMax policy shows very high sample efficiency: it
+achieves higher episodic reward even though it observes
+much less episodes, for the same running time.
+7. Related Work
+Our work intersects several fields of the RL literature:
+Softmax Operator. The softmax policy became a canonical
+part of PG to the point where theoretical results of PG focus
+specifically on it (Zhang et al., 2021; Mei et al., 2020b; Li
+et al., 2021; Schulman et al., 2017; Haarnoja et al., 2018).
+Even though we focus on a tree extension to the softmax
+policy, the methodology we propose is general and can be
+easily applied to other discrete or continuous parameterized
+policies as in (Mei et al., 2020a; Miahi et al., 2021).
+Tree Search. Planning with a TS is the process of using a
+forward model to consider possible future trajectories and
+decide on the best action at the root. One famous such algo-
+rithm is Monte-Carlo TS (MCTS; Browne et al. 2012) used
+in AlphaGo (Silver et al., 2016) and MuZero (Schrittwieser
+et al., 2020). Other principal algorithms such as Value Itera-
+tion, Policy Iteration and DQN were also shown to give an
+improved performance with a tree search extensions (Efroni
+et al., 2019; Dalal et al., 2021).
+Risk Aversion. Many works considered an exponential
+utility function for risk aversion (Chen et al., 2007; Garcıa
+& Fern´andez, 2015; Fei et al., 2021). This utility function
+is the same as E-SoftTreeMax formulation from (4), but we
+have it directly in the policy instead of the objective.
+Reward-free RL. We showed that the gradient variance is
+minimized when the transitions induced by the behavior
+policy πb are uniform. This is expressed by the second
+eigenvalue of the transition matrix P πb. This notion of
+uniform exploration is common to the reward-free RL setup
+(Jin et al., 2020). Several such works considered the same
+second eigenvalue in their analysis (Liu & Brunskill, 2018;
+Tarbouriech & Lazaric, 2019).
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+Figure 3. Reward and Gradient variance: GPU SoftTreeMax (single worker) vs PPO (256 GPU workers). The blue reward plots
+show the average of 50 evaluation episodes. The red variance plots show the average gradient variance of the corresponding training runs,
+averaged over five seeds. The dashed lines represent the same for PPO. Note theat the variance y-axis is in log-scale. The reward and
+variance are negatively correlated and mirror each other in almost all games. This demonstrates the necessity to lower the variance of PG
+for improving performance. We limit the training run-time in all experiments to one week. The deeper the TS, the slower each step and
+less steps are finished by the end of the training run. This explains the non-monotone performance and variance as a function of depth.
+Figure 4. Training curves: GPU SoftTreeMax (single worker) vs PPO (256 GPU workers). The plots show average reward and
+standard deviation over five seeds. The x-axis is the wall-clock time. The runs ended after a maximum of 200M time-steps, and after no
+longer than one week. The standard PPO finished in less than one week. The training curves correspond to the evaluation runs in Figure 3.
+8. Discussion
+Planning in RL is typically carried out with value-based
+algorithms due to its seamless integration with the Bellman
+operator, leaving aside the popular class of PG methods. In
+this work, we introduced for the first time a differentiable
+parametric policy that combines TS with PG. We prove that
+SoftTreeMax is essentially an exponential variance reduc-
+tion technique and provide novel insight on how to choose
+the expansion policy to minimize the gradient variance. It
+is an open question whether optimal variance reduction cor-
+responds to the appealing regret properties tackled by UCT
+(Kocsis & Szepesv´ari, 2006).
+Mitigating
+the
+known
+sample
+inefficiency
+issue,
+SoftTreeMax
+achieves
+better
+performance
+than
+the
+widely used PPO with multiple workers and softmax policy.
+Our method can be further applied to continuous control
+tasks, or in tasks where the forward model is learned with
+some estimation error. Other possible future directions
+are: (i) to study the implications of sampling trajectories
+instead of directly calculating their expectation; (ii) analyze
+the convergence rate of SoftTreeMax, and (iii) to extend
+SoftTreeMax to adaptively changing depths to optimize
+run-time and performance.
+
+Asteroids
+Gopher
+Krull
+Breakout
+15000
+9000
+5000
++10-5
+800
+10-5
+10-5
+p
+10-7
+10000
+8000 +
+10-6
+10-6
+10-7
+600
+601)
+10-9
+5000
+ 10-7
+7000-
+3000
+10-7
+400
+SoftTreeMax Reward
+6
+2
+4
+4
+6
+8
+2
+4
+8
+2
+6
+8
+2
+PPO Reward
+Depth
+Depth
+Depth
+Depth
+SoftTreeMax Variance
+Phoenix
+VideoPinball
+KungFuMaster
+NameThisGame
+PPO Variance
+800000 +
+10-5
+10-5
+20000
+75000
+600000
+10-6
+10-6
+15000
+50000
+400000
+: 601)
+10-7
+ 10-7 25000-
+10-)
+10000
+10-8
+200000-
+40000卡
+9
+2
+2
+4
+6
+2
+4
+6
+8
+2
+4
+8
+4
+6
+8
+Depth
+Depth
+Depth
+DepthAsteroids
+Breakout
+Gopher
+Krull
+8000-
+6000
+400
+ 3000
+6000
+Rewar
+4000
+200
+R 2000
+2000
+4000
+PPO
+10
+SoftTreeMax Depth 2
+100
+0
+100
+0
+100
+0
+100
+SoftTreeMax Depth 3
+Time [hours]
+Time [hours]
+Time [hours]
+Time [hours]
+SoftTreeMax Depth 5
+SoftTreeMax Depth 6
+KungFuMaster
+NameThisGame
+Phoenix
+VideoPinball
+SoftTreeMax Depth 8
+60000
+30000
+200000
+10000
+g
+ 40000
+20000
+Rewa
+5000
+10000
+20000
+0
+100
+0
+100
+0
+100
+0
+100
+Time [hours]
+Time [hours]
+Time [hours]
+Time [hours]SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
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+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+A. Appendix
+A.1. Proofs
+A.1.1. PROOF OF LEMMA 4.1 – BOUND ON THE POLICY GRADIENT VARIANCE
+For any parametric policy πθ and function Q : S × A → R,
+Var (∇θ log πθ(a|s)Q(s, a)) ≤ max
+s,a [Q(s, a)]2 max
+s
+||∇θ log πθ(·|s)||2
+F ,
+where ∇θ log πθ(·|s) ∈ RA×dim(θ) is a matrix whose a-th row is ∇θ log πθ(a|s)⊤.
+Proof. The variance for a parametric policy πθ is given as follows:
+Var (∇θ log πθ(a|s)Q(a, s)) =Es∼µπθ ,a∼πθ(·|s)
+�
+∇θ log πθ(a|s)⊤∇θ log πθ(a|s)Q(s, a)2�
+−
+Es∼ρπθ ,a∼πθ(·|s) [∇θ log πθ(a|s)Q(s, a)]⊤ Es∼µπθ ,a∼πθ(·|s) [∇θ log πθ(a|s)Q(s, a)] ,
+where Q(s, a) is the currently estimated Q-function and µπθ is the stationary distribution induced by following the policy
+πθ. Since the second term we subtract is always positive (it is of quadratic form v⊤v) we can bound the variance by the first
+term:
+Var (∇θ log πθ(a|s)Q(a, s)) ≤Es∼µπθ ,a∼πθ(·|s)
+�
+∇θ log πθ(a|s)⊤∇θ log πθ(a|s)Q(s, a)2�
+=
+�
+s
+µπθ(s)
+�
+a
+πθ(a|s)∇θ log πθ(a|s)⊤∇θ log πθ(a|s)Q(s, a)2
+≤ max
+s,a
+�
+[Q(s, a)]2 πθ(a|s)
+� �
+s
+µπθ(s)
+�
+a
+∇θ log πθ(a|s)⊤∇θ log πθ(a|s)
+≤ max
+s,a [Q(s, a)]2 max
+s
+�
+a
+∇θ log πθ(a|s)⊤∇θ log πθ(a|s)
+= max
+s,a [Q(s, a)]2 max
+s
+||∇θ log πθ(·|s)||2
+F .
+A.1.2. PROOF OF LEMMA 4.2 – VECTOR FORM OF C-SOFTTREEMAX
+In vector form, (3) is given by
+πC
+d,θ(·|s) =
+exp
+�
+β
+�
+Cs,d + γdPs (P πb)d−1 Θ
+��
+1⊤
+A exp
+�
+β
+�
+Cs,d + γdPs (P πb)d−1 Θ
+��,
+(10)
+where
+Cs,d = Rs + Ps
+�d−1
+�
+h=1
+γh (P πb)h−1
+�
+Rπb.
+(11)
+Proof. Consider the vector ℓs,· ∈ R|A|. Its expectation satisfies
+Eπbℓs,·(d; θ) = Eπb
+�d−1
+�
+t=0
+γtrt + γdθ(sd)
+�
+= Rs +
+d−1
+�
+t=1
+γtPs(P πb)t−1Rπb + γdPs(P πb)d−1Θ.
+As required.
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+A.1.3. PROOF OF LEMMA 4.3 – GRADIENT OF C-SOFTTREEMAX
+The C-SoftTreeMax gradient of dimension A × S is given by
+∇θ log πC
+d,θ = βγd �
+IA − 1A(πC
+d,θ)⊤�
+Ps (P πb)d−1 ,
+where for brevity, we drop the s index in the policy above, i.e., πC
+d,θ ≡ πC
+d,θ(·|s).
+Proof. The (j, k)-th entry of ∇θ log πC
+d,θ satisifes
+[∇θ log πC
+d,θ]j,k =
+∂ log(πC
+d,θ(aj|s))
+∂θ(sk)
+= βγd[Ps(P πb)d−1]j,k −
+�
+a
+�
+exp
+�
+β
+�
+Cs,d + γdPs (P πb)d−1 Θ
+���
+a βγd �
+Ps(P πb)d−1�
+a,k
+1⊤
+A exp
+�
+β
+�
+Cs,d + γdPs (P πb)d−1 Θ
+��
+= βγd[Ps(P πb)d−1]j,k − βγd �
+a
+πC
+d,θ(a|s)
+�
+Ps(P πb)d−1�
+a,k
+= βγd[Ps(P πb)d−1]j,k − βγd �
+(πC
+d,θ)⊤Ps(P πb)d−1�
+k
+= βγd[Ps(P πb)d−1]j,k − βγd �
+1A(πC
+d,θ)⊤Ps(P πb)d−1�
+j,k .
+Now, moving back to matrix form, we obtain the lemma.
+A.1.4. PROOF OF THEOREM 4.4 – EXPONENTIAL VARIANCE DECAY OF C-SOFTTREEMAX
+The C-SoftTreeMax policy gradient is bounded by
+Var
+�
+∇θ log πC
+d,θ(a|s)Q(s, a)
+�
+≤ 2 A2S2β2
+(1 − γ)2 γ2d|λ2(P πb)|2(d−1).
+Proof. We use Lemma 4.1 directly. First of all, it is know that when the reward is bounded in [0, 1], the maximal value of
+the Q-function is
+1
+1−γ as the sum as infinite discounted rewards. Next, we bound the Frobenius norm of the term achieved in
+Lemma 4.3, by applying the eigen-decomposition on P πb:
+P πb = 1Sµ⊤ +
+S
+�
+i=2
+λiuiv⊤
+i ,
+(12)
+where µ is the stationary distribution of P πb, and ui and vi are left and right eigenvectors correspondingly.
+||βγd �
+IA,A − 1Aπ⊤�
+Ps(P πb)d−1||F = βγd||
+�
+IA,A − 1Aπ⊤�
+Ps
+�
+1Sµ⊤ +
+S
+�
+i=2
+λd−1
+i
+uiv⊤
+i
+�
+||F
+(Ps is stochastic)
+= βγd||
+�
+IA,A − 1Aπ⊤�
+�
+1Aµ⊤ +
+S
+�
+i=2
+λd−1
+i
+Psuiv⊤
+i
+�
+||F
+(projection nullifies 1Aµ⊤)
+= βγd||
+�
+IA,A − 1Aπ⊤�
+� S
+�
+i=2
+λd−1
+i
+Psuiv⊤
+i
+�
+||F
+(triangle inequality)
+≤ βγd
+S
+�
+i=2
+||
+�
+IA,A − 1Aπ⊤� �
+λd−1
+i
+Psuiv⊤
+i
+�
+||F
+(matrix norm sub-multiplicativity)
+≤ βγd|λd−1
+2
+|
+S
+�
+i=2
+||IA,A − 1Aπ⊤||F ||Ps||F ||uiv⊤
+i ||F
+= βγd|λd−1
+2
+|(S − 1)||IA,A − 1Aπ⊤||F ||Ps||F
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+Now we can bound the norm ||IA,A − 1Aπ⊤||F by direct calculation:
+||IA,A − 1Aπ⊤||2
+F = Tr
+��
+IA,A − 1Aπ⊤� �
+IA,A − 1Aπ⊤�⊤�
+(13)
+= Tr
+�
+IA,A − 1Aπ⊤ − π1⊤
+A + π⊤π1A1⊤
+A
+�
+(14)
+= A − 1 − 1 + Aπ⊤π
+(15)
+≤ 2A
+(16)
+And from the Cauchy-Schwartz inequality:
+||Ps||2
+F =
+�
+a
+�
+s
+[[Ps]a,s]2 =
+�
+a
+||[Ps]a,·||2
+2 ≤
+�
+a
+||[Ps]a,·||1||[Ps]a,·||∞ ≤ A.
+So:
+Var
+�
+∇θ log πC
+d,θ(a|s)Q(s, a)
+�
+≤ max
+s,a [Q(s, a)]2 max
+s
+||∇θ log πC
+d,θ(·|s)||2
+F
+≤
+1
+(1 − γ)2 ||βγd �
+IA,A − 1Aπ⊤�
+Ps(P πb)d−1||2
+F
+≤
+1
+(1 − γ)2 β2γ2d|λ2(P πb)|2(d−1)S2(2A2)
+Which obtains the desired bound.
+A.1.5. A LOWER BOUND ON C-SOFTTREEMAX GRADIENT (RESULT NOT IN THE PAPER)
+For completeness we also supply a lower bound on the Frobenius norm of the gradient. Note that this result does not
+translate to the a lower bound on the variance since we have no lower bound equivalence of Lemma 4.1.
+Lemma A.1. The Frobenius norm on the gradient of the policy is lower-bounded by:
+||∇θ log πC
+d,θ(·|s)||F ≥ C · βγd|λ2(P πb)|(d−1).
+(17)
+Proof. We begin by moving to the induced l2 norm by norm-equivalence:
+||βγd �
+IA,A − 1Aπ⊤�
+Ps(P πb)d−1||F ≥ ||βγd �
+IA,A − 1Aπ⊤�
+Ps(P πb)d−1||2
+Now, taking the vector u to be the eigenvector of the second eigenvalue of P πb:
+||βγd �
+IA,A − 1Aπ⊤�
+Ps(P πb)d−1||2 ≥ ||βγd �
+IA,A − 1Aπ⊤�
+Ps(P πb)d−1u||2
+= βγd||
+�
+IA,A − 1Aπ⊤�
+Psu||2
+= βγd|λ2(P πb)|(d−1)||
+�
+IA,A − 1Aπ⊤�
+Psu||2
+Note that even though Psu can be 0, that is not the common case since we can freely change πb (and therefore the
+eigenvectors of P πb).
+A.1.6. PROOF OF LEMMA 4.5 – VECTOR FORM OF E-SOFTTREEMAX
+For d ≥ 1, (4) is given by
+πE
+d,θ(·|s) =
+Es,d exp(βγdΘ)
+1⊤
+AEs,d exp(βγdΘ),
+(18)
+where
+Es,d = Ps
+d−1
+�
+h=1
+�
+D
+�
+exp[βγhR]
+�
+P πb�
+(19)
+with R being the |S|-dimensional vector whose s-th coordinate is r(s).
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+Proof. Recall that
+ℓs,a(d; θ) = r(s) +
+d−1
+�
+t=1
+γtr(st) + γdθ(sd).
+(20)
+and, hence,
+exp[βℓs,a(d; θ)] = exp
+�
+β
+�
+r(s) +
+d−1
+�
+t=1
+γtr(st) + γdθ(sd)
+��
+.
+(21)
+Therefore,
+E[exp βℓs,a(d; θ)] = E
+�
+exp
+�
+β
+�
+r(s) +
+d−1
+�
+t=1
+γtr(st)
+��
+E
+�
+exp
+�
+β
+�
+γdθ(sd)
+����s1, . . . , sd−1
+�
+�
+(22)
+= E
+�
+exp
+�
+β
+�
+r(s) +
+d−1
+�
+t=1
+γtr(st)
+��
+P πb(·|sd−1)
+�
+exp(βγdΘ)
+(23)
+= E
+�
+exp
+�
+β
+�
+r(s) +
+d−2
+�
+t=1
+γtr(st)
+��
+exp[βγd−1r(sd−1)]P πb(·|sd−1)
+�
+exp(βγdΘ).
+(24)
+By repeatedly using iterative conditioning as above, the desired result follows. Note that exp(βr(s)) does not depend on the
+action and is therefore cancelled out with the denominator.
+A.1.7. PROOF OF LEMMA 4.6 – GRADIENT OF E-SOFTTREEMAX
+The E-SoftTreeMax gradient of dimension A × S is given by
+∇θ log πE
+d,θ = βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1
+Es,dD(exp(βγdΘ))
+1⊤
+AEs,d exp(βγdΘ)
+,
+where for brevity, we drop the s index in the policy above, i.e., πE
+d,θ ≡ πE
+d,θ(·|s).
+Proof. The (j, k)-th entry of ∇θ log πE
+d,θ satisfies
+[∇θ log πE
+d,θ]j,k =
+∂ log(πE
+d,θ(aj|s))
+∂θ(sk)
+=
+∂
+∂θ(sk)
+�
+log[(Es,d)⊤
+j exp(βγdΘ)] − log[1⊤
+AEs,d exp(βγdΘ)]
+�
+= βγd(Es,d)j,k exp(βγdθ(sk))
+(Es,d)⊤
+j exp(βγdΘ)
+− βγd1⊤
+AEs,dek exp(βγdθ(sk))
+1⊤
+AEs,d exp(βγdΘ)
+= βγd(Es,dek exp(βγdθ(sk)))j
+(Es,d)⊤
+j exp(βγdΘ)
+− βγd1⊤
+AEs,dek exp(βγdθ(sk))
+1⊤
+AEs,d exp(βγdΘ)
+= βγd
+�
+e⊤
+j
+e⊤
+j Es,d exp(βγdΘ) −
+1⊤
+A
+1⊤
+AEs,d exp(βγdΘ)
+�
+Es,dek exp(βγdθ(sk)).
+Hence,
+[∇θ log πE
+d,θ]·,k = βγd �
+D(Es,d exp(βγdΘ))−1 − (1⊤
+AEs,d exp(βγdΘ))−11A1⊤
+A
+�
+Es,dek exp(βγdθ(sk))
+From this, it follows that
+∇θ log πE
+d,θ = βγd �
+D
+�
+πE
+d,θ
+�−1 − 1A1⊤
+A
+� Es,dD(exp(βγdΘ))
+1⊤
+AEs,d exp(βγdΘ)
+.
+(25)
+The desired result is now easy to see.
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+A.1.8. PROOF OF THEOREM 4.7 — EXPONENTIAL VARIANCE DECAY OF E-SOFTTREEMAX
+There exists α ∈ (0, 1) such that, for any function Q : S × A → R,
+Var
+�
+∇θ log πE
+d,θ(a|s)Q(s, a)
+�
+∈ O
+�
+β2γ2dα2d�
+.
+If all rewards are equal (r ≡ const), then α = |λ2(P πb)|.
+Proof. Let d ≥ 2. Recall that
+Es,d = Ps
+d−1
+�
+h=1
+�
+D
+�
+exp[βγhR]
+�
+P πb�
+,
+(26)
+and that R refers to the S-dimensional vector whose s-th coordinate is r(s). Define
+Bi =
+�
+P πb
+if i = d − 1,
+D−1(P πbMi+1)P πbD(Mi+1)
+if i = 1, . . . , d − 2,
+(27)
+and the vector
+Mi =
+�
+exp(βγd−1R)
+if i = d,
+exp(βγiR) ◦ P πbMi+1
+if i = 1, . . . , d − 2,
+(28)
+where ◦ denotes the element-wise product. Then,
+Es,d = PsD(M1)
+d−1
+�
+i=1
+Bi.
+(29)
+It is easy to see that each Bi is a row-stochastic matrix, i.e., all entries are non-negative and Bi1S = 1S.
+Next, we prove that all non-zeros entries of Bi are bounded away from 0 by a constant. This is necessary to apply the next
+result from (Chatterjee & Seneta, 1977). The j-th coordinate of Mi satisfies
+(Mi)j = exp[βγiRj]
+�
+k
+[P πb]j,k(Mi+1)k ≤ ∥ exp[βγiR]∥∞∥Mi+1∥∞.
+(30)
+Separately, observe that ∥Md−1∥∞ ≤ ∥ exp(βγd−1R)∥∞. Plugging these relations in (28) gives
+∥M1∥∞ ≤
+d−1
+�
+h=1
+∥ exp[βγhR]∥∞ =
+d−1
+�
+h=1
+∥ exp[βR]∥γh
+∞ = ∥ exp[βR]∥
+�d−1
+h=1 γh
+∞
+≤ ∥ exp[βR]∥
+1
+1−γ
+∞ .
+(31)
+Similarly, for every 1 ≤ i ≤ d − 1, we have that
+∥Mi∥∞ ≤
+d−1
+�
+h=i
+∥ exp[βR]∥γh
+∞ ≤ ∥ exp[βR]∥
+1
+1−γ
+∞ .
+(32)
+The jk-th entry of Bi = D−1(P πbMi+1)P πbD(Mi+1) is
+(Bi)jk =
+P πb
+jk [Mi+1]k
+�|S|
+ℓ=1 P πb
+jℓ [Mi+1]ℓ
+≥
+P πb
+jk
+�|S|
+ℓ=1 P πb
+jℓ [Mi+1]ℓ
+≥
+P πb
+jk
+∥ exp[βR]∥
+1
+1−γ
+∞
+.
+(33)
+Hence, for non-zero P πb
+jk , the entries are bounded away from zero by the same. We can now proceed with applying the
+following result.
+Now, by (Chatterjee & Seneta, 1977, Theorem 5) (see also (14) in (Mathkar & Borkar, 2016)), limd→∞
+�d−1
+i=1 Bi exists and
+is of the form 1Sµ⊤ for some probability vector µ. Furthermore, there is some α ∈ (0, 1) such that ε(d) :=
+��d−1
+i=1 Bi
+�
+−
+1S µ⊤ satisfies
+∥ε(d)∥ = O(αd).
+(34)
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+Pick linearly independent vectors w2, . . . , wS such that
+µ⊤wi = 0 for i = 2, . . . , d.
+(35)
+Since �S
+i=2 αiwi is perpendicular to µ for any α2, . . . αS and because µ⊤ exp(βγdΘ) > 0, there exists no choice of
+α2, . . . , αS such that �S
+i=2 αiwi = exp(βγdΘ). Hence, if we let z1 = 1S and zi = D(exp(βγdΘ))−1wi for i = 2, . . . , S,
+then it follows that {z1, . . . , zS} is linearly independent. In particular, it implies that {z1, . . . , zS} spans RS.
+Now consider an arbitrary unit norm vector z := �S
+i=1 cizi ∈ RS s.t. ∥z∥2 = 1. Then,
+∇θ log πE
+d,θz = ∇θ log πE
+d,θ
+S
+�
+i=2
+cizi
+(36)
+= βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1
+Es,dD(exp(βγdΘ))
+1⊤
+AEs,d exp(βγdΘ)
+S
+�
+i=2
+cizi
+(37)
+= βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1
+Es,d
+1⊤
+AEs,d exp(βγdΘ)
+S
+�
+i=2
+ciwi
+(38)
+= βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1 �
+1Sµ⊤ + ε(d)
+�
+1⊤
+AEs,d exp(βγdΘ)
+S
+��
+i=2
+ciwi
+(39)
+= βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1
+ε(d)
+1⊤
+AEs,d exp(βγdΘ)
+S
+�
+i=2
+ciwi
+(40)
+= βγd �
+IA − 1A(πE
+d,θ)⊤� D
+�
+πE
+d,θ
+�−1
+ε(d)D(exp(βγdΘ))
+1⊤
+AEs,d exp(βγdΘ)
+(z − c11S),
+(41)
+where (36) follows from the fact that ∇θ log πE
+d,θz1 = ∇θ log πE
+d,θ1S = 0, (37) follows from Lemma 4.6, (38) holds
+since zi = D(exp(βγdΘ))−1wi, (40) because µ is perpendicular wi for each i, while (41) follows by reusing zi =
+D(exp(βγdΘ))−1wi relation along with the fact that z1 = 1S.
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+From (41), it follows that
+∥∇θ log πE
+d,θz∥ ≤ βγd∥ε(d)∥
+�������
+�
+IA − 1A(πE
+d,θ)⊤�
+D
+�
+πE
+d,θ
+�−1
+1⊤
+AEs,d exp(βγdΘ)
+�������
+∥D(exp(βγdΘ))∥ ∥z − c11S∥
+(42)
+≤ βγdαd(∥IA∥ + ∥1A(πE
+d,θ)⊤∥)
+�������
+D
+�
+πE
+d,θ
+�−1
+1⊤
+AEs,d exp(βγdΘ)
+�������
+exp(βγd max
+s
+θ(s))∥z − c11S∥
+(43)
+≤ βγdαd(1 +
+√
+A)
+�������
+D
+�
+πE
+d,θ
+�−1
+1⊤
+AEs,d exp(βγdΘ)
+�������
+exp(βγd max
+s
+θ(s))∥z − c11S∥
+(44)
+≤ βγdαd(1 +
+√
+A)
+��D−1(Es,d exp(βγdΘ))
+�� exp(βγd max
+s
+θ(s))∥z − c11S∥
+(45)
+≤ βγdαd(1 +
+√
+A)
+1
+mins[Es,d exp(βγdΘ]s
+exp(βγd max
+s
+θ(s))∥z − c11S∥
+(46)
+≤ βγdαd(1 +
+√
+A)
+exp(βγd maxs θ(s))
+exp(βγd mins θ(s)) mins |M1|∥z − c11S∥
+(47)
+≤ βγdαd(1 +
+√
+A)
+exp(βγd maxs θ(s))
+exp(βγd mins θ(s)) exp(β mins r(s))∥z − c11S∥
+(48)
+≤ βγdαd(1 +
+√
+A) exp(β[max
+s
+θ(s) − min
+s
+θ(s) − min
+s
+r(s)])∥z − c11S∥.
+(49)
+Lastly, we prove that ∥z−c11S∥ is bounded independently of d. First, denote by c = (c1, . . . , cS)⊤ and ˜c = (0, c2, . . . , cS)⊤.
+Also, denote by Z the matrix with zi as its i-th column. Now,
+∥z − c11S∥ = ∥
+S
+�
+i=2
+cizi∥
+(50)
+= ∥Z˜c∥
+(51)
+≤ ∥Z∥∥˜c∥
+(52)
+≤ ∥Z∥∥c∥
+(53)
+= ∥Z∥∥Z−1z∥
+(54)
+≤ ∥Z∥∥Z−1∥,
+(55)
+where the last relation is due to z being a unit vector. All matrix norms here are l2-induced norms.
+Next, denote by W the matrix with wi in its i-th column. Recall that in (35) we only defined w2, . . . , wS. We now set
+w1 = exp(βγdΘ). Note that w1 is linearly independent of {w2, . . . , wS} because of (35) together with the fact that
+µ⊤w1 > 0. We can now express the relation between Z and W by Z = D−1(exp(βγdΘ))W. Substituting this in (55), we
+have
+∥z − c11S∥ ≤ ∥D−1(exp(βγdΘ))W∥∥W −1D(exp(βγdΘ))∥
+(56)
+≤ ∥W∥∥W −1∥∥D(exp(βγdΘ))∥∥D−1(exp(βγdΘ))∥.
+(57)
+It further holds that
+∥D(exp(βγdΘ))∥ ≤ max
+s
+exp
+�
+βγdθ(s)
+�
+≤ max{1, exp[β max
+s
+θ(s)])},
+(58)
+where the last relation equals 1 if θ(s) < 0 for all s. Similarly,
+∥D−1(exp(βγdΘ))∥ ≤
+1
+mins exp (βγdθ(s)) ≤
+1
+min{1, exp[β mins θ(s)])}.
+(59)
+
+SoftTreeMax: Exponential Variance Reduction in Policy Gradient via Tree Search
+Furthermore, by the properties of the l2-induced norm,
+∥W∥2 ≤
+√
+S∥W∥1
+(60)
+=
+√
+S max
+1≤i≤S ∥wi∥1
+(61)
+=
+√
+S max{exp(βγdΘ), max
+2≤i≤S ∥wi∥1}
+(62)
+≤
+√
+S max{1, exp[β max
+s
+θ(s)], max
+2≤i≤S ∥wi∥1)}.
+(63)
+Lastly,
+∥W −1∥ =
+1
+σmin(W)
+(64)
+≤
+�S−1
+�
+i=1
+σmax(W)
+σi(W)
+�
+1
+σmin(W)
+(65)
+= (σmax(W))S−1
+�S
+i=1 σi(W)
+(66)
+= ∥W∥S−1
+| det(W)|.
+(67)
+The determinant of W is a sum of products involving its entries. To upper bound (67) independently of d, we lower bound
+its determinant by upper and lower bounds on the entries [W]i,1 that are independent of d, depending on their sign:
+min{1, exp[β min
+s
+θ(s)])} ≤ [W]i,1 ≤ max{1, exp[β max
+s
+θ(s)])}.
+(68)
+Using this, together with (55), (57), (58), (59), and (63), we showed that ∥z − c11S∥ is upper bounded by a constant
+independent of d. This concludes the proof.
+

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+AI Maintenance:
+A Robustness Perspective
+Pin-Yu Chen and Payel Das
+IBM Research
+pin-yu.chen@ibm.com and daspa@us.ibm.com
+Abstract—With the advancements in machine learning (ML) methods and compute resources,
+artificial intelligence (AI) empowered systems are becoming a prevailing technology. However,
+current AI technology such as deep learning is not flawless. The significantly increased model
+complexity and data scale incur intensified challenges when lacking trustworthiness and
+transparency, which could create new risks and negative impacts. In this paper, we carve out AI
+maintenance from the robustness perspective. We start by introducing some highlighted
+robustness challenges in the AI lifecycle and motivating AI maintenance by making analogies to
+car maintenance. We then propose an AI model inspection framework to detect and mitigate
+robustness risks. We also draw inspiration from vehicle autonomy to define the levels of AI
+robustness automation. Our proposal for AI maintenance facilitates robustness assessment,
+status tracking, risk scanning, model hardening, and regulation throughout the AI lifecycle,
+which is an essential milestone toward building sustainable and trustworthy AI ecosystems.
+1. Introduction
+Just like the indispensable role of cars in the
+modern world, AI-empowered technology, and
+ML-based systems and algorithms are bringing
+revolutionary changes and far-reaching impacts
+on our life, society, and environment, if not
+happening already. As AI models are perceived
+as a new “vehicle” to a better future, this article
+aims to stress the importance of formalizing and
+practicing AI maintenance from the robustness
+perspective, by drawing analogies in the model
+development and deployment between car and AI.
+Towards achieving trustworthiness and sustain-
+ability for AI, this article is motivated by the fol-
+lowing question: Cars require regular inspection,
+maintenance, and continuous status monitoring,
+why should AI technology be any different?
+Robustness in AI often entails multiple mean-
+ings depending on the context and use cases.
+In this article, we study robustness from the
+perspective of the generalization capability of an
+AI model in adversarial and unseen scenarios.
+In general, the performance of an AI model is
+evaluated in the average case, by comparing the
+model predictions on a set of data samples to their
+ground-truth labels and then using the average
+prediction result as a performance metric, such
+as the top-1 classification accuracy measuring the
+fraction of correct model prediction on the most-
+likely (top-1) class over a dataset. In contrast, the
+adversarial scenario evaluates the model perfor-
+mance in the worst case among all possible and
+plausible changes (often pre-specified) to the data
+and AI model, by assuming a virtual adversary is
+in place. Moreover, the unseen scenario evaluates
+the model performance on new data samples that
+are drawn from a different data distribution than
+the seen data samples during training (but not
+necessarily the worst-case distribution), possibly
+caused by natural data/label shifts, and real-world
+observational noises, among others.
+The rationale for studying AI maintenance
+© preprint
+1
+arXiv:2301.03052v1  [cs.LG]  8 Jan 2023
+
+from the robustness viewpoint is motivated by
+the rapidly intensified demand for inspecting and
+preventing failure modes for AI models, in or-
+der to understand the limitations and prepare AI
+technology for the real world against malicious
+attempts and contiguous data changes. According
+to a recent Gartner report1, 30% of cyberattacks
+by 2022 will involve data poisoning, model theft
+or adversarial examples (see [1] for an overview
+of these new risks centered on machine learn-
+ing). However, the industry seems underprepared.
+In a survey of 28 organizations spanning small
+and large organizations, 25 organizations did not
+know how to secure their AI/ML systems [2].
+Unlike car insurances that cover damage and
+liability, the risk of lacking robustness in AI
+models can be further amplified if cyber insurance
+providers impose stringent requirements when the
+root cause is related to AI failure modes2. More-
+over, AI maintenance is closely related to action
+plans for enhancing trustworthiness in safety-
+related ML applications, such as fulfilling the
+milestones and objectives defined in the roadmap
+of the European Union Aviation Safety Agency
+(EASA)3, the AI/ML Software as a Medical
+Device Action Plan defined by U.S. Food &
+Drug Administration4, and the NIST AI Risk
+Management Framework5.
+To gain insights into AI maintenance, this ar-
+ticle first introduces major robustness challenges
+in the AI lifecycle for model development and
+deployment. Then, we make analogies of the
+commonality between car and AI maintenance.
+Finally, we propose the conceptual framework
+named “AI model inspector” for holistic robust-
+ness inspection and enhancement. Similar to the
+definitions of driving automation for vehicle au-
+tonomy, we define six levels of AI robustness
+towards facilitating qualitative and quantitative
+assessment of AI technology throughout the en-
+tire lifecycle.
+1https://www.gartner.com/smarterwithgartner/
+gartner-top-10-strategic-technology-trends-for-2020
+2https://hbr.org/2020/04/the-case-for-ai-insurance
+3https://www.easa.europa.eu/newsroom-and-events/news/
+easa-releases-its-concept-paper-first-usable-guidance-level-1-machine-0
+4https://www.fda.gov/medical-devices/
+software-medical-device-samd/artificial-intelligence-and-machine-learning-software-medical-device
+5https://www.nist.gov/itl/ai-risk-management-framework
+2. Robustness Challenges in AI
+Lifecycle
+Figure 1 provides an overview of robustness
+inspection pipeline in the AI lifecycle (left panel)
+and the highlighted robustness challenges (right
+panel). The AI lifecycle is recurring between
+two phases: model development and deployment.
+The model development phase consists of two
+states: (i) data collection and processing, and
+(ii) model training. Data collection and process-
+ing include typical data operations such as data
+acquisition and labeling, feature normalization,
+filtering, anonymization, and data augmentation.
+Model training involves machine learning model
+selection, algorithm development, system design,
+and optimization. Between states (i) and (ii), data
+sanitization inspects the data fidelity and performs
+mitigation steps (e.g., deleting problematic data
+samples or correcting mislabeled samples) prior
+to model training. After model development, the
+AI lifecycle enters the state of (iii) model deploy-
+ment, in which the trainable model parameters are
+frozen for use. Between (ii) and (iii), performance
+validation inspects and reduces the gap between
+model training and deployment. If the deployed
+model undergoes significant performance degra-
+dation, possibly due to naturally occurring data
+shifts or malicious attempts, the AI lifecycle will
+re-enter the model development phase to collect
+new data or update the model. Between (iii) and
+(i), continuous monitoring inspects the perfor-
+mance status of the currently deployed model
+and gives a notice upon observing significant
+performance degradation or detecting anomalous
+events.
+There are different types of robustness chal-
+lenges in the model development and deployment
+phases that can lead to model misbehavior and
+degraded performance, varied by their objectives,
+feasible actions on intervening in the AI model,
+and knowledge about the AI model. In the ad-
+versarial scenario, the robustness challenges can
+be related to a “threat model” specifying what
+an attacker can know and do to compromise the
+AI model. In the unseen scenario, the robustness
+challenges are associated with the domain gener-
+alization capability between the development and
+deployment phases. Figure 1 (right panel) lists
+two highlighted robustness challenges for each
+2
+
+Figure 1. Left: Schematic illustration of robustness inspection pipeline (data sanitization, performance
+validation, and continuous monitoring) in the AI lifecycle consisting of three major states: data collection and
+processing, model training, and model deployment. The model development phase includes data collection
+and processing and model training. Right: Highlighted robustness challenges in the AI lifecycle. In the model
+development phase, the robustness challenges assume the training data are subject to manipulation prior to
+model training. In the model deployment phase, the robustness challenges have no access to the training data
+but may assume some knowledge of the deployed model such as the model architecture and the associated
+model parameters. Based on the categorization of states in the AI lifecycle, the chart can be extended to
+incorporate other robustness challenges and other trustworthiness dimensions such as safety, privacy, etc.
+phase, which are detailed as follows.
+2.1. Robustness challenges in development
+phase
+Data poisoning concerns the model perfor-
+mance when trained on noisy data. The source of
+noise may come from imperfect data collection
+and processing such as incorrect data annotation,
+data bias and imbalance, and context-irrelevant
+spurious features. The noise may also be inten-
+tionally introduced to the training data by adding
+a set of poisoned data samples for the purpose of
+undermining the model performance in the de-
+ployment phase. For example, making the target
+model has low classification errors in develop-
+ment but high classification errors in deployment.
+Such intentional data poisoning attacks usually
+assume the ability to manipulate the training data
+and have access or some partial knowledge about
+the model details and training procedure [3].
+Backdoor is a Trojan attack targeting machine
+learning [4]. It works by injecting some pattern
+(a trigger) with modified labels to a subset of
+training data. Due to the memorization effect of
+state-of-the-art machine learning models such as
+neural networks, models trained on the tampered
+dataset will contain a backdoor. In the deployment
+phase, backdoored models will allow an attacker
+to gain control of the model output in the presence
+of the designated trigger, regardless of the actual
+content of the data input. However, in the absence
+of the trigger, the backdoored model will behave
+like a normal model trained on the untampered
+training dataset. Therefore, backdoor attacks are
+stealthy because the tampered model will not
+misbehave if the backdoor is inactivated. This
+challenge can be amplified in distributed and de-
+centralized machine learning paradigms involving
+multiple parties exchanging limited information
+about their local private data, such as federated
+learning [5].
+2.2. Robustness challenges in deployment
+phase
+The deployment phase takes a fully-tuned
+model in the development phase and freezes the
+model for subsequent data inference tasks. A
+deployed model is called a white-box model if its
+details are transparent to a user (e.g., releasing a
+deep learning model with its model architecture
+3
+
+(i) Data
+collection &
+Robustness
+processing
+Challenges in
+Al Lifecylce
+Continuous
+Data
+Model
+Model
+Monitoring
+Sanitization
+Development
+Deployment
+Performance
+Out-of-
+(ili) Model
+(ii) Model
+Validation
+Data
+Adversarial
+Backdoor
+Distribution
+deployment
+training
+Poisoning
+Examples
+Generalization
+Al Lifecycle
+robustness inspectionand pre-trained weights). Otherwise, if model
+details are unknown (or partially known) to a user,
+it is called a black-box (gray-box) model, such
+as a prediction application programming interface
+(API) or proprietary software that only gives
+model prediction results and does not reveal other
+details. For robustness assessment, the white-box
+mode enables full-stack system debugging and
+internal penetration testing, while the black-box
+mode allows practical vulnerability and informa-
+tion leakage analysis based on user access.
+Adversarial examples are carefully crafted
+data samples that cause prediction evasion when
+compared to the original unmodified data samples
+[6]. The easiness in prediction evasion reflects the
+model sensitivity against small changes in data
+inputs, such as a human-imperceptible additive
+perturbation. The robustness challenges of adver-
+sarial examples are often associated with safety-
+critical and security-related AI applications, such
+as autonomous driving cars, identification and
+recognition, and malware detection because their
+existence can be interpreted as counter-examples
+that violate the required robustness constraints. In
+the black-box setting, adversarial examples can be
+generated by iteratively modifying a data input
+based only on the model’s prediction output [7].
+Out-of-distribution generalization refers to the
+characterization of model performance when the
+input data samples undergo certain semantic-
+preserving transformations that deviate from the
+seen data distribution during model training. In
+contrast, in-distribution generalization refers to
+the model performance on data samples or in-
+stances drawn from the same distribution as the
+training data or environments. The quest for out-
+of-distribution generalization is motivated by re-
+taining robust predictions against natural varia-
+tions (their effect can be either observable or hid-
+den). The examples include distributional shifts
+between development and deployment phases,
+data/label drifts in online data streaming, com-
+mon corruptions caused by measurement/device
+errors, and data-invariant operations made by
+image rotation or scaling. An ideal model in
+deployment should generalize well or has the
+ability to quickly recognize and adapt to unseen
+data samples that are out-of-distribution yet share
+similar contexts to the in-distribution data seen
+during training.
+3. Analogies between Car and AI
+Maintenance
+As AI-empowered algorithms and systems are
+often perceived as a powerful yet mysterious tech-
+nology to end users, we believe making analogies
+to (autonomous) cars can deliver better trans-
+parency and a more comprehensive understand-
+ing of AI technology’s utilities and limitations.
+Towards formalizing and standardizing the notion
+of AI maintenance, we aim to draw connections
+to a more familiar case – car maintenance – as
+AI and car share many commonalities in model
+development and deployment. The development
+of new car models is a resource-intensive process
+(e.g., electric cars). It is taken for granted that
+essential regulatory and law requirements such as
+reliability and safety are fully certified throughout
+the development process, to avoid catastrophic
+failures, fatal damage, and critical product recalls.
+Similarly, AI model development can be quite ex-
+pensive, especially when it comes to the training
+of foundation models [8] that require pre-training
+on large-scale datasets with neural networks con-
+sisting of a massive number of trainable parame-
+ters. Take the Generative Pre-trained Transformer
+3 (GPT-3) [9] as an example, which is one of
+the largest language models ever trained to date.
+GPT-3 has 175 billion parameters and is trained
+on a dataset consisting of 499 billion tokens. The
+estimated training cost is about 4.6 million US
+dollars even with the lowest priced GPU cloud
+on the market in 20206. Having invested so much,
+one would expect the resulting AI model is risk-
+proof and robust to be deployed.
+In deployment, car maintenance involves reg-
+ular mechanical and electrical inspection, perfor-
+mance testing and certification, automobile part
+replacement, and repair. We argue that many
+familiar concepts in car maintenance can be well-
+mapped to AI models. In what follows, we make
+analogies between car and AI to facilitate the
+consolidation of AI maintenance for robustness.
+Table 1 summarizes the key terms that share
+analogies between car and AI maintenance for ro-
+bustness. In what follows, we divide those terms
+into four categories and discuss their connections.
+6https://lambdalabs.com/blog/demystifying-gpt-3/
+4
+
+Table 1. Analogies between car and AI models for maintenance and robustness divided into four categories.
+Category
+Car
+AI
+Model descriptions
+and performance
+characterization
+user manual
+model specification
+automobile parts
+machine learning modules
+warrant
+robustness checkpoints
+transmission efficiency
+memory/data/power efficiency
+Systematic inspection
+and monitoring
+collision test & safety report
+internal robustness assessment
+mechanical and electrical inspection
+penetration testing and debugging
+problematic status warning
+operational errors
+health state monitoring
+model behavior tracking
+Fix and update
+repair
+model fix and update
+wheel alignment
+model calibration
+winter tire
+model hardening
+flat tire response
+fast adaptation
+Education and
+societal impacts
+driver licence
+AI ethics and value alignment
+sustainability
+green and righteous AI
+3.1. Model descriptions and performance
+characterization
+The “user manual” provides instructions for
+an AI system, with descriptions specifying nec-
+essary information for transparency and account-
+ability, such as data and model training details,
+privacy, usability, and impact statements regard-
+ing recommended uses and possible misuse. The
+“automobile parts” in AI means functional and
+configurable modules in the machine learning
+pipeline that can be modified and ideally stan-
+dardized for the ease of model fix and update. The
+“warrant” in AI means qualitative and quantitative
+performance checkpoints in the development pro-
+cess. The “transmission efficiency” in AI relates
+to how the model scales with data, memory,
+and power, such as floating-point operations per
+second (FLOPS).
+3.2. Systematic inspection and monitoring
+During model development, the “collision
+test” for AI refers to internal comprehensive ro-
+bustness assessment, white-hat hacking, and red-
+teaming to identify limitations and hidden issues,
+similar to comprehensive road testing and car
+reviews. The results can be used to generate a
+“safety report” providing a quantified level of
+robustness in adversarial and unseen scenarios.
+The “mechanical and electrical inspection” for
+AI means penetration testing and debugging of
+the entire system (e.g., the software and hard-
+ware supporting AI technology) using probing
+and active measurement. The “problematic status
+warning” refers to real-time operational abnormal
+event detection during deployment, such as erro-
+neous instances or malfunctioning. The “health
+state monitoring” means continuous tracking of
+model behaviors, such as identifying the emer-
+gence of adversarial threats and data drifts.
+3.3. Fix and update
+After inspecting and identifying errors and
+risks, the “repair” for AI models refers to mit-
+igation strategies to fix, update, and re-certify
+the underlying model. The “wheel-alignment” for
+AI means model calibration, the “winter tire”
+means hardening the model with a more robust
+module, and the “flat tire response” means fast
+adaption of an AI model in the face of model
+performance degradation and anomalous events.
+Depending on the severity of the found robustness
+risks, user demand, and enforced regulation for
+AI technology, model fix and update for AI main-
+tenance can have differentiated services at varying
+costs, ranging from simple model patching and
+quick problem fixing, module replacement, partial
+model upgrade, to model rebuild.
+3.4. Education and societal impacts
+The “driver license” for AI means education
+on the ethics and value alignment when using
+AI technology, to understand its capabilities and
+limitations. The “sustainability” for AI involves
+gaining environmental awareness such as greener
+AI models with reduced energy consumption, as
+well as achieving positive societal impacts, in
+order to fulfill social responsibility and prevent
+possible misuse.
+4. AI Model Inspector
+Towards practicing and realizing the notion
+of AI maintenance, in this section we propose a
+5
+
+Figure 2. The AI model inspector framework consists of detection and mitigation stages. The model under
+inspection first takes a series of robustness testing and checkpoints, including procedural and operational
+assessment, passive evaluation on representative datasets, and active probing by generating new instances
+on-the-fly to find failure modes. In the detection stage, the inspector extracts statistics and runs a diagnosis
+to identify possible risks in robustness. In the mitigation stage, the inspector employs model fix and update to
+mitigate the identified robustness risks, and then re-assesses the model using the same robustness checklist.
+Finally, the inspector returns a risk-mitigated model. The entire process is analog to car inspection, fixing, and
+cleaning for car maintenance.
+methodology called AI model inspector, which is
+a conceptual pipeline for proactive detection and
+mitigation of robustness issues throughout the AI
+lifecycle. We also highlight two case studies on
+different robustness challenges to illustrate how
+the AI model inspector can be realized. Finally,
+as motivated by vehicle autonomy, we define
+different levels of AI robustness.
+4.1. Robustness inspection: detection and
+mitigation
+Figure 2
+shows the pipeline of AI model
+inspector consisting of two stages: detection and
+mitigation. First, a user using the AI mainte-
+nance service provides a model and/or some
+data samples for robustness inspection. The in-
+spection takes a series of robustness testing and
+checkpoints in both qualitative and quantitative
+manners, including procedural and operational
+assessment, passive model performance evalua-
+tion on representative datasets, and active probing
+by generating new instances on-the-fly to find
+failure modes. Qualitative assessment includes
+soliciting system characterization and problem
+descriptions from the model operator to gain a
+comprehensive understanding of the scope and
+details of model development and deployment,
+such as what model and data are used for training,
+how the model is deployed, how much informa-
+tion is known to a user, what types of robustness
+challenges are of top concerns, to name a few.
+Based on the qualitative assessment, quantitative
+analysis includes running the corresponding di-
+agnosis and reporting the numerical results and
+summary by generating and leveraging proper test
+cases and datasets for performance evaluation.
+Specifically, in the detection stage, the inspec-
+tor extracts discriminative statistics and runs a
+diagnosis to identify possible risks in robustness.
+Then, in the mitigation stage, the inspector em-
+ploys model fix and update to mitigate the iden-
+tified robustness risks, such as model finetuning
+and re-training, adding or replacing some mod-
+ules in the AI system, and re-assessing the model
+using the same robustness checklist. Finally, the
+inspector returns a risk-mitigated model. The en-
+tire process is analog to car maintenance in terms
+of car inspection, fixing, and cleaning. The no-
+tion of differentiated services in car maintenance
+can also be mapped to the varying demand and
+6
+
+Al model for
+Mitigation
+Detection
+inspection
+大一石
+Robustness
+No robustness
+Car inspection
+ Car fix
+Car wash
+checklist
+issuesfoundcost of AI maintenance, such as fast scanning,
+thorough inspection, quick patching, and detailed
+fix and update. We note that the usage of the AI
+model inspector is continuous rather than one-
+shot. Based on the recurrence of the states in
+the AI lifecycle, a model will repeatedly undergo
+several transitions between the states of data col-
+lection & processing, model training, and model
+deployment. Moreover, a model can be fixed but
+broken again later. This is analogous to the notion
+of weariness and fatigue testing in predictive car
+maintenance – after inspection, some parts need
+to be updated or replaced on a regular basis to
+ensure the model remains in good condition.
+Based on the robustness challenges shown in
+Figure 1, we make the following two examples
+that realize the concept of the AI model inspector.
+Backdoor detection and mitigation: In the de-
+tection stage, the inspector adopts the Trojan net
+detector proposed in [10], which uses a limited
+number of untampered clean data samples (as
+few as one sample per class) to derive a dis-
+criminate statistic for discerning a trained neural
+network classifier has any hidden backdoor. The
+detector can even achieve data-free detection for
+convolutional neural networks. After detection,
+the inspector can adopt the mitigation strategy of
+model sanitization proposed in [11] to remove the
+backdoor by finetuning the model parameters.
+Anomalous input detection and mitigation:
+Given a data input to an AI model under inspec-
+tion, the inspector can use internal data represen-
+tations (e.g., similarity to training data), domain
+knowledge (e.g., innate data characteristics and
+physical rules), or external knowledge checking
+(e.g., searching and reasoning over a knowl-
+edge graph or a database) to determine whether
+the data input is anomalous or not. Here, the
+anomaly encompasses different robustness chal-
+lenges, such as adversarial examples and out-
+of-distribution samples. For instance, the innate
+temporal dependency in audio data is used in
+[12] to detect audio adversarial examples for
+automatic speech recognition, and many distance
+metrics based on the internal data representations
+extracted from the model have been proposed to
+detect out-of-distribution samples [13]. In addi-
+tion to filtering out anomalous inputs, the in-
+spector can further take mitigation strategies to
+update the model and strengthen its robustness
+against anomalous inputs. For instance, the self-
+progressing robust training method proposed in
+[14] can further strengthen a trained model for
+enhanced adversarial robustness by instructing the
+model to mitigate the self-discovered ambiguity
+during model finetuning.
+4.2. Adversarial Machine Learning for
+Robustness
+Cars like the Mars Exploration Rovers can
+successfully execute the assigned task on new
+and unseen terrain because they were developed
+in comprehensive simulated environments. For AI
+models, one can incorporate the failure examples
+generated from model inspection tools to improve
+the robustness in unseen and even adversarial en-
+vironments. This methodology is known as adver-
+sarial machine learning, by introducing a virtual
+adversary in the AI lifecycle to help create better
+and more robust models. In the development
+phase, the role of the virtual adversary is to simu-
+late the out-of-distribution or worst-case scenarios
+and generate new challenging cases to help the
+model generalize better in unseen and adversarial
+environments. In the deployment case, the role
+of the virtual adversary is to employ proactive
+robustness evaluation and risk discovery, in order
+to prevent real damage and negative impacts.
+One typical example is adversarial training [15]
+which exploits self-generated adversarial exam-
+ples during model training to strengthen adver-
+sarial robustness against adversarial inputs in the
+deployment phase. We refer the readers to [16] for
+recent advances in adversarial machine learning
+for AI robustness.
+4.3. Roadmap towards the levels of AI
+robustness
+Inspired by the definitions for six levels of
+driving automation for autonomous vehicles7, we
+define six levels of AI robustness to facilitate
+technical progress tracing, risk quantification, and
+inspection, model auditing, and standardization.
+Table 2 compares the defined levels for vehicle
+autonomy and AI robustness, respectively. The
+level of robustness quantifies the progress in the
+soundness of machine intelligence for robustness.
+As the level increases, it signifies the practice
+7https://www.sae.org/standards/content/j3016 202104/
+7
+
+Table 2. Comparisons between the levels of vehicle autonomy versus AI robustness.
+Level
+Vehicle Autonomy
+AI Robustness
+0
+no driving automation
+no robustness (standard training)
+1
+driver assistance
+generalization under distribution shifts
+2
+partial driving automation
+robustness against single risk
+3
+conditional driving automation
+robustness against multiple risks
+4
+high driving automation
+universal robustness to known risks
+5
+full driving automation
+human-aligned and augmented robustness
+and guarantee of robustness in a more practical
+and comprehensive manner. For AI robustness,
+an increased level means broader coverage of
+robustness risks under consideration. We believe
+formalizing the levels of AI robustness can be
+useful for the discussion and practice of AI
+standardization related to robustness, security, and
+safety, such as ISO/IEC JTC 1/WG 13 on Trust-
+worthiness8 and ISO/TC 22/SC 32/WG 14 on
+Safety and Artificial Intelligence9.
+Level 0 means the original robustness ob-
+tained from a standard model training process
+without any risk mitigation operations. Level 1
+concerns the generalization capability on natu-
+rally occurring shifted data distributions, such
+as maintaining robust predictions against dis-
+tributional changes caused by spurious features
+that are irrelevant to the actual semantic context
+(e.g., classifying traffic signs with altering sky
+backgrounds). Level 2 considers the worst-case
+robustness against single risk (e.g., adversarial
+examples), and Level 3 extends to multiple risks,
+such as the multi-objective (but selected) ro-
+bustness to adversarial examples, common data
+corruptions, and spurious correlations [17]. Level
+4 guarantees universal robustness to all known
+risks. Here universal robustness means joint ef-
+fectiveness on all known robustness risks. Finally,
+Level 5 aligns robustness with human-centered
+values and user feedback, and it has the capability
+to automatically augment new robustness that
+is complimentary to existing robustness require-
+ments. Depending on the requirements (e.g., law
+regulation) and contexts of the applications, dif-
+ferent levels can be necessitated as pre-requisite
+before deployment. For example, some high-risk
+AI applications should pass the criterion of higher
+levels – similar to the necessary requirements
+8https://www.iso.org/committee/45020.html
+9https://standards.iteh.ai/catalog/tc/iso/
+6ec701ad-7678-442d-b186-a84b9ba2bbdf/iso-tc-22-sc-32
+for different driving automation conditions (e.g.,
+driving on highways versus urban environments).
+It is worth noting that the assessment of
+level-1 robustness can likely be accomplished by
+static evaluation on a representative dataset or
+benchmark. However, moving forward to level 2
+and above, the validation of worst-case robustness
+performance also requires model intervention,
+such as active model scanning and probing for
+finding failure cases. Moreover, AI model inspec-
+tor takes proactive steps for detecting and mit-
+igating potential robustness risks, which differs
+from existing frameworks such as Factsheets [18],
+Model Cards [19], and Datasheets for Datasets
+[20] that only employ passive model character-
+ization and specification. Finally, in addition to
+maintenance for AI, one can also adopt AI to
+improve maintenance, such as predictive main-
+tenance that takes preventive care to AI models
+based on historical records and risk forecasting.
+5. Concluding Remarks
+This article discusses a novel maintenance
+framework for robustness in AI technology based
+on analogies to the development and deployment
+of car models. To instill and improve trustworthi-
+ness in the AI lifecycle, we propose an automated
+and scalable solution based on the principle of
+AI model inspector for detecting and mitigating
+potential risks when lacking robustness. Inspired
+by vehicle autonomy, we also define different
+AI robustness levels for formalizing, evaluating,
+standardizing, and regulating risk-proof AI mod-
+els. As AI technology is transforming our life,
+society, and environment with greater width and
+depth and at a faster speed than cars, we believe
+the quest for AI maintenance is imminent and
+necessary. Beyond robustness, the AI model in-
+spector framework can also be extended to incor-
+porate other dimensions of trustworthy AI, such
+as fairness, explainability, privacy, accountability,
+8
+
+and uncertainty quantification.
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+pp. 86–92, 2021.
+9
+

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+arXiv:2301.08640v1  [astro-ph.HE]  20 Jan 2023
+Draft version January 23, 2023
+Typeset using LATEX manuscript style in AASTeX631
+Tracing the Evolution of SMBHs and Stellar Objects in Galaxy Mergers: An
+Multi-mass Direct N-body Model
+Shuo Li,1, 2 Shiyan Zhong,3 Peter Berczik,4, 5, 6 Rainer Spurzem,1, 4, 7 Xian Chen,2, 7 and
+F.K. Liu2, 7
+1National Astronomical Observatories and Key Laboratory of Computational Astrophysics,
+Chinese Academy of Sciences, 20A Datun Rd., Chaoyang District, Beijing 100012, China
+2Department of Astronomy, School of Physics, Peking University,
+Yiheyuan Lu 5, Haidian Qu, Beijing 100871, China
+3Yunnan Observatories, Chinese Academy of Sciences,
+396 Yang-Fang-Wang, Guandu District, 650216, Kunming, Yunnan, China
+4Astronomisches Rechen-Institut, Zentrum f¨ur Astronomie, University of Heidelberg,
+M¨ochhofstrasse 12-14, Heidelberg 69120, Germany
+5Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E¨otv¨os Lor´and Research Network
+(ELKH), MTA Centre of Excellence, Konkoly Thege Mikl´os ´ut 15-17, 1121 Budapest, Hungary
+6Main Astronomical Observatory, National Academy of Sciences of Ukraine,
+27 Akademika Zabolotnoho St., 03680 Kyiv, Ukraine
+7Kavli Institute for Astronomy and Astrophysics, Peking University,
+Yiheyuan Lu 5, Haidian Qu, Beijing 100871, China
+ABSTRACT
+By using direct N-body numerical simulations, we model the dynamical co-evolution
+of two supermassive black holes (SMBHs) and the surrounding stars in merging galaxies.
+In order to investigate how different stellar components evolve during the merger, we
+generate evolved stellar distributions with an initial mass function. Special schemes
+have also been developed to deal with some rare but interesting events, such as tidal
+disruption of main sequence stars, the plunge of low mass stars, white dwarfs, neutron
+stars and stellar mass black holes, and the partial tidal disruption of red giants or
+
+2
+Li et al.
+asymptotic giant branch stars.
+Our results indicate that the formation of a bound
+supermassive black hole binary (SMBHB) will enhance the capture rates of stellar
+objects by the SMBHs. Compared to the equal stellar mass model, the multi-mass
+model tends to result in a higher average mass of disrupted stars. Instead of being
+tidally disrupted by the SMBH, roughly half of the captured main sequence stars will
+directly plunge into the SMBH because of their small stellar radius. Giant stars, on the
+other hand, can be stripped of their envelopes if they are close enough to the SMBH.
+Though most remnants of the giant stars can survive after the disruption, a small
+fraction still could plunge into the SMBH quickly or after many orbital periods. Our
+results also indicate significant mass segregation of compact stars at the beginning of
+the merger, and then this effect is destroyed as the two SMBHs form a bound binary.
+Keywords: Galaxies: evolution — Galaxies: interactions — Galaxies: kinematics and
+dynamics — Galaxies: nuclei — Methods: numerical
+1. INTRODUCTION
+Supermassive black hole binaries (SMBHBs) are predicted as the descendents of the hierarchical
+galaxy formation model (Begelman et al. 1980; Volonteri et al. 2003). Over the past few decades,
+more and more observational evidence has indicated that most massive galaxies, if not all, have a
+supermassive black hole (SMBH) hidden in the center (Kormendy & Ho 2013). Since massive galaxies
+could undergo several mergers in their evolutionary history, it is natural to predict the existence of
+SMBHBs in merging or merged galaxies.
+Besides, many investigations find that there is a close
+connection between the SMBH and its host galaxy (Magorrian et al. 1998; Ferrarese & Merritt 2000;
+Gebhardt et al. 2000; Tremaine et al. 2002; Kormendy & Ho 2013). Therefore it is essential to find
+out how SMBHBs evolve in galaxy mergers.
+In a merging galaxy, two SMBHs will first approach each other mainly through dynamical friction.
+However, as two SMBHs get closer, this effect gets weaker and weaker. When two SMBHs are close
+enough to form a bound binary system, the dynamical friction is not efficient enough to drive the
+
+MULTI-MASS N-BODY MODEL
+3
+SMBHB coalesce within Hubble time in most of the massive galaxies (Begelman et al. 1980). A
+hard SMBHB can eject surrounding stars to transfer their orbital energy and angular momentum,
+which may be efficient to drive two SMBHs coalesce quickly (Saslaw et al. 1974; Mikkola & Valtonen
+1992; Quinlan 1996). But it needs enough close encounter stars to be ejected by the SMBHB, which
+may be not always the case, because stars scattered into the vicinity of the SMBHB through two-
+body relaxation may be inefficient in spherical stellar distributions (Begelman et al. 1980; Yu 2002;
+Milosavljevi´c & Merritt 2003; Berczik et al. 2005). Fortunately, both gas dynamics and more realistic
+stellar dynamics other than spherical two-body relaxation can avoid this problem (Gould & Rix 2000;
+Chatterjee et al. 2003; Merritt & Poon 2004; Berczik et al. 2006; Preto et al. 2011; Khan et al. 2011).
+There are many observational evidence to confirm the above scenario, such as dual active galactic
+nuclei (AGNs), jet reorientation in X-shaped radio galaxies, double-peaked emission lines, quasi-
+periodic outbursts in some blazars and so on (Liu & Wu 2002; Komossa et al. 2003; Liu 2004;
+Shen et al. 2013; Komossa et al. 2020; Tang et al. 2021).
+Most of these are indirect evidence of
+SMBHBs and they are in gaseous environments. In gas poor environments, hard SMBHBs are very
+difficult to detect. Besides emitting electromagnetic waves, a close SMBHB has strong gravitational
+wave (GW) emission, which could be detected by the ongoing Pulsar Timing Arrays (PTAs) and
+the planned space borne GW detectors such as the Laser Interferometer Space Antenna (LISA)1
+, Taiji Program and Tianqin Project2 (Foster & Backer 1990; Verbiest et al. 2016; Mei et al. 2021;
+Luo et al. 2021). However, after years of effort, though PTAs may already get some clue on the
+stochastic GW background, the specific detection is still missing (Shannon et al. 2015; Babak et al.
+2016; Arzoumanian et al. 2016, 2020).
+There are also other processes connected to SMBHs and
+SMBHBs, which could generate GW emission for LISA or similar instruments. For instance, com-
+pact stellar objects can inspiral to SMBHs with detectable GW emissions for LISA, which is the
+so-called extreme mass ratio inspirals (EMRIs) (Amaro-Seoane 2018, and references therein). Simi-
+lar events should also exist in the SMBHB system. But it has not been well studied. Nevertheless,
+1 https://www.elisascience.org/
+2 http://tianqin.sysu.edu.cn/en/
+
+4
+Li et al.
+due to limited spatial resolutions of GW detectors, even if a GW signal from the SMBHB could be
+detected, it is still difficult to locate its host galaxy. The inconsistency between theoretical expecta-
+tion on SMBHBs and GW observations may be due to many processes that have not been clarified.
+It is important to design observations that can be used to best constrain the evolution of SMBHBs.
+A dormant SMBH can be temporarily illuminated by tidal disruption events (TDEs). If a star
+closely approaches the SMBH by less than a critical distance, it will be torn into debris by the
+SMBH, which may prompt temporary flares with periods from days to years (Hills 1975; Rees 1988;
+Evans & Kochanek 1989; Guillochon & Ramirez-Ruiz 2013). We use for the critical distance tidal
+radius
+rt ⋍ µr∗(MBH/m∗)1/3,
+(1)
+where µ is a dimensionless parameter of order unity which reflects the stellar structure, r∗, m∗ and
+MBH are the stellar radius, the stellar mass and the mass of black hole (BH), respectively. Since the
+first identified event in the 1990s, dozens of TDEs have been reported, with emission range from γ-ray
+to radio bands (Komossa & Bade 1999; Komossa 2015; Gezari 2021, and references therein). Similar
+events can also happen in SMBHB systems, but light curves may be different from normal single
+SMBH TDEs. Due to the perturbation of the companion SMBH, a TDE in SMBHB system could have
+repeated gaps or be truncated, which has been investigated theoretically and numerically (Liu et al.
+2009; Ricarte et al. 2016; Coughlin et al. 2017; Vigneron et al. 2018). With the expansion of the
+TDE sample in these years, a few SMBHB TDE candidates also have been founded in observation
+(Liu et al. 2014; Shu et al. 2020; Huang et al. 2021). With more and more optical and X-ray transient
+surveys, such as the Large Synoptic Survey Telescope (LSST)3, the All-Sky Automated Survey for
+Supernovae (ASAS-SN)4, and the Einstein Probe (EP)5, join the game, the sample of the SMBHB
+TDE could significantly increase in the near future.
+3 https://www.lsst.org
+4 http://www.astronomy.ohio-state.edu/asassn/index.shtml
+5 http://ep.bao.ac.cn/
+
+MULTI-MASS N-BODY MODEL
+5
+There are some attempts to investigate the tidal disruption rate (TDR) of SMBHBs. Theoret-
+ical analyses and numerical scattering experiments indicate that, due to the perturbation of the
+companion SMBH and the triaxial stellar distribution, TDR in galaxy merger remnants could be
+enhanced from a few times to a few orders of magnitudes (Ivanov et al. 2005; Chen et al. 2009;
+Wegg & Nate Bode 2011; Liu & Chen 2013). However, since the interaction of two SMBHs and sur-
+rounding stars is chaotic during the formation of the SMBHB, these results are limited and can not
+fully reveal the underlying physical processes. In order to investigate the dynamical co-evolution of
+SMBHBs and stars in galaxy mergers with more depth, we used GPU accelerated direct N −body
+simulations to analyze a series of models on the TDR evolution of SMBHBs in both major mergers
+Li et al. (2017, hereafter Paper I) and minor mergers Li et al. (2019, hereafter Paper II). Both equal
+and unequal mass models indicate significantly enhanced TDRs during two SMBHs forming bound
+binary systems, which can be considered as a possible explanation for the high detection rates of
+TDEs preferred in E+A galaxies (Arcavi et al. 2014). However, for simplicity, it was assumed in
+both papers that all stars are solar type stars with the same mass. In reality, for example, giant
+stars will be partially tidally disrupted by the SMBH at relatively large distances. Compact ob-
+jects (neutron stars, black holes), on the other hand, could directly plunge into the SMBH without
+disruption. Even main sequence stars could have different fates after considering different masses.
+According to Eq. 1, massive main sequence stars with larger stellar radius correspond to larger tidal
+radius. Low mass main sequence stars usually correspond to smaller tidal radius. Massive stars
+with large envelopes may as well have only partial mass loss from TDE (see e.g.Zhong et al. (2022)).
+Finally, instead of a tidal disruption with significant flare, many main sequence stars near the low
+mass end actually will be entirely swallowed by the SMBH, because their tidal radii are very close to
+the Schwarzschild radius of the SMBH.
+In this work we improve our model by using a realistic distribution of stellar properties (mass and
+radius) instead of equal mass stars as in previous papers such as e.g. Li et al. (2019). In a dry major
+merger case, the typical evolution time is ∼ 1 Gyr (Colpi 2014). This period of time is long enough
+for a stellar system to evolve from the zero-age main sequence to different stellar components. For
+
+6
+Li et al.
+this reason, our stellar population is initialized with an age of ∼ 1 Gyr to account for the lifetime of
+the galaxies before their merger. We developed a special scheme to deal with TDEs of different types
+of stars, which includes variation of the tidal disruption radius as a function of stellar parameters,
+partial tidal disruption of giant stars, and direct plunges of compact objects (see for details next
+section and Fig. 1). With these modifications we are able to trace the co-evolution of all kinds of
+stars and SMBHs in galaxy mergers. This paper is organized as follows. We introduce our simulation
+models in Section 2. Our simulation results on different types of stars are presented in Section 3. In
+Section 4 we discuss how to extrapolate our results to more realistic systems, and some observational
+implications are also provided. A short summary is allocated to Section 5.
+2. THE DIRECT N -BODY MODEL WITH MORE REALISTIC STELLAR OBJECTS
+Our numerical model base on a direct N-body code ϕ -Grape/ϕ -GPU, which is an accu-
+rate GPU accelerated code with fourth-order Hermite integrator and efficient parallel scheme
+(Makino & Aarseth 1992; Berczik et al. 2005; Harfst et al. 2007).
+ϕ -Grape/ϕ -GPU has been
+proved to be an efficient tool on investigating the dynamical co-evolution of SMBH and stars in
+both single galaxy and galaxy mergers (Berczik et al. 2006; Gualandris & Merritt 2008; Khan et al.
+2011; Preto et al. 2011). It can be also adapted to investigate the tidal disruption evolution, with
+a simplified tidal disruption scheme included (Zhong et al. 2014; Li et al. 2017; Panamarev et al.
+2018). Recently, Khan et al. (2018) have tried to study the coalescence time of supermassive black
+holes in galaxy mergers with post-Newtonian (PN) terms and stellar mass function included. In this
+work, we try to introduce a more realistic model, which can self-consistently study how different type
+stellar objects, such as main sequence(MS) star, red giant (RG), asymptotic giant branch(AGB),
+white dwarf (WD), neutron star (NS), stellar mass black hole (BH), interact with SMBHs in galaxy
+mergers.
+2.1. Initial Mass Function and Stellar Evolution
+Compared with previous equal stellar mass models, a more realistic multi-mass model with initial
+mass function is adopted. We assume that the stellar mass of a star cluster ranges from 0.1 M⊙ to
+
+MULTI-MASS N-BODY MODEL
+7
+100 M⊙, and the initial mass function follows a multiple-part power-law as Kroupa (2001) suggested.
+This model corresponds to an average initial mass m∗ ∼ 0.6 M⊙.
+For simplicity, we are focusing on dry mergers with gas poor environments, which are usually
+dominated by early type galaxies, and the starbursts induced by mergers are not significant. It is
+reasonable to assume that all the stars have evolved for a relatively long time as the initial condition.
+We evolve all the stars for 1 Gyr by using the stellar evolution package SSE (Hurley et al. 2000). After
+the evolution, our model with largest particle number N = 106 has 4484 giant branch stars (includes
+RGs, core He burning stars, early AGBs and thermally pulsing AGBs), 29811 WDs(includes C/O
+WDs and O/Ne WDs), 5246 NSs and 1884 BHs. And the average mass decreased to m∗ ∼ 0.4 M⊙ due
+to the mass loss during the stellar evolution. The same scheme as Panamarev et al. (2019) adopted
+is involved to model the natal kick during the formation of NSs and BHs. The kick amplitude of NSs
+is represented by a Maxwellian distribution with 1D velocity dispersion σ = 265 km s−1 (Hobbs et al.
+2005). For BHs, both the mass and kick velocity sensitively depend on the ”fallback” of debris,
+those materials failed to escape during the explosion of the progenitors (Colgate 1971; Chevalier
+1989; Zhang et al. 2008). We take this effect into account by following the scheme suggested by
+Belczynski et al. (2002).
+As a result, a group of stars with different stellar components including NSs and BHs with initial
+kick velocities have been generated. In order to model a dense star cluster, the next step is to assign
+position and velocity to every star according to a proper stellar mass distribution.
+2.2. Dense Star Cluster Model
+The dynamical parameters of the initial condition are similar to those in Paper I. The stellar
+distribution of the dense star cluster around the SMBH is represented by a Dehnen model (Dehnen
+1993).
+ρ(r) = 3 − γ
+4π
+MaD
+rγ(r + aD)4−γ ,
+(2)
+
+8
+Li et al.
+where aD, M and γ denote the scaling radius, the total mass of the galaxy/nucleus and the density
+profile index, respectively.
+Here we adopt the units G = M = aD = 1, and assume γ = 1 in
+the following discussion for simplicity, because models with more steep central density profiles are
+very time consuming in the integration. The influence of γ has been carefully discussed in Paper I,
+with equal stellar mass models. The general results should be still informative here. According to
+results in Paper I, steep density profiles usually correspond to significantly higher TDRs, and the
+boosted TDRs in phase II are common for all models with different γ. However, the magnitude of
+the enhancement of the averaged TDRs in phase II only weakly depends on γ. Detailed discussions
+can be found in Paper I. The relation between numerical and physical quantities can be derived as
+[T]=
+�GM
+a3
+D
+�−1/2
+=1.491 × 107(2
+1
+3−γ − 1)3/2
+�
+M
+109 M⊙
+�−1/2 � r1/2
+1 kpc
+�3/2
+yr,
+(3)
+[V]=
+�GM
+aD
+�1/2
+=65.58 × (2
+1
+3−γ − 1)−1/2
+�
+M
+109 M⊙
+�1/2 � r1/2
+1 kpc
+�−1/2
+km s−1,
+(4)
+[R]=aD = (2
+1
+3−γ − 1)
+� r1/2
+1 kpc
+�
+kpc,
+(5)
+[ ˙M]=M/[T]
+=67.07 × (2
+1
+3−γ − 1)−3/2
+�
+M
+109 M⊙
+�3/2 � r1/2
+1 kpc
+�−3/2
+M⊙/ yr.
+(6)
+The SMBH is represented by a heavy particle with mass MBH = 0.01 at the center. The Dehnen
+model above does not considered the influence of SMBH and multiple stellar mass distribution. Sim-
+ilar to Paper I, to relax the SMBH with surrounding stars, we integrate the entire system for dozens
+of N-body unit time before the simulation, which is roughly the two-body relaxation timescale at the
+influence radius of the SMBH in the model. After this relaxation procedure, the mass segregation
+of heavy stellar components in the central region is significant. It has been confirmed by the result
+
+MULTI-MASS N-BODY MODEL
+9
+in the left panel of Fig. 4 Based on this template model, two identical galaxies/nuclei are set in a
+parabolic orbit with initial separation d ∼ 20 and the first pericenter ∼ 1. The pericenter distance
+here is for the convenience of comparison with Paper I. A closer encounter will lead to faster evolu-
+tion, but the general results should be similar. All the integrations are executed on the laohu GPU
+cluster in National Astronomical Observatories of China (NAOC).
+2.3. Scheme of Close Encounters with SMBHs
+The interactions between merging SMBHs and the surrounding stars, especially those stars close
+to SMBHs, are the focus of this work. In order to carefully investigate these ”close encounters” we
+involve some special treatments. In general, there are four kinds of close encounters in our model:
+normal stars tidally disrupted by the SMBH, main sequence stars with relatively light mass swallowed
+by the SMBH without tidal flares, giant branch stars partially disrupted, and compact stars, such
+as NSs and BHs, forming EMRIs or the extreme mass ratio bursts(EMRBs, more discussions can be
+found in Seciton 4.2).
+Fig. 1 demonstrates what will happen if an ordinary star passes by a SMBH within the tidal radius.
+Most of stars in our model are MS stars. If the stellar mass of a star is massive enough, there will
+be a typical TDE. Conversely, the tidal radius of a relatively light MS star or WD could be smaller
+than the Schwarzschild radius. The star will be directly swallowed by the SMBH without flare. The
+critical swallow radius of a BH, comparing to the Schwarzschild radius rsch, can be enlarged when
+we take into account the eccentricity of the intruder star. In this work we adopt the pericenter of
+the marginally stable orbit rMSO as the critical boundary. A star will be marked as a plunge event
+when it’s tidal radius and separation to the SMBH are both smaller than rMSO (Cutler et al. 1994;
+Gair et al. 2005)
+rMSO = 3 + e
+1 + ersch,
+(7)
+where rsch = 2GMBH/c2 is the Schwarzschild radius, e is the eccentricity of the star. rMSO determines
+the minimum pericenter distance of a test particle with fixed eccentricity.
+A test particle with
+
+10
+Li et al.
+Red Giant
+TP AGB
+Early AGB
+Naked He  
+Giant Branch
+Core He 
+Burning
+Naked 
+He MS
+C/O WD
+Naked He 
+Hertzsprung Gap
+O/Ne WD
+He WD
+Intruding 
+Stars
+Giant TD
+(GTD)
+Normal TD
+Standard TD
+(STD)
+Swallow
+(PLG)
+Figure 1. The fate of different types of stars after close encounters with SMBHs.
+pericenter distance smaller than rMSO plunges directly into the black hole. If e = 0 the rMSO is
+equivalent to the innermost stable circular orbit(ISCO). For extreme hyperbolic situations with e →
+∞, only orbits with pericenter distances larger than the rsch can survive. For simplicity, the mass
+and linear momentum of both the disrupted and swallowed stars will directly add to the SMBH in
+our model. Here we only take into account the linear momentum, and the angular momentum is not
+considered because we can not trace the spin evolution of the SMBH without the PN approximations
+included.
+In addition to main sequence stars, there are many giant branch stars (GBs), such as RGs and
+AGBs. These stars, with their compact cores and diffuse envelopes, usually have very large radii
+compared with MS stars, which consequently corresponds to significantly large tidal radii. In most
+cases a giant tidal disruption (GTD) will not cause a complete disruption. Inversely, the stellar core,
+and even a large fraction of the envelope of the star can survive (MacLeod et al. 2012). We assume
+that the envelope will be completely striped and accreted by the SMBH immediately after the GTD,
+
+MULTI-MASS N-BODY MODEL
+11
+and the remnant could be considered as a production of a fast evolved giant star. According to the
+predicted evolution paths proposed by Hurley et al. (2000), we make a simplified evolution scheme of
+GTDs in Fig. 1. Depending on the initial mass of the disrupted star, GTDs in this scheme may result
+in different types of remnants, such as naked helium main sequence stars, WDs and naked helium
+stars in the Hertzsprung gap. More details can be found in the discussion of Hurley et al. (2000).
+Compact stars, on the other hand, are simply assumed to be directly swallowed by the SMBH if
+their separations to the SMBH are smaller than the ISCO. In that case, the linear momentum and
+mass of the compact stars will be added to the SMBH. Therefore the capture criteria of compact stars
+is different from other stars. Our integration does not include the post-Newtonian approximation
+because it is very difficult to be involved in three body problems.
+The orbits of those compact
+stars with heavier masses, especially when they are close to ISCO, are not accurate enough. The
+eccentricity evolution due to GW emission can not be well traced in this work. We prefer to leave
+this problem in future works. For this reason we did not adopt the rMSO as the criteria for simplicity.
+2.4. From numerical models to the reality
+Direct N-body simulations are limited by particle resolution. Even taking into account the accel-
+eration of Heterogeneous Computing, the maximum particle number that can be managed by direct
+N-body simulation with reasonable integration time is only several million, which is obviously less
+than the number of stars in a typical galaxy. A direct consequence of limited particle resolution is
+that the TDEs are so rare in the simulation that we can not collect enough events to make statistical
+analyses. For example, if there is a galaxy with total mass 109 M⊙, and the half mass radius is
+r1/2 = 1 kpc, then a length scaling relation can be set up by Eq. 5. For a solar type star, the tidal
+radius is ∼ 10−5 pc corresponding to ∼ 10−8 in this simulation unit, which is a very tiny scale in the
+simulation. The collected TDEs in such configuration will be only a few in the entire simulation.
+In order to collect more TDEs in the simulation, we have to adopt a larger ”tidal radius”, which is
+rt ∼ 10−4. And the Schwarzschild radius also adopts the same scaling to make sure that rt/rsch is
+the same both in simulation and reality. By integrating several models with different tidal radii, we
+can try to extrapolate the simulation result to reality. This scheme has been proved to be feasible in
+
+12
+Li et al.
+Paper I. Since rt ∝ M1/3
+BH , while rMSO ∝ MBH, it is not recommended to make direct extrapolations
+from simulation results to other real systems with different MBH. Otherwise the results of the normal
+TDEs and those swallow events could be inconsistent. For this reason, we have to fix the mass of the
+SMBH in our simulations. In the following discussion, we set the total mass of each galaxy/nucleus
+is 109 M⊙, and the mass of each SMBH is 107 M⊙.
+The limited particle resolution also induced other artificial effects. Some dynamical processes are
+N dependent. For instance, the two-body relaxation timescale follows tr ∝ N/ ln N, which means the
+two-body relaxation in our model will be much faster than the case of a real galaxy. In a spherical
+stellar system with a SMBH in the center, two-body relaxation is the most important mechanism to
+scatter stars into the tidal disruption loss-cone, the phase space region corresponding to the orbits
+of stars can be tidally disrupted (Frank & Rees 1976; Lightman & Shapiro 1977). The process that
+scatters stars into loss-cone is the so-called loss-cone refilling. The situation in merging galaxies is
+quite different. In principle, as we did in Paper I, we can divide the dynamical evolution of the
+SMBHB in a galaxy merger into three phase. In phase I, two galaxies and their central SMBHs have
+so large separation that the interaction between central SMBH and surrounding stars is roughly the
+same as the case of a single SMBH in an isolated galaxy. That means the loss-cone refilling will
+be dominated by two-body relaxation. Thus the TDEs rate will be N dependent in phase I. As
+two SMBHs get more and more close to each other, the perturbation becomes stronger and stronger
+in phase II. As a result, the perturbation will dominate the loss-cone refilling, which will result in
+an approximate N independent TDEs rate. In phase III, two SMBHs form a compact binary and
+the loss-cone refilling is very complicated. By carefully analyzing the different situations in different
+phases, it is possible to make a credible extrapolation based on analytical models and our numerical
+models with different N.
+Detailed discussions can be found in Paper I and II. There are more
+discussions in Section 4.1.
+In addition to the artificial effects mentioned above, the stellar evolution model also needs to
+consider the influence of the limited particle resolution.
+For instance, if we use 106 particles to
+represent a dense star cluster, the total mass of stars in the Kroupa model should be ∼ 6 × 105 M⊙.
+
+MULTI-MASS N-BODY MODEL
+13
+However, we have to adapt this cluster to represent a galaxy with 109 stars. That means every particle
+in our model actually corresponds to a group of 1000 stars in the real galaxy. A straightforward
+discrepancy between our model and the real galaxy is that they have different two-body relaxation
+timescale.
+For example, we can assume that all the stars in the cluster have evolved for 1 Gyr.
+In our model, due to the limited particle number, the system could be well relaxed.
+While the
+same thing should not happen in a real dense star cluster with 109 stars. Therefore we need to
+have a proper scaling relation between the stellar evolution timescale in reality and the two-body
+relaxation timescale in our model.
+A simplified solution is to make sure that the ratio between
+relaxation timescale and stellar evolution time scale in the real dense star cluster should be equal
+to the ratio in our model (Panamarev et al. 2019). This scaling relation is essential for models with
+stellar evolution included (more discussions can be found in Panamarev et al. (2019)). However, as
+we mentioned before, we are focusing on the merging phase, which corresponds to a relatively short
+period of time. Consequently, in dry mergers with all stars that have already evolved for quite a
+long time, the stellar evolution during the merger could be neglected. Instead of including stellar
+evolution, it is more important to have an initial model with properly evolved stars before the merger,
+which is adopted here.
+3. RESULTS
+3.1. Dynamical evolution of SMBHs and surrounding stars
+Compared with previous major merger models with equal mass stars, we have more realistic models
+with different stellar components. Fig. 2 demonstrates the dynamical evolution of SMBHs in merger
+galaxies, which are numerically integrated by our models with N = 2×106 particles. Unless otherwise
+specified, all results in following discussions are based on this model. For better comparison, we have
+two integrations with and without initial mass function (IMF), which are represented by blue solid
+lines and red dashed lines, respectively. The left panel represents the evolution of dBH, the separation
+of two SMBHs, and the right panel demonstrates the evolution of the semi-major axis a.
+Since
+bound SMBHBs are not formed in phase I, the time begins from t ∼ 80 in the figure. As the results
+
+14
+Li et al.
+0
+50
+100
+150
+200
+t ([T])
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+10
+1
+d
+BH
+([R])
+80
+100
+120
+140
+160
+180
+200
+t ([T])
+0
+100
+200
+300
+400
+500
+1/a
+EQ
+MS
+Figure 2. The dynamical evolution of two SMBHs in merging galaxies. Red dashed and blue solid lines
+correspond to equal stellar mass model and multi-mass model, respectively. The left panel represents the
+separation evolution of two SMBHs. The right panel represents the semi-major axis evolution of the SMBHB,
+after a bound binary system is formed.
+indicated, though the evolutions of two SMBHs in different models are roughly the same in phase I,
+there is a significant difference in phase II. The SMBHB in the model with IMF evolves slower than
+the case of the equal mass model, which looks inconsistent with Khan et al. (2018). However, our
+integrations only continued to ∼ 100 N-body time unit, which is far less than Khan et al. (2018) did.
+According to the Fig. 1 and Fig. 2 in their paper, the difference between equal mass and multi-mass
+models is not obvious in the early stages of the SMBHB formation.
+The evolution of Lagrangian radii is demonstrated in Fig. 3. A Lagrangian radius is a sphere with
+the center of the stellar system. Stars inside this sphere occupy a fixed fraction of the total mass of
+the system. By tracing the evolution of several Lagrangian radii with different mass fractions, we
+can trace the dynamical evolution of the star cluster. Here dashed lines and solid lines denote equal
+mass and multi-mass models respectively. Different colors correspond to 0.1%, 1%, 10%, 50%, and
+90% of the total mass respectively, which have been marked in the legend. It should be noticed that
+the 1% curve is close to the influence radius, because the stellar mass inside this radius is close to the
+mass of the central SMBH. The upper left panel of the figure represents the Lagrangian radii relative
+to the center of mass. Since two galaxies are far away from each other, there are only a few stars
+around the center of mass, which results in the large Lagrangian radii at the beginning. The upper
+
+MULTI-MASS N-BODY MODEL
+15
+right panel demonstrates the Lagrangian radii relative to one of the SMBH. The inner region around
+the SMBH of the equal mass model is slightly more compact than the multi-mass model, which is
+consistent with a similar multi-mass model with single BH (Baumgardt et al. 2004). The bottom
+panels are the corresponding average mass inside each Lagrangian radii. In the equal mass model,
+all average masses are equal to 10−6, the single particle mass in the simulation. In the multi-mass
+model, the average masses inside larger radii, which correspond to all Lagrangian radii equal and
+larger than 10% in the figure, are very close to 10−6. However, this is not the case in the inner region.
+According to bottom panels, the average mass inside Lagrangian radii of mass fraction 0.1% and
+1% are significantly heavier than the average mass of the entire system, which can be considered as
+the consequence of the mass segregation at the inner region. In the bottom left panel, the average
+masses of the inner region are close to 10−6 at the beginning of the integration, because the center
+of mass is at the midpoint where the stellar density is very low. The mass segregation effect is more
+significant in the plot relative to one SMBH, which is demonstrated in the bottom right panel. The
+inner region average masses are significantly higher than the equal mass model at the beginning of
+the integration. However, during the formation of the bound SMBHB, the central average masses
+decline sharply. The strong interaction between the two SMBHs and the violent evolution of stars
+in the central region suppress the mass segregation quickly. After the two SMBHs form a compact
+binary, the average mass at the central region starts to recover with gradually increasing.
+Fig. 4 demonstrates the evolution of Lagrangian radii and corresponding average masses of different
+components relative to one of the SMBH. We classified stars into three groups: normal stars (NORM,
+solid lines) including MS star and WD, giant branch stars (GB, dotted lines) including RGs and AGBs
+and compact stars (CPT, dashed lines) including NSs and BHs. Different colors in the figure denote
+different mass fractions of corresponding component. Two vertical gray lines divide the evolution
+into three phases. The criteria for dividing three phases are based on the fluctuations of the TDE
+rate. Other criteria, for instance, a criteria based on dynamical evolution, will only slightly change
+the position and period of phase II, and will not lead to significantly different results. More details
+can be found in Paper II. Since NORM stars correspond to the largest fraction, the Lagrangian radii
+
+16
+Li et al.
+0
+50
+100
+150
+200
+t ([T])
+10
+−2
+10
+−1
+10
+0
+10
+1
+10
+2
+r
+Lagr.
+([r])
+0
+50
+100
+150
+200
+t ([T])
+10
+−2
+10
+−1
+10
+0
+10
+1
+10
+2
+r
+Lagr.
+([r])
+0.1%EQ
+0.1%MS
+1%EQ
+1%MS
+10%EQ
+10%MS
+50%EQ
+50%MS
+90%EQ
+90%MS
+0
+50
+100
+150
+200
+t ([T])
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+M
+avg
+([M])
+×10
+−6
+0
+50
+100
+150
+200
+t ([T])
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+M
+avg
+([M])
+×10
+−6
+Figure 3. The evolution of Lagrangian radii and corresponding average mass. The upper left panel is
+the Lagrangian radii relative to the center of mass, and the upper right panel corresponds to one of the
+SMBH. Dashed lines and solid lines denote equal mass and multi-mass models respectively. The bottom left
+and right panels demonstrate corresponding average mass inside each Lagrangian radius. Different colors
+represent different mass fractions of the total mass.
+of NORM stars are almost the same as the radii of all the stars. In the left panel, CPT stars with
+heavier mass show significant mass segregation in the central region, especially before the formation
+of SMBHB. Due to the heating of SMBHB, all the Lagrangian radii in the central region expand
+after two SMBHs form a bound system. Especially in the phase II, a large fraction of CPT stars in
+
+MULTI-MASS N-BODY MODEL
+17
+0
+50
+100
+150
+200
+t ([T])
+10
+−2
+10
+−1
+10
+0
+r
+Lagr.
+([r])
+PI
+PII
+PIII
+0
+50
+100
+150
+200
+t ([T])
+10
+−6
+10
+−5
+10
+−4
+M
+avg
+([M])
+0.1%NORM
+0.1%GB
+0.1%CPT
+1%NORM
+1%GB
+1%CPT
+10%NORM
+10%GB
+10%CPT
+Figure 4. The evolution of Lagrangian radii and corresponding average masses of different components
+relative to one of the SMBH. Solid lines, dotted lines and dashed lines denote normal stars, GBs, and compact
+stars, respectively. Different colors represent different mass fractions of the total mass of each component.
+The entire evolution has been divided into three phases by two vertical gray lines. The left panel is the
+evolution of Lagrangian radii, and the right panel is the evolution of average mass inside each Lagrangian
+radius.
+the central region are kicked out or swallowed by the SMBHs, which result in a significant jump of
+the Lagrangian radii inside the influence radius. This result is consistent with Gualandris & Merritt
+(2012), who has numerically investigated the evolution of SMBHBs in multi-component merging
+galaxies. According to the results in the right panel, only CPT stars at central region have significant
+evolution on average mass. NORM and GB stars roughly keep a constant average mass during the
+integration, with latter corresponding to heavier value. Similar to the Fig. 3, the average mass of
+CPT stars at inner region has a sharp drop in phase II. Since the drops of other two components are
+not significant, the sharp decline of the average mass in phase II should be mainly contributed by
+CPT stars.
+3.2. Tidal disruptions in multi-mass models
+In our multi-component simulation there are 13991 TDEs/swallow events within the 200 N-
+body time unit, and ∼ 94% are made by MS stars. The fraction of GB, WD, NS and BH are,
+respectively, ∼ 2.3%, ∼ 2.7%, ∼ 0.7% and ∼ 0.3%. Similar to the equal mass model, the tidal
+
+18
+Li et al.
+0
+50
+100
+150
+200
+t ([T])
+0
+1
+2
+3
+4
+5
+̇
+M([
+̇
+M])
+×10
+−4
+EQ
+MS
+0
+50
+100
+150
+200
+t ([T])
+0
+100
+200
+300
+400
+̇
+N
+0
+50
+100
+150
+200
+t ([T])
+0.8
+1.0
+1.2
+1.4
+1.6
+M
+avg
+([M])
+×10
+−6
+Figure 5. The evolution of the total tidal disruption/swallowed rate in the multi-mass model and the equal
+mass model. The red dashed and blue solid lines represent the result of equal mass model and multi-mass
+model respectively. The left panel and middle panel are, respectively, the evolution of the mass accretion
+rate and the event rate. The right panel is the average mass of disrupted/swallowed stars.
+disruption in multi-mass models also can be divided into three phases, which correspond to before,
+during and after the formation of bound SMBHBs. Fig. 5 demonstrates the total capture rate, which
+includes both disrupted stars and swallowed normal stars or compact stars, in the multi-mass model
+and the equal mass model. The red dashed and blue solid lines represent the result of equal mass
+model and multi-mass model respectively. The left panel and middle panel are, respectively, the
+evolution of the mass accretion rate and the event rate. In the simulation, we assume that all the
+mass of the disrupted stars will be accreted into the SMBH. That means, the evolution of the mass
+accretion rate and the event rate in the equal mass models are the same thing. However, it is different
+in multi-mass models. As a result, the peak mass accretion rates of the equal mass and multi-mass
+model in phase II are at the same level. While their event rates are different. Compared with the
+multi-mass model which has peak event rate ˙N = 275/[T], the equal mass model has significantly
+higher peak event rate ˙N = 421.5/[T]. According to this result, though the disrupted/swallowed
+stars are less than the case in the equal mass model, the disrupted/swallowed stars in the multi-mass
+model are preferred at the high mass end. It has been confirmed by the right panel of Fig. 5, which
+represents the average mass of disrupted/swallowed stars. This effect is mainly due to the larger
+tidal radii of heavier stars, and the mass segregation may also have some contributions.
+
+MULTI-MASS N-BODY MODEL
+19
+3.2.1. Tidal disruptions of main sequence stars
+Tidal disruption of MS stars dominate the TDEs in our models. As discussed in Section 2.3, MS
+stars with relatively low mass generally correspond to small radii. Stars with tidal radius smaller
+than rMSO will be essentially swallowed when they get close enough to the SMBH. According to our
+simulation results with the largest particle number, there are ∼ 45% MS stars, with masses range
+from ∼ 0.1 M⊙ to ∼ 0.4 M⊙, that have been swallowed by the SMBH. It should be noticed that
+all the WDs will be swallowed by the SMBH too. But they only contribute less than 3% of the
+total disruption/swallow events. Other models with different particle numbers give similar results.
+In general, nearly half of captured stars will directly plunge into the SMBH without flares. Fig. 6
+demonstrates the disruption/swallow evolution of MS stars. The orange and green lines in the figure
+denote stars with standard tidal disruption flare (STD) and stars with plunge orbits without flare
+(PLG), respectively. Blue lines and gray lines denote the evolution of all stars and the separation
+of two SMBHs, respectively. The entire evolution is divided into three phases by two vertical gray
+lines. The left y-axis represents the separation, and the right y-axis in the left and right panels are,
+respectively, the mass accretion rate and event number rate.
+Fig. 6 indicates that, due to the strong perturbation and the rapid evolution of the stellar dis-
+tribution around two SMBHs, both swallowed stars and normal disrupted stars have significantly
+enhanced rate in phase II. And there will be a rate peak every time two SMBHs get close. This result
+is consistent with the previous equal mass model in Paper I. From the right panel we can easily see
+that the normal TDEs and the plunge cases have very close event rates. However, it is obvious that
+the plunge stars only contribute a small fraction of the mass accretion rate in the left panel, because
+they are dominated by the stars at the low-mass end.
+3.2.2. Tidal disruptions of giant branch stars
+GB stars usually have very large radii, which means a GB star has a very large tidal radius. Usually,
+there is a dense core with significant mass concentrated in a GB star. During the tidal disruption,
+instead of being totally disrupted by the SMBH, only the envelope of the GB star will be striped
+
+20
+Li et al.
+0
+50
+100
+150
+200
+t ([T])
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+10
+1
+d
+BH
+([R])
+PI
+PII
+PIII
+0
+1
+2
+3
+4
+5
+̇
+M([
+̇
+M])
+×10
+−4
+0
+50
+100
+150
+200
+t ([T])
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+10
+1
+d
+BH
+([R])
+d
+BH
+ALL
+STD
+PLG
+0
+50
+100
+150
+200
+250
+300
+350
+̇
+N
+Figure 6. The tidal disruption/swallow evolution of MS stars. The left panel and right panel are, respec-
+tively, the evolution of mass accretion rate and the event rate. Blue, orange and green solid lines represent the
+result of all of MS stars, standard TDE, and the MS stars which directly plunge into the SMBH, respectively.
+Two vertical gray lines divide the evolution into three phases.
+away, leading to a light accretion with only a fraction of the stellar mass. As demonstrated in Fig. 7,
+the tidal disruption evolution of GB stars is similar to MS stars. The evolution can be divided into
+three phases and there is a significant enhanced rate in phase II. Here blue and orange lines represent
+the evolution of CPT and GB stars respectively, the rest of the legend and labels are the same as
+Fig. 6.
+As mentioned in Section 2.3, after the disruption, the remnant stars usually are WDs or naked
+helium MS stars. Most of them will fly by the SMBH and not be swallowed. However, according to
+our result, there is ∼ 14% remnants will finally plunge into the SMBH. Some of them will quickly
+plunge into the SMBH within one orbital period, and the rest may be captured by the SMBH first
+and finally swallowed with a time delay ∆t. Fig. 8 represents the distribution of these accreted
+remnant stars.
+The x-axis and y-axis, respectively, represent ”t1stTD”, the time that stars been
+tidally disrupted by the SMBH and the time delay ∆t between the first tidal disruption and the final
+plunge. The mass of remnants after the first tidal disruption has been denoted in different colors,
+with a color bar in solar mass. The size of the filled circles represents the mass of striped envelopes
+during GTD with the range from 2.6 × 10−5 M⊙ to 1.8 M⊙. According to the figure, most of the
+plunge events of the remnants are recorded soon after the first tidal disruption. There are some
+
+MULTI-MASS N-BODY MODEL
+21
+0
+50
+100
+150
+200
+t ([T])
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+10
+1
+d
+BH
+([R])
+PI
+PII
+PIII
+0
+1
+2
+3
+4
+5
+6
+7
+8
+̇
+M([
+̇
+M])
+×10
+−5
+0
+50
+100
+150
+200
+t ([T])
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+10
+0
+10
+1
+d
+BH
+([R])
+CPT
+GB
+0
+2
+4
+6
+8
+10
+12
+14
+̇
+N
+Figure 7. The tidal disruption evolution of CPT stars and GB stars. The left panel and right panel are,
+respectively, the evolution of mass accretion rate and the event rate. Blue and orange solid lines represent
+the result of CPT and GB stars respectively. Two vertical gray lines divide the evolution into three phases.
+exceptions which can survive for a relatively long time after the first tidal disruption. And most of
+such large time delay cases are in early phases. After a compact SMBHB formed in phase III, the
+remnants will be more easily kicked by the companion SMBH and avoid the final plunge. Since our
+integration terminated at t = 200, there may be some plunge events at t > 200 missed. In addition,
+we also find that ∼ 17 stars in Fig. 8 have been stripped off more than half of their mass during the
+GTD.
+3.2.3. SMBHB and compact objects
+Fig. 7 demonstrates the swallow rate evolution of CPT stars, which is similar to GB stars. According
+to the heavier mass, though CPT plunge events are not as frequent as GTD, their contributions on
+mass accretion is more significant than GB stars. Fig. 9 represents the mass distribution of swallowed
+NSs and BHs. Most swallowed NSs concentrate in the low mass end, while the BHs are the opposite.
+It should be noted that, since general relativity effects are not included in all of these integrations, the
+orbits of compact stars very close to SMBHs are not accurate. In principle, by using post-Newtonian
+approximation we can integrate the orbital evolution of a two-body system with relatively high
+accuracy. However, such an approximation is not easy to be adopted in a triple system which we
+
+22
+Li et al.
+0
+25
+50
+75
+100
+125
+150
+175
+200
+t
+1stTD
+−10
+0
+10
+20
+30
+40
+50
+60
+70
+Δt
+PI
+PII
+PIII
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+M
+PLG
+[M
+⊙
+]
+Figure 8. The GB stars which have been finally swallowed by the SMBH after a tidal disruption. The
+x-axis and y-axis represent the time that stars have been tidally disrupted by the SMBH and the time delay
+between the tidal disruption and the final plunge. Two vertical gray lines divide the evolution into three
+phases. The color bar denotes the mass of the tidal disruption remnants which finally plunge into the SMBH,
+in solar mass.
+are discussing here. Bonetti et al. (2016) has achieved an improvement with corrections up to 2.5PN
+order. We prefer to solve this problem with similar methods in future work. This simplification will
+not make a significant influence on the TDR evolution. Because most of disrupted stars are tend to
+be concentrated on the low mass end which was relatively less affected. The rate may only be slightly
+underestimated, but the evolution should be the same. While for CPT stars, specially those with
+orbits very close to SMBHs, the absence of the PN corrections may lead to the suppressed EMRIs
+formation.
+4. DISCUSSION
+
+MULTI-MASS N-BODY MODEL
+23
+1.4
+1.6
+1.8
+2.0
+2.2
+m * [M⊙]
+0
+5
+10
+15
+20
+25
+⊙0
+⊙5
+N
+NS
+⊙
+4
+5
+6
+7
+8
+9
+10
+m * [M⊙]
+0
+2
+4
+6
+8
+10
+12
+N
+BH
+Figure 9.
+The mass distribution of swallowed NSs and BHs.
+The left panel and the right panel are,
+respectively, the distribution of swallowed NSs and BHs. The masses of CPT stars are in solar mass and
+the y-axis represents the event counts.
+4.1. Extrapolations to galaxies
+As discussed in Section 2.4, the simulation results here can not be directly used to estimate the
+tidal disruption rate evolution in a galaxy. Due to the limited particle resolution, extrapolations
+based on the simulation results are crucial. Here we follow the same scheme as Paper I. We choose
+N = 5 × 105 for each galaxy and the average tidal radius rt = 5 × 10−4 as fiducial parameters.
+In order to investigate the rt and N dependence, we vary rt with fixed fiducial N, and vary N
+with fiducial rt individually. The rt varies in [10−5, 10−4.5, 10−4, 10−3.5, 10−3], and the N varies in
+[1.25 × 105, 2.5 × 105, 5 × 105, 106]. In every model, the average tidal disruption accretion rates and
+event rates in three phases are calculated individually.
+Fig. 10 demonstrates the corresponding simulation results of phase I and II for different stellar
+components. The red, green and blue dots represent the tidal disruption accretion rates of STD,
+PLG and CPT stars respectively. Limited by the particle resolution, even the model with largest
+star particles does not record enough events of GTD and CPT for statistical study in some phases.
+Especially in phase II, due to the short time periods, only STD and PLG stars have enough recorded
+events for statistical study. The left and middle panels in the figure demonstrate the rt dependence
+
+24
+Li et al.
+0
+5
+10
+r
+t
+([r])
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+̇
+M([
+̇
+M])
+×10
+−4
+×10
+−4
+PI
+STD
+PLG
+CPT
+0
+500
+1000
+N
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+̇
+M([
+̇
+M])
+×10
+−4
+×10
+3
+PI
+0
+5
+10
+r
+t
+([r])
+0
+1
+2
+3
+4
+̇
+M([
+̇
+M])
+×10
+−4
+×10
+−4
+PII
+Figure 10. The disruption rate extrapolations of phase I and II. The red, green and blue dots represent
+the tidal disruption accretion rates of STD, PLG and CPT events based on simulation results respectively.
+The red solid, the green dashed and the blue dotted lines represent the corresponding fitting results based
+on analytical estimations. The left and middle panels demonstrate the rt dependence and N dependence in
+phase I, respectively. And the right panel denotes the rt dependence in phase II.
+and N dependence in phase I, respectively. And the right panel represents the rt dependence in
+phase II.
+Based on the discussions in Paper I and II, the tidal disruption in phase I is dominated by two-body
+relaxation, and the accretion rate can be well estimated by
+˙M ∝
+� N
+ln Λ
+�α
+rβ
+t ,
+(8)
+where α and β can be fitted through numerical simulation results, and ln Λ is the Coulomb logarithm
+and can be estimated by
+ln Λ ≈ ln
+�MBH
+2m∗
+�
+,
+(9)
+where m∗ is the average mass of stars (Preto et al. 2004).
+However, unlike the equal stellar mass models, the mass accretion rates in multi-mass models can
+not intuitively reflect the event rate. As a rough estimation, we can simply assume ˙N = ˙M/m∗. The
+TDE rate in phase I can be estimated by
+˙N ∝ N
+�ln Λ
+N
+�ζ
+rξ
+t,
+(10)
+where ζ and ξ can be fitted through numerical simulation results.
+
+MULTI-MASS N-BODY MODEL
+25
+The tidal disruption loss cone refilling in phase II is almost full, which means the tidal disruption
+mass accretion rate weakly depends on N. Therefore the rate can be approximated as a power law of
+rt. But the event rate ˙N should be divided by m∗, or equivalent to multiplying by N. The situation
+is very complicated in phase III. Two-body relaxation, perturbation of companion SMBH and the
+triaxial stellar distributions have significant contributions to the loss cone refilling. It is very difficult
+to analytically estimate the disruption rate in phase III. Therefore the extrapolation in phase III is
+not reliable.
+With above analytical models, and assuming that different stellar components roughly follow the
+same relations, we can make rough extrapolations for different stellar components in phase I and II.
+4.1.1. Phase I
+Following the similar scheme in Paper I, combining with simulation results, the tidal disruption
+mass accretion rate of STD stars in phase I can be estimated as
+˙MSTD ∼ 1.48 ×
+� N
+ln Λ
+�−0.57
+r0.48
+t
+,
+(11)
+which can be extrapolated as
+˙MSTD ∼ 2.7 × 10−2
+� N
+ln Λ
+�−0.57 � r1/2
+1 kpc
+�−1.98 �
+rt
+10−6pc
+�0.48
+M⊙ yr−1.
+(12)
+The event rate is
+˙NSTD ∼ 0.84N
+� N
+ln Λ
+�−0.57
+r0.48
+t
+,
+(13)
+which can be extrapolated as
+˙NSTD ∼ 1.5 × 10−2 M⊙
+m∗
+� N
+ln Λ
+�−0.57 � r1/2
+1 kpc
+�−1.98 �
+rt
+10−6pc
+�0.48
+yr−1.
+(14)
+For our fiducial model with M = 109 M⊙, MBH = 107 M⊙, r1/2 = 1 kpc, and according to our
+simulation results with m∗ = 0.43 M⊙ corresponding to stellar radius r∗ = 0.40R⊙, we can derive
+the mass accretion rate of the disrupted stars should be ∼ 1.0 × 10−6M⊙ yr−1, and the event rate is
+∼ 1.3 × 10−6yr−1.
+
+26
+Li et al.
+Similarly, the mass accretion rate of PLG stars can be estimated as
+˙MPLG ∼ 0.16 ×
+� N
+ln Λ
+�−0.47
+r0.51
+t
+,
+(15)
+with corresponding event rate
+˙NPLG ∼ 0.32N
+� N
+ln Λ
+�−0.47
+r0.51
+t
+.
+(16)
+The extrapolation of the mass accretion rate and the event rate are
+˙MPLG ∼ 1.6 × 10−3
+� N
+ln Λ
+�−0.47 � r1/2
+1 kpc
+�−2.01 �
+rt
+10−6pc
+�0.51
+M⊙ yr−1,
+(17)
+˙NPLG ∼ 3.2 × 10−3 M⊙
+m∗
+� N
+ln Λ
+�−0.47 � r1/2
+1 kpc
+�−2.01 �
+rt
+10−6pc
+�0.51
+yr−1.
+(18)
+In our fiducial model, the mass accretion rate should be ∼ 4.0 × 10−7M⊙ yr−1, and the event rate is
+∼ 1.8 × 10−6yr−1.
+GTD and CPT events are not popular in all simulations. GTD events are quite rare. However, we
+can still find some CPT stars swallowed by SMBHs in phase I. Although the data is not very good
+for statistical study, we can still try some extrapolations, which gives the mass accretion rate
+˙MCPT ∼ 62.74 ×
+� N
+ln Λ
+�−1.04
+r0.55
+t
+,
+(19)
+and the event rate
+˙NCPT ∼ 7.18N
+� N
+ln Λ
+�−1.04
+r0.55
+t
+.
+(20)
+The extrapolation relations of the mass accretion rate and the event rate are
+˙MCPT ∼ 0.29
+� N
+ln Λ
+�−1.04 � r1/2
+1 kpc
+�−2.05 �
+rt
+10−6pc
+�0.55
+M⊙ yr−1,
+(21)
+˙NCPT ∼ 3.3 × 10−2 M⊙
+m∗
+� N
+ln Λ
+�−1.04 � r1/2
+1 kpc
+�−2.05 �
+rt
+10−6pc
+�0.55
+yr−1.
+(22)
+And the corresponding accretion rate and the event rate of the fiducial model are, respectively,
+˙M ∼ 1.6 × 10−9M⊙ yr−1 and ˙N ∼ 4.4 × 10−10yr−1. It should be noticed that our results here are
+purely Newtonian. Since the post-Newtonian approach is not included in our model, the results here
+can only be considered as a crude approximation.
+
+MULTI-MASS N-BODY MODEL
+27
+0
+2
+4
+6
+8
+10
+12
+rt([r])
+0
+5
+10
+15
+20
+25
+30
+35
+̇N
+×10−4
+PI
+STD
+PLG
+̇PT
+0
+200 400 600 800 10001200
+N
+0
+5
+10
+15
+20
+25
+30
+35
+̇N
+×103
+PI
+0
+2
+4
+6
+8
+10
+12
+rt([r])
+0
+20
+40
+60
+80
+100
+120
+140
+̇N
+×10−4
+PII
+0
+200 400 600 800 10001200
+N
+0
+20
+40
+60
+80
+100
+120
+140
+̇N
+×103
+PII
+Figure 11. The disruption event rate extrapolations of phase I and II. The legends are similar to the
+Fig. 10. The first and the second rows demonstrate the results of phase I and phase II, respectively. The
+left and right panels in each row demonstrate the rt dependence and N dependence, respectively.
+Fig. 10 and 11 demonstrate the extrapolation fitting results of the mass accretion rate and event
+rate respectively. Red, green and blue dots in two figures represent simulation results of STD, PLG
+and CPT stars respectively. Red solid, green dashed and blue dotted lines are, respectively, fitting
+results of STD, PLG and CPT stars based on the derived extrapolation formulas in Section 4.1.1 and
+4.1.2. It should be aware that, due to very limited event records, the extrapolation fitting results of
+swallowed CPT stars in PI are not as good as STD or PLG stars. Compare the two figures, the STD
+
+28
+Li et al.
+and PLG have similar event rates in phase I. But the STD stars correspond to significantly higher
+mass accretion rates. Our extrapolation results of the fiducial model also indicate that more than
+half events are contributed by PLG stars, which may not induce any observational effects. But more
+than half accreted mass is contributed by STD stars.
+4.1.2. Phase II
+Both analytical estimations and our numerical simulations indicate that the tidal disruption rate
+in phase II does not significantly depend on N. The accretion rate of STD stars can be estimated by
+˙MSTD ∼ 0.11r0.82
+t
+,
+(23)
+with extrapolation relation
+˙MSTD ∼ 2.2 × 10−6
+� r1/2
+1 kpc
+�−2.32 �
+rt
+10−6pc
+�0.82
+M⊙ yr−1,
+(24)
+which corresponds to ˙M ∼ 5.0 × 10−6M⊙ yr−1 for our fiducial model.
+The event rate is
+˙NSTD ∼ 5.97 × 10−2Nr0.82
+t
+,
+(25)
+and the extrapolation can be wrote as
+˙NSTD ∼ 1.2 × 10−6 M⊙
+m∗
+� r1/2
+1 kpc
+�−2.32 �
+rt
+10−6pc
+�0.82
+yr−1,
+(26)
+which corresponds to ˙N ∼ 6.3 × 10−6yr−1 for the fiducial model.
+Similarly, the mass accretion rate of PLG stars can be estimated by
+˙MPLG ∼ 2.69 × 10−2r0.85
+t
+,
+(27)
+with extrapolation relation
+˙MPLG ∼ 3.3 × 10−7
+� r1/2
+1 kpc
+�−2.35 �
+rt
+10−6pc
+�0.85
+M⊙ yr−1.
+(28)
+For our fiducial model, that corresponds to ˙M ∼ 7.6 × 10−7M⊙ yr−1.
+
+MULTI-MASS N-BODY MODEL
+29
+Table 1. Averaged mass accretion rates and event rates in different phases.
+Stage
+˙
+MSTD
+˙NSTD
+˙MPLG
+˙NPLG
+˙MCPT
+˙NCPT
+×10−6M⊙/yr
+×10−6/yr
+×10−6M⊙/yr
+×10−6/yr
+×10−9M⊙/yr
+×10−9/yr
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+PI
+1.0
+1.3
+0.4
+1.8
+1.6
+0.4
+PII
+5.0
+6.3
+0.8
+3.5
+-
+-
+Note—Col.(1): Stage of the evolution.
+Col.(2): Mass accretion rate of disrupted STD stars.
+Col.(3): Event rate of disrupted STD stars. Col.(4): Mass accretion rate of swallowed PLG
+stars. Col.(5): Event rate of swallowed PLG stars. Col.(6): Mass accretion rate of swallowed
+CPT stars. Col.(7): Event rate of swallowed CPT stars.
+The event rate of PLG stars is
+˙NPLG ∼ 5.36 × 10−2Nr0.85
+t
+.
+(29)
+and the extrapolation can be wrote as
+˙NPLG ∼ 6.5 × 10−7 M⊙
+m∗
+� r1/2
+1 kpc
+�−2.35 �
+rt
+10−6pc
+�0.85
+yr−1,
+(30)
+which corresponding to ˙N ∼ 3.5×10−6yr−1 for the fiducial model. Obviously, both the mass accretion
+rate and the event rate in phase II have significant increases. The rate of STD has increased several
+times compared with PI, which is consistent with the result in Paper I and II.
+Since the period of phase II in all of our models is around 20 N-body time units, there are not many
+recorded events contributed by CPT and GTD stars. The extrapolation can not be well managed
+for them.
+Table. 1 summarizes the extrapolation results of averaged mass accretion rates and event rates in
+phase I and II, for disrupted/swallowed STD, PLG and CPT stars respectively. Since the swallowed
+CPT stars in phase II are relatively rare, we do not estimate their rates in the table.
+4.2. Delectability by space based GW instruments
+
+30
+Li et al.
+As strong GW sources, SMBHBs could be directly detected by PTAs or space born detectors such
+as LISA, TaiJi and TianQin in the future. There are many discussions on this topic, which is out
+of the scope of this paper. Here we want to discuss EMRIs, another kind of typical GW source
+produced by SMBHs which could be detected by LISA/TaiJi/TianQin. Such events could also be
+prompted around a SMBHB. However, a typical EMRI should spiral the SMBH for many orbits. It
+needs accurate integrations with general relativity effects considered, which is not included in our
+integrations. Therefore, it is not reliable to estimate the rate of EMRIs based on our models. Another
+interesting event are EMRBs, which are GW bursts from stellar objects passing by a SMBH with
+very small pericenter distances. It has been considered as the precursor to EMRIs, because many
+EMRBs will lose their energy and angular momentum through GW radiation and finally evolve into
+EMRIs (Rubbo et al. 2006). Hopman et al. (2007) demonstrates that the burst rates for stellar BHs
+and MSs/WDs are, respectively, 1 yr−1 and 0.1 yr−1 in the Milky Way. If extragalactic sources could
+be included, a detector like LISA could manage the detection out to ∼ 100 Mpc for a 10 M⊙ BH,
+with event rate of ∼ 0.2 yr−1 (Berry & Gair 2013b,c). Recently, Han et al. (2020) calculates the event
+rates of very extreme mass ratio bursts with a mass ratio about 10−8. They especially considered the
+contribution of plunge stellar objects such as brown dwarfs with unbound orbits. Their estimation
+indicates that, for small stellar objects with mass ∼ 0.1 M⊙, the space based facilities could detect
+the bursts inside 10 Mpc, with corresponding event rate 4 − 8 yr−1.
+In our model, as discussed in Section 4.1, the rates of compact stars such as NSs and BHs getting
+swallowed by SMBHs are quite low. However, there are many low mass main sequence stars that
+plunge into SMBHs. According to the result of our largest simulation, the average mass of swallowed
+stars is ∼ 0.2 M⊙, which have similar mass as Han et al. (2020) discussed. In principal, the signal-
+to-noise ratio (SNR) of such kind of EMRBs is proportional to m∗R−1M2/3
+BH, with R is the distance
+from the source to observers (Berry & Gair 2013b). With carefully numerical estimations, Han et al.
+(2020) finds that the SNRs of plunging events with 0.1 M⊙ stars in the Galactic Center can be up
+to ten thousands for LISA. If we consider a 0.2 M⊙ star swallowed by an 107 M⊙ SMBH, the SNR
+can be still as large as ∼ 8 at ∼ 50 Mpc. However, according to estimations in Section 4.1, the event
+
+MULTI-MASS N-BODY MODEL
+31
+rates of the PLG are only around 10−6 yr−1 in phase I and II. Since there are not too many galaxies
+inside ∼ 50 Mpc, it is unlikely to detect such kind of EMRBs with space borne GW detectors in the
+near future.
+5. SUMMARY
+We investigated full and partial tidal disruption of stars and direct plunges into supermassive black
+holes (MSBH) in nuclear star clusters during and after a galaxy merger. For that a full direct N-
+body simulation has been used with stars obtained from a realistic stellar mass distribution, evolved
+for ∼ 1 Gyr (to account for the age of the galaxies before the merger), surrounding two SMBH
+situated in the centres of the two merging galaxies.
+With the stellar evolution included, there are different stellar components, from low mass main
+sequence stars to white dwarfs, neutron stars and stellar mass black holes. Different stellar objects,
+according to their properties like mass and radius, have different fates after the close encounter with
+central SMBHs. Compared to the equal mass model, the multi-mass model with similar parameters
+tends to have similar or even slightly higher mass accretion rate. However, although their event
+rates in phase I and III (before and after the galaxy merger) are similar, the multi-mass model has
+lower event rate compared to the equal mass model in phase II (the short time while the merger is
+dynamically ongoing), which indicates that the disrupted/swallowed stars in multi-mass model prefer
+high mass end.
+In the multi-mass model, if a main sequence star is heavy enough, its close encounter with a SMBH
+may lead to a standard TDE. Otherwise a light main sequence star may correspond to a tiny tidal
+radius which makes it inside the marginally bound orbits. That will result in a plunge event. During
+the galaxy merger, due to the perturbation of the companion SMBH and stars around it, both STD
+and PLG event rates will be enhanced in phase II. STD and PLG have similar event rates. But STD
+events correspond to significantly higher mass accretion rate, because the PLG events are preferring
+to low mass stars. Since PLG events may not have any observational signatures, probably nearly
+half of the SMBH tidal capture events are invisible.
+
+32
+Li et al.
+Post sequence stars, such as RGs or AGB stars, could be partially disrupted. There might be a
+core that survived after a GTD which stripped away the envelope. The remnant could escape or
+plunge into the SMBH. Our largest numerical simulation gets ∼ 10% GTD stars finally plunge into
+SMBHs after their disruption. Some of them can have bound orbits around the SMBH and survive
+for many orbits.
+CPT stars show significant mass segregation in the central region during phase I. Due to the heating
+of the newly formed SMBHB, the Lagrangian radii in the central region expand quickly during phase
+II. Since our integrations are totally Newtonian, the orbits of compact stars close to the SMBH are
+not accurate. According to our limited results, the rate of compact stars getting swallowed by SMBHs
+in phase II could be significantly enhanced. And most swallowed NSs concentrate on the low mass
+end, while BHs are the opposite.
+
+MULTI-MASS N-BODY MODEL
+33
+We
+are
+grateful
+to
+the
+support
+of
+the
+National
+Natural
+Science
+Foundation
+of
+China
+(NSFC11988101,NSFC11303039), the Key International Partnership Program of the Chinese
+Academy of Sciences (CAS) (No.114A11KYSB20170015), and the Strategic Priority Research Pro-
+gram (Pilot B) Multiwavelength gravitational wave universe of CAS (No.XDB23040100). We (SL,
+PB, RS) acknowledge support by CAS through the Silk Road Project at National Astronomical Ob-
+servatories (NAOC) of China, and the support by Key Laboratory of Computational Astrophysics.
+The computations have been done on the Laohu supercomputer at the Center of Information and
+Computing at NAOC, CAS, funded by the Ministry of Finance of People’s Republic of China under
+the grant ZDY Z2008−2. LS and RS acknowledge the support of Yunnan Academician Workstation
+of Wang Jingxiu (No. 202005AF150025). LS acknowledges support from the K.C.Wong Education
+Foundation. PB acknowledges the special support by the CAS President’s International Fellowship
+for Visiting Scientists (PIFI) program during his stay in NAOC, CAS. XC acknowledges the support
+of the National Natural Science Foundation of China (No. 11873022). The work of PB was sup-
+ported by the Volkswagen Foundation under the special stipend No. 9B870 and the grant No. 97778.
+PB acknowledge the support within the grant No. AP14870501 of the Science Committee of the
+Ministry of Science and Higher Education of Kazakhstan. The work of PB was supported under
+the special program of the NRF of Ukraine Leading and Young Scientists Research Support - ”As-
+trophysical Relativistic Galactic Objects (ARGO): life cycle of active nucleus”, No. 2020.02/0346.
+PB thanks the support from the ACIISI, Consejer´ıa de Econom´ıa, Conocimiento y Empleo del Gob-
+ierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference
+PROID2021010044.
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9dFLT4oBgHgl3EQfCC4D/content/tmp_files/2301.11973v1.pdf.txt

@@ -0,0 +1,307 @@
+1 
+GAIN-SWITCHED VCSEL AS A QUANTUM ENTROPY SOURCE: 
+THE PROBLEM OF QUANTUM AND CLASSICAL NOISE 
+1,2R.А. Shakhovoy✉, 1E.I. Maksimova,  
+1QRate, 100 Novaya str., Skolkovo, Russia 
+2NTI Center for Quantum Communications, National University of Science and Technology 
+1MISiS, 4 Leninsky prospekt, Moscow, Russia  
+✉r.shakhovoy @goqrate.com 
+ 
+Abstract. We consider the problem of quantum noise extraction from polarization swapping 
+in a gain-switched VCSEL. The principle of operation of a quantum random number generator is 
+based on the generation of laser pulses with one of two orthogonal polarization states, followed by 
+digitization of polarization-resolved pulses with a comparator. At intensity values of laser pulses 
+close to the threshold value of the comparator, the contribution of the classical noise of the 
+photodetector will have a crucial role in making a decision on the choice of a logical zero or one. 
+We show how to evaluate the contribution of classical noise and how to calculate the quantum 
+reduction factor required for post-processing. 
+Keywords: quantum random number generators, vertical surface emitting laser, quantum 
+noise extraction 
+Funding: This work was supported by Russian Science Foundation (grant no. 17-71-20146). 
+ 
+Introduction 
+Random number generators (RNGs) play a primary role in modern cryptographic 
+applications. Due to the development of quantum cryptography, a special place among RNGs 
+occupy now quantum RNGs (QRNGs), which use various quantum sources of entropy. Over the 
+past 15-20 years, a number of approaches have been proposed to obtain quantum randomness; 
+however, optical QRNGs, which employ laser noise, have gained the most popularity. Laser noise 
+can be associated with various effects, e.g., with temperature-related fluctuations of the laser cavity 
+length or with pump fluctuations. However, at relatively high frequencies, laser noise is associated 
+mainly with spontaneous emission occurring due to zero-point oscillations of an electromagnetic 
+field, which have purely quantum nature and are generally considered to have the properties of 
+white noise. Due to this, laser noise can be employed as a high-frequency source of quantum 
+entropy. 
+The main difference between optical QRNGs based on laser noise lies in how the noise is 
+measured. Thus, interference-based optical QRNGs use phase noise of laser radiation, which is 
+converted into amplitude fluctuations in the interferometer and then is readily measured with 
+conventional photodetectors. In lasers that do not have fixed polarization of light, one may also 
+use fluctuations of the polarization state in addition to phase fluctuations. Such an approach can 
+be used, e.g., in a vertical-cavity surface-emitting laser (VCSEL). A VCSEL-based QRNG 
+employing spontaneous polarization switching was first described in [1], where the author 
+demonstrated the random bit generation rate up to 2 Mbps. Recently, we discussed a simple optical 
+scheme of a QRNG based on a gain-switched VCSEL, which allows generating the sequence of 
+random “on-off” pulses at several gigahertz [2]. Experimental demonstration of theoretical 
+calculations performed there have been published in [3]. In the present article, we consider the 
+problem of quantum noise extraction from polarization swapping in a gain-switched VCSEL. We 
+use the approach developed in [4], namely, we introduce for the QRNG under consideration the 
+quantum reduction factor containing information on the amount of classical noise “falling” into 
+the digitized random sequence due to fluctuations in the photodetector. We also describe how this 
+classical noise can be filtered out with the post-processing procedure. 
+ 
+Simulations 
+A simplified scheme of the proposed QRNG is shown in Fig. 1(a). A gain-switched VCSEL 
+is driven by a high-frequency laser driver, which is, in turn, controlled by the computer or FPGA. 
+Laser output is followed by the polarization filter (PF) that allows obtaining polarization-resolved 
+
+2 
+optical pulses, which are converted into the electrical signal via a broadband photodetector (PD). 
+Random bits are obtained by digitizing pulses with a comparator, whose threshold voltage is 
+calculated in the FPGA, which also performs post-processing procedures including randomness 
+extraction. 
+ 
+Fig. 1. (a) A simplified scheme of a VCSEL-based QRNG.  
+(b) Pulses at the comparator input and the results of the digitization 
+ 
+In Fig. 1(b), we simulated the digitization process of polarization-resolved laser pulses. It 
+was assumed in simulations that the polarizer in Fig. 1(a) passes to the photodetector the x -linear 
+polarization. Laser pulses were simulated with VCSEL rate equations given in [1]; the calculated 
+signal was then processed with a low-pass digital filter (30 GHz bandpass) to simulate the finite 
+bandwidth of the photodetector. The level of the comparator threshold (
+th
+V ) is shown by the dash-
+dotted line in Fig. 1(b); red circles correspond to the moments of the comparator latch actuation. 
+The result of digitization (‘0’-s or ‘1’-s) is shown in the corresponding frames (each time frame is 
+shown by the blue rectangle). 
+Generally, a laser pulse at the VCSEL output contains both polarization components ( x and
+y ), such that polarization state of a given pulse can be referred to as “quasi-elliptical”. Relative 
+contribution of orthogonal components is a random quantity; however, it depends on the width of 
+the pulse and the rate of relaxation processes (transients). To demonstrate this, we calculated 
+probability density function (PDF) of the normalized integral signal 
+xS  at three different repetition 
+rates (Fig. 2(a)). In the ideal case, we would get two peaks at the values 
+0
+xS =
+ and 
+1
+xS = , which 
+means that all optical power goes into one particular linear polarization ( y  and x respectively). 
+However, due to the finiteness of transients, 
+xS  could take intermediate values between 0 and 1. 
+The influence of transients becomes more prominent when decreasing the pulse width, which is 
+clearly seen in Fig. 2(a), where the area under the PDF curve in the middle of the histogram is 
+increased when increasing the pulse repetition rate from 2.5 to 7 GHz. Polarization-resolved laser 
+pulses ( x-pulses in our case) that fall into this intermediate region are the most affected by classical 
+(non-quantum) noises of the photodetector; therefore, ‘0’-s and ‘1’-s resulted from digitization of 
+such pulses can be considered as “untrusted” bits. The proportion of these bits can be thought of 
+as a quantum reduction factor 
+, whose value determines how much the raw random sequence 
+should be “compressed” using the randomness extractor. We propose the following formula to 
+find 
+: 
+ 
+)
+(
+1
+1
+H
+P  
+(1) 
+
+0
+0
+0
+Vmax
+th
+0
+2
+3
+4
+Time, ns3 
+where H  is the min-entropy of the raw random sequence, and P  is the probability to obtain the 
+pulse with the 
+x
+S  value inside some “window” around the middle of the probability distribution, 
+whose width is proportional to the relative r.m.s. value of the photodetector noise σ  (“measured” 
+in terms of the normalized value 
+x
+S ). 
+We also calculated the dependence of 
+ on the photodetector’s noise σ  at different pulse 
+repetition rates (see Fig. 2(b)). One can see that the reduction factor 
+ grows when increasing the 
+pulse repetition rate and begins to grow faster with increasing σ . It means that it does not make 
+much sense to increase the repetition rate of laser pulses if the photodetector is quite noisy. 
+ 
+ 
+Fig. 2. Probability densities of the normalized integral signal 
+x
+S  (a) and dependences of the reduction 
+factor 
+ on the photodetector’s noise σ  (b) at different pulse repetition rates. 
+ 
+Post-processing 
+The digitized random sequence in the proposed scheme must be subjected to the randomness 
+extraction procedure with the reduction factor 
+, defined by (1). We may, however, use a 
+deterministic extractor, e.g., the von Neumann extractor [5], which extracts randomness regardless 
+the value of the reduction factor. The von Neumann extractor discards repeated bits in a sequence 
+and replaces the two-bit words '01' and '10' with bits '0' and '1', respectively. Unfortunately, this 
+extractor reduces the length of a sequence by at least 4 times, which is not very efficient. Therefore, 
+one generally uses instead a seeded extractor. In cryptographic applications, an extractor with a 
+seed is generally implemented in the form of 2-universal hash functions, whose efficiency is 
+guaranteed by the leftover hash lemma [6]. A common way to implement 2-universal hashing is 
+to multiply the input raw sequence by a random binary matrix [7]. Without loss of generality, one 
+may always use for these purposes random Boolean Toeplitz matrices, which allow significantly 
+saving the seed length. In our case, the randomness extractor is then divided into three steps: 
+1) 
+For a “raw” binary sequence of length n, determine the length of the output 
+sequence m by the formula: m
+n
+=
+. 
+2) 
+Generate the Toeplitz matrix using the “seed” of length 
+1
+m
+n
++
+−  bits. 
+3) 
+Multiply the Toeplitz matrix by the raw sequence. This yields the resulting random 
+sequence. 
+It is important to discuss the method of obtaining the seed. By default, it is assumed that the 
+seed is obtained from a strong source of entropy, i.e., one that allows getting truly random bits. If 
+the RNG being developed is not a strong source of entropy, then an additional source of entropy 
+must be used. We propose, however, the following algorithm to obtain the seed. When switching-
+on the QRNG, the system buffers a raw random sequence of a given (relatively small) length. 
+Then, this sequence is subjected to a deterministic extractor, e.g., the von Neumann extractor. The 
+random sequence obtained after the extractor can be now used as a seed in hashing algorithms. 
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1.0
+1.2
+1.4
+1.6
+0.0
+0.5
+1.0
+0
+2
+4
+6
+(a)
+5 GHz
+7 GHz
+2.5 GHz
+PDF
+2.5 GHz
+(b)
+0.0
+0.5
+1.0
+5 GHz
+0.0
+0.5
+1.0
+7 GHz
+
+4 
+“Equipped” with such a procedure, the QRNG under consideration is an autonomous source of 
+entropy that does not need an additional entropy source, i.e., the device can operate even in the 
+absence of a pre-memorized random sequence. 
+One of the common ways to test the quality of randomness, is to perform statistical tests, 
+e.g., NIST tests [8]. Unfortunately, we did not have an access to real (obtained in the experiment) 
+random numbers; however, we have a fairly detailed theoretical model, which may be used to 
+follow the whole route the laser noise “travels” from spontaneous emission to the sequence of 
+random bits. For this, we simulated 106 laser pulses similar to those shown in Fig. 1(b). To 
+“digitize” them, the energy of each pulse (area under the pulse) was calculated and compared with 
+a certain threshold energy. The obtained random bits were then grouped into k -bit words (we put 
+8
+k =
+), which we denote as [ ]
+x i . The sequence of [ ]
+x i  were processed with a second-order FIR 
+filter according to the following formula: [ ]
+mod( [ ] 2 [
+1]
+[
+2],2 )
+k
+y i
+x i
+x i
+x i
+=
++
+−
++
+−
+, where each [ ]
+y i  
+is an i -th output word. The obtained data were then concatenate to a (“filtered”) random bit string. 
+After filtering, we processed the data with the randomness extractor described above. Finally, we 
+performed NIST tests with random bits; the results of the test are summarized in Fig. 3. As one 
+can see, the obtained random sequence successfully passed all the test. 
+ 
+Fig. 3. NIST statistical test result. 
+ 
+Conclusion 
+We propose here an approach for quantum noise extraction from polarization swapping in a 
+gain-switched VCSEL and proposed a simple method to get a seed for hashing the raw random 
+sequence without an additional entropy source. The discussed algorithms allow developing a fully 
+autonomous QRNG with proven “quantumness” of generated random bits. 
+ 
+Acknowledgments 
+Authors are grateful to Vladimir Meshkov for valuable comments. 
+ 
+REFERENCES 
+1. V. N. Chizhevsky, Bistable vertical cavity laser with periodic pump modulation as a random 
+bits generator, Optics and Spectroscopy, 108 (3) (2010) 343-346. 
+2. R. Shakhovoy, E. Maksimova, V. Sharoglazova, M. Puplauskis and Y. Kurochkin, Fast 
+and compact VCSEL-based quantum random number generator, Journal of Physics: Conference 
+Series, 1984 (1) (2021) 012005. 
+3. A. Quirce and A. Valle, Quantum random number generation based on polarization switching 
+in gain-switched VCSELs, Optics Express, 30 (7) (2022) 10513-10527. 
+4. R. Shakhovoy, D. Sych, V. Sharoglazova, A. Udaltsov, A. Fedorov and Y. Kurochkin, 
+Quantum noise extraction from the interference of laser pulses in optical quantum random number 
+generator, Optics Express, 28 (5) (2020) 6209-6224. 
+5. J. von Neumann, Various Techniques Used in Connection With Random Digits, J. Res. Nat. 
+Bur. Stand. Appl. Math. Series, 3 (1951) 36-38. 
+6. N. Nisan and A. Ta-Shma, Extracting Randomness: A Survey and New Constructions, Journal 
+of Computer and System Sciences, 58 (1) (1999) 148-173. 
+0,001
+0,01
+0,1
+1
+p – value
+MonobitFrequencyTest
+BlockFrequencyTest90Blocks
+RunsTest
+LongestRunsOnes10000
+BinaryMatrixRankTest
+SpectralTest
+NonOverlappingTemplateMatching
+OverlapingTemplateMatching
+MaurersUniversalStatisticTest
+LinearComplexityTest
+SerialTest
+ApproximateEntropyTest
+CumulativeSumsTest
+RandomExcursionsTest
+RandomExcursionsVariantTest
+
+5 
+7. H. Krawczyk, LFSR-based Hashing and Authentication, In: Proceedings of the Advances in 
+Cryptology — CRYPTO ’94, Berlin, Heidelberg, 1994, 129-139. 
+8. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, 
+D. Banks, A. Heckert, J. Dray, and S. Vo, A statistical test suite for random and pseudorandom 
+number generators for cryptographic applications, NIST Special Publication 800-22 revision 1a, 
+(2010). 
+ 
+ 
+ 
+

9dFLT4oBgHgl3EQfCC4D/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf,len=202
+page_content='1  GAIN-SWITCHED VCSEL AS A QUANTUM ENTROPY SOURCE:  THE PROBLEM OF QUANTUM AND CLASSICAL NOISE  1,2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Shakhovoy✉, 1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Maksimova,  1QRate, 100 Novaya str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', Skolkovo, Russia  2NTI Center for Quantum Communications, National University of Science and Technology  1MISiS, 4 Leninsky prospekt, Moscow, Russia  ✉r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='shakhovoy @goqrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='com  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' We consider the problem of quantum noise extraction from polarization swapping  in a gain-switched VCSEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The principle of operation of a quantum random number generator is  based on the generation of laser pulses with one of two orthogonal polarization states, followed by  digitization of polarization-resolved pulses with a comparator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' At intensity values of laser pulses  close to the threshold value of the comparator, the contribution of the classical noise of the  photodetector will have a crucial role in making a decision on the choice of a logical zero or one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  We show how to evaluate the contribution of classical noise and how to calculate the quantum  reduction factor required for post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Keywords: quantum random number generators, vertical surface emitting laser, quantum  noise extraction  Funding: This work was supported by Russian Science Foundation (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 17-71-20146).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Introduction  Random number generators (RNGs) play a primary role in modern cryptographic  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Due to the development of quantum cryptography, a special place among RNGs  occupy now quantum RNGs (QRNGs), which use various quantum sources of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Over the  past 15-20 years, a number of approaches have been proposed to obtain quantum randomness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  however, optical QRNGs, which employ laser noise, have gained the most popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Laser noise  can be associated with various effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', with temperature-related fluctuations of the laser cavity  length or with pump fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' However, at relatively high frequencies, laser noise is associated  mainly with spontaneous emission occurring due to zero-point oscillations of an electromagnetic  field, which have purely quantum nature and are generally considered to have the properties of  white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Due to this, laser noise can be employed as a high-frequency source of quantum  entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  The main difference between optical QRNGs based on laser noise lies in how the noise is  measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Thus, interference-based optical QRNGs use phase noise of laser radiation, which is  converted into amplitude fluctuations in the interferometer and then is readily measured with  conventional photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' In lasers that do not have fixed polarization of light, one may also  use fluctuations of the polarization state in addition to phase fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Such an approach can  be used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', in a vertical-cavity surface-emitting laser (VCSEL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' A VCSEL-based QRNG  employing spontaneous polarization switching was first described in [1], where the author  demonstrated the random bit generation rate up to 2 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Recently, we discussed a simple optical  scheme of a QRNG based on a gain-switched VCSEL, which allows generating the sequence of  random “on-off” pulses at several gigahertz [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Experimental demonstration of theoretical  calculations performed there have been published in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' In the present article, we consider the  problem of quantum noise extraction from polarization swapping in a gain-switched VCSEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' We  use the approach developed in [4], namely, we introduce for the QRNG under consideration the  quantum reduction factor containing information on the amount of classical noise “falling” into  the digitized random sequence due to fluctuations in the photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' We also describe how this  classical noise can be filtered out with the post-processing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Simulations  A simplified scheme of the proposed QRNG is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' A gain-switched VCSEL  is driven by a high-frequency laser driver, which is, in turn, controlled by the computer or FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Laser output is followed by the polarization filter (PF) that allows obtaining polarization-resolved  2  optical pulses, which are converted into the electrical signal via a broadband photodetector (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Random bits are obtained by digitizing pulses with a comparator, whose threshold voltage is  calculated in the FPGA, which also performs post-processing procedures including randomness  extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' (a) A simplified scheme of a VCSEL-based QRNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  (b) Pulses at the comparator input and the results of the digitization  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 1(b), we simulated the digitization process of polarization-resolved laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' It  was assumed in simulations that the polarizer in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 1(a) passes to the photodetector the x -linear  polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Laser pulses were simulated with VCSEL rate equations given in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' the calculated  signal was then processed with a low-pass digital filter (30 GHz bandpass) to simulate the finite  bandwidth of the photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The level of the comparator threshold (  th  V ) is shown by the dash-  dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 1(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' red circles correspond to the moments of the comparator latch actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  The result of digitization (‘0’-s or ‘1’-s) is shown in the corresponding frames (each time frame is  shown by the blue rectangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Generally, a laser pulse at the VCSEL output contains both polarization components ( x and  y ), such that polarization state of a given pulse can be referred to as “quasi-elliptical”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Relative  contribution of orthogonal components is a random quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' however, it depends on the width of  the pulse and the rate of relaxation processes (transients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' To demonstrate this, we calculated  probability density function (PDF) of the normalized integral signal  xS  at three different repetition  rates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' In the ideal case, we would get two peaks at the values  0  xS =  and  1  xS = , which  means that all optical power goes into one particular linear polarization ( y  and x respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  However, due to the finiteness of transients,  xS  could take intermediate values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  The influence of transients becomes more prominent when decreasing the pulse width, which is  clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 2(a), where the area under the PDF curve in the middle of the histogram is  increased when increasing the pulse repetition rate from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='5 to 7 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Polarization-resolved laser  pulses ( x-pulses in our case) that fall into this intermediate region are the most affected by classical  (non-quantum) noises of the photodetector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' therefore, ‘0’-s and ‘1’-s resulted from digitization of  such pulses can be considered as “untrusted” bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The proportion of these bits can be thought of  as a quantum reduction factor  , whose value determines how much the raw random sequence  should be “compressed” using the randomness extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' We propose the following formula to  find  :  )  (  1  1  H  P  (1)  0  0  0  Vmax  th  0  2  3  4  Time, ns3  where H\uf0a5  is the min-entropy of the raw random sequence, and P  is the probability to obtain the  pulse with the  x  S  value inside some “window” around the middle of the probability distribution,  whose width is proportional to the relative r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' value of the photodetector noise σ  (“measured”  in terms of the normalized value  x  S ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  We also calculated the dependence of  on the photodetector’s noise σ  at different pulse  repetition rates (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' One can see that the reduction factor  grows when increasing the  pulse repetition rate and begins to grow faster with increasing σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' It means that it does not make  much sense to increase the repetition rate of laser pulses if the photodetector is quite noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Probability densities of the normalized integral signal  x  S  (a) and dependences of the reduction  factor  on the photodetector’s noise σ  (b) at different pulse repetition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Post-processing  The digitized random sequence in the proposed scheme must be subjected to the randomness  extraction procedure with the reduction factor  , defined by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' We may, however, use a  deterministic extractor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', the von Neumann extractor [5], which extracts randomness regardless  the value of the reduction factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=" The von Neumann extractor discards repeated bits in a sequence  and replaces the two-bit words '01' and '10' with bits '0' and '1', respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Unfortunately, this  extractor reduces the length of a sequence by at least 4 times, which is not very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Therefore,  one generally uses instead a seeded extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' In cryptographic applications, an extractor with a  seed is generally implemented in the form of 2-universal hash functions, whose efficiency is  guaranteed by the leftover hash lemma [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' A common way to implement 2-universal hashing is  to multiply the input raw sequence by a random binary matrix [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Without loss of generality, one  may always use for these purposes random Boolean Toeplitz matrices, which allow significantly  saving the seed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' In our case, the randomness extractor is then divided into three steps:  1)  For a “raw” binary sequence of length n, determine the length of the output  sequence m by the formula: m  n  =  \uf047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  2)  Generate the Toeplitz matrix using the “seed” of length  1  m  n  +  −  bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  3)  Multiply the Toeplitz matrix by the raw sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' This yields the resulting random  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  It is important to discuss the method of obtaining the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' By default, it is assumed that the  seed is obtained from a strong source of entropy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', one that allows getting truly random bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' If  the RNG being developed is not a strong source of entropy, then an additional source of entropy  must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' We propose, however, the following algorithm to obtain the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' When switching-  on the QRNG, the system buffers a raw random sequence of a given (relatively small) length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Then, this sequence is subjected to a deterministic extractor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', the von Neumann extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The  random sequence obtained after the extractor can be now used as a seed in hashing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  0  2  4  6  (a)  5 GHz  7 GHz  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='5 GHz  PDF  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='5 GHz  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  5 GHz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='0  7 GHz  4  “Equipped” with such a procedure, the QRNG under consideration is an autonomous source of  entropy that does not need an additional entropy source, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', the device can operate even in the  absence of a pre-memorized random sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  One of the common ways to test the quality of randomness, is to perform statistical tests,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=', NIST tests [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Unfortunately, we did not have an access to real (obtained in the experiment)  random numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' however, we have a fairly detailed theoretical model, which may be used to  follow the whole route the laser noise “travels” from spontaneous emission to the sequence of  random bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' For this, we simulated 106 laser pulses similar to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' To  “digitize” them, the energy of each pulse (area under the pulse) was calculated and compared with  a certain threshold energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The obtained random bits were then grouped into k -bit words (we put  8  k =  ), which we denote as [ ]  x i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The sequence of [ ]  x i  were processed with a second-order FIR  filter according to the following formula: [ ]  mod( [ ] 2 [  1]  [  2],2 )  k  y i  x i  x i  x i  =  +  −  +  −  , where each [ ]  y i  is an i -th output word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The obtained data were then concatenate to a (“filtered”) random bit string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  After filtering, we processed the data with the randomness extractor described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' Finally, we  performed NIST tests with random bits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' the results of the test are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' As one  can see, the obtained random sequence successfully passed all the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' NIST statistical test result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Conclusion  We propose here an approach for quantum noise extraction from polarization swapping in a  gain-switched VCSEL and proposed a simple method to get a seed for hashing the raw random  sequence without an additional entropy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content=' The discussed algorithms allow developing a fully  autonomous QRNG with proven “quantumness” of generated random bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
+page_content='  Acknowledgments  Authors are grateful to Vladimir Meshkov for valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFLT4oBgHgl3EQfCC4D/content/2301.11973v1.pdf'}
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+Graphlets over Time: A New Lens for Temporal Network Analysis
+Deukryeol Yoon
+KAIST AI
+Seoul, South Korea
+deukryeol.yoon@kaist.ac.kr
+Dongjin Lee
+KAIST EE
+Daejeon, South Korea
+dongjin.lee@kaist.ac.kr
+Minyoung Choe
+KAIST AI
+Seoul, South Korea
+minyoung.choe@kaist.ac.kr
+Kijung Shin
+KAIST AI & EE
+Seoul, South Korea
+kijungs@kaist.ac.kr
+ABSTRACT
+Graphs are widely used for modeling various types of interactions,
+such as email communications and online discussions. Many of
+such real-world graphs are temporal, and specifically, they grow
+over time with new nodes and edges.
+Counting the instances of each graphlet (i.e., an induced sub-
+graph isomorphism class) has been successful in characterizing local
+structures of graphs, with many applications. While graphlets have
+been extended for temporal graphs, the extensions are designed
+for examining temporally-local subgraphs composed of edges with
+close arrival times, instead of long-term changes in local structures.
+In this paper, as a new lens for temporal graph analysis, we
+study the evolution of distributions of graphlet instances over time
+in real-world graphs at three different levels (graphs, nodes, and
+edges). At the graph level, we first discover that the evolution
+patterns are significantly different from those in random graphs.
+Then, we suggest a graphlet transition graph for measuring the
+similarity of the evolution patterns of graphs, and we find out a
+surprising similarity between the graphs from the same domain. At
+the node and edge levels, we demonstrate that the local structures
+around nodes and edges in their early stage provide a strong signal
+regarding their future importance. In particular, we significantly
+improve the predictability of the future importance of nodes and
+edges using the counts of the roles (a.k.a., orbits) that they take
+within graphlets.
+1
+INTRODUCTION
+Graphs are a simple yet powerful tool, and thus they have been
+used for representing various types of interactions: email commu-
+nications, online Q/As, research collaborations, to name a few. Due
+to newly formed interactions, such real-world graphs are temporal,
+i.e., they evolve over time with new nodes and edges. Many studies
+have examined the dynamics of real-world temporal graphs and re-
+vealed interesting patterns, including densification [26], shrinking
+diameter [26], and temporal locality in triangle formation [23].
+Graphlets have been widely employed for analyzing local struc-
+tures of graphs. Graphlets [36] are defined as the sets of isomorphic
+small subgraphs with a predefined number of nodes. Specifically,
+the relative counts of the instances of different graphlets effec-
+tively characterize the local structures of graphs, with successful
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+Conference’17, July 2017, Washington, DC, USA
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+Hep
+pTPh
+He 
+h 
+Patent 
+Enron -
+EU -
+College -
+Math -
+Ask-
+Stack -
+I 
+-
+1• 
+| 
+- 1.0
+l卜 0.5 
+- 0.0
+(a) Similarity between graphs w.r.t.
+graphlet transitions
+(classification accuracy = 97.2%)
+HepPh 
+HepTh 
+Patent 
+Enron -
+EU­
+College -
+Math -
+Ask­
+Stack -
+1.0 
+0.5 
+- 0.0
+(b) Similarity between graphs w.r.t.
+graphlet occurrences
+(classification accuracy = 83.3%)
+Figure 1: Real-world temporal graphs from the same do-
+main share similar evolution patterns captured by transi-
+tions between graphlets. The figures show the pairwise sim-
+ilarity between 9 graphs from 3 domains (distinguished by
+text colors) with respect to the transitions between graphlets
+(see (a)) and the occurrences of graphlets (see (b)). The do-
+mains of graphs can be classified more accurately in (a) than
+in (b). Specifically, with the best thresholds of similarity, the
+classification accuracy is 97.2% in (a) and 83.3% in (b). See
+Section 3.2 for details about the similarity measures.
+applications in graph classification [31, 32], community detection
+[5, 10, 39], anomaly detection [20], and node embedding [24, 28, 42].
+As temporal graphs are pervasive, the concept of graphlets has
+been generalized in a number of ways for temporal graph analysis.
+Temporal network motifs [21, 35] are sets of temporal subgraphs
+that are (a) identical not just topologically but also temporally, (b)
+composed of a fixed number of nodes, and (c) temporally local,
+i.e., composed of edges whose arrival times are close enough (see
+Section 6 for details). Due to the last condition, they are suitable
+for analyzing short-term changes of graphs but not for long-term
+changes in local structures, which are the focus of this paper.
+In this paper, we examine the long-term evolution of local struc-
+tures captured by graphlets, as a new lens for temporal graph analy-
+sis, in nine real-world temporal graphs from three different domains.
+Our analysis is at three levels: graphs, nodes, and edges.
+At the graph level, we first investigate the changes in the distribu-
+tions of graphlet instances over time. We find out that the evolution
+patterns are distinguished from those in randomized graphs that
+are obtained by randomly shuffling edges. Moreover, the evolution
+patterns in graphs from the same domain share some common
+characteristics. In order to compare the evolution patterns in a sys-
+tematic way, we introduce graphlet transition graphs, which encode
+transitions between graphlets due to changes in graphs. As shown
+in Figure 1(a), graphs from the same domain share similar graphlet-
+transition patterns, which facilitates accurate graph classification,
+although the sizes of the graphs vary.
+At the node and edge levels, we investigate how local structures
+around each node and edge in their early stage signal their future
+importance. Specifically, as local structures, we consider node roles
+arXiv:2301.00310v1  [cs.SI]  1 Jan 2023
+
+1
+2
+3
+4
+5
+6
+8
+7
+9
+13
+10
+11
+12
+(a) 13 graphlets
+1
+2
+3
+4
+5
+6
+8
+7
+9
+10
+11
+12
+13
+15
+14
+16
+17
+18
+19
+20
+23
+21
+22
+24
+25
+28
+26
+27
+30
+29
+(b) 30 node roles (also known as, node orbits)
+7
+19
+25
+1
+2
+3
+4
+5
+6
+8
+9
+10
+11
+12
+13
+15
+14
+16
+17
+18
+20
+23
+21
+22
+24
+28
+26
+27
+30
+29
+(c) 30 edge roles (also known as, edge orbits)
+Figure 2: (a) The 13 graphlets [36] with three nodes. (b) The 30 node roles [36] within the graphlets (see the positions of black
+nodes). (c) The 30 edge roles within the graphlets (see the positions of edges from a red node to a blue node).
+(formally, node automorphism orbits [36]) and edge roles [17], which
+are roughly sets of symmetric positions of nodes and edges within
+graphlets. We also demonstrate that the counts of the roles taken
+by each node and edge in their early stage are more informative
+than previously-used features [41], and they are complementary
+to simple global features (e.g., total counts of nodes and edges) for
+the task of predicting future centralities (specifically, in-degree,
+betweenness [15], closeness [9], and PageRank [34]).
+We summarize our contributions as follows:
+• Patterns: We make several interesting observations about the
+temporal evolution of graphlets: a surprising similarity in graphs
+from the same domain and local-structural signals regarding the
+future importance of nodes and edges.
+• Tool: We introduce graphlet transition graphs, which is an ef-
+fective tool for measuring the similarity of local dynamics in
+temporal graphs of different sizes.
+• Prediction: We enhance the prediction accuracy of the future
+importance of nodes and edges by introducing role-based local
+features, which are complementary to global features.
+Reproducibility: The code and the datasets are available at https:
+//github.com/deukryeol-yoon/graphlets-over-time.
+In Section 2, we introduce basic concepts, notations, and datasets.
+In Section 3, we present our graph-level analysis. In Section 4 and
+Section 5, we present our node-level and edge-level analyses. In
+Section 6, we present a brief survey of related works. In Section 7,
+we conclude our work.
+2
+BASIC CONCEPTS, NOTATIONS, AND DATA
+In this section, we first introduce some basic concepts and notations.
+Then, we describe the nine datasets used in this paper.
+2.1
+Basic Concepts and Notations
+Temporal Graph: A temporal graph G = (V, E, T) consists of a
+set of nodes V, a set of directed edges E := {𝑒1, · · · ,𝑒|E |}, and a
+multiset of edge arrival times T := [𝑡1, · · · ,𝑡|E |]. For each directed
+edge 𝑒𝑖 ∈ E, we use 𝑡𝑖 ∈ T to denote the arrival time of 𝑒𝑖. We use
+𝑢 → 𝑣 to denote a directed edge from a node 𝑢 to a node 𝑣, and the
+nodes 𝑢 and 𝑣 are adjacent if 𝑢 → 𝑣 or 𝑣 → 𝑢 exists. From now on,
+Table 1: Table of symbols.
+Notation
+Definition
+G = (V, E, T)
+temporal graph with nodes V, edges E, and times T
+G(𝑡) = (V (𝑡), E (𝑡))
+snapshot of G at time 𝑡
+˜G = (V, E, ˜T)
+a temporal graph randomized from G
+˜G(𝑡) = ( ˜
+V (𝑡), ˜E (𝑡))
+snapshot of ˜G at time 𝑡
+𝑚(𝑡)
+𝑖
+(𝑣)
+count of node role 𝑖 at a node 𝑣 in G(𝑡)
+we will use the term edge to indicate a directed edge when there is
+no ambiguity.
+Randomized Graph: A randomized graph ˜G = (V, E, ˜T) of G =
+(V, E, T) is obtained by assigning arrival times in T to edges in E
+uniformly at random in a one-to-one manner. For each edge 𝑒𝑖 ∈ E,
+we use ˜𝑡𝑖 ∈ ˜T to denote the arrival time assigned to it.
+Snapshot: We define the snapshot at time 𝑡 of G = (V, E, T)
+as G(𝑡) = (V (𝑡), E (𝑡)) where E (𝑡) := {𝑒𝑖 ∈ E : 𝑡𝑖 ≤ 𝑡} and
+V (𝑡) ⊆ V is the endpoints of any edge in E (𝑡). That is, G(𝑡)
+consists of the nodes and edges arriving at time𝑡 or earlier. Similarly,
+the snapshot at time 𝑡 of ˜G = (V, E, ˜T) is ˜G(𝑡) = ( ˜
+V (𝑡), ˜E (𝑡))
+where ˜E (𝑡) := {𝑒𝑖 ∈ E : ˜𝑡𝑖 ≤ 𝑡} and ˜
+V (𝑡) is the endpoints of
+any edge in ˜E (𝑡). We define the neighbors of a node 𝑣 ∈ V (𝑡) in
+a snapshot G(𝑡) as the nodes adjacent to 𝑣 in G(𝑡). We define the
+degree of a node 𝑣 ∈ V (𝑡) in a snapshot G(𝑡), which is denoted by
+𝑑 (𝑡) (𝑣), as the number of directed edges whose endpoints include
+𝑣 in G(𝑡). We simply use 𝑑(𝑣) to denote the degree of the node 𝑣 in
+the last snapshot G(𝑡|E|).
+Induced Subgraphs: A subgraph of a snapshot G(𝑡) = (V (𝑡), E (𝑡))
+is induced if and only if it consists of a subset of V (𝑡) and all of the
+edges connecting pairs of the nodes in the subset. Two subgraphs
+H and H ′ are isomorphic if there exists a one-to-one mapping 𝑓
+between the nodes of both graphs such that there exists an edge
+from a node 𝑢 to a node 𝑣 in H if and only if there exists an edge
+from the node 𝑓 (𝑢) to the node 𝑓 (𝑣) in H ′.
+Graphlets: A graphlet is the set of induced subgraphs that are
+isomorphic to each other. In this paper, we limit our attention to
+the 13 graphlets consisting of three connected nodes. An induced
+
+Table 2: Summary of nine real-world temporal graphs used
+throughout this paper.
+Domain
+Dataset
+|𝑉 |
+|𝐸𝑇 |
+Period
+Citation
+HepPh
+34, 565
+346, 849
+9 years
+HepTh
+18, 477
+136, 190
+10 years
+Patent
+3, 774, 362
+16, 512, 782
+25 years
+Email/Message
+Enron
+55, 655
+209, 203
+24 years
+EU
+986
+24, 929
+1.5 years
+College
+1, 899
+20, 296
+0.5 years
+Online Q/A
+Askubuntu
+159, 316
+262, 106
+6 years
+Mathoverflow
+24, 818
+90, 489
+7 years
+Stackoverflow
+2, 601, 977
+16, 266, 395
+8 years
+subgraph is called an instance of graphlet 𝑘 if it is isomorphic to
+the 𝑘-th graph in Figure 2(a).
+Node Roles: Consider an induced subgraph H with a node set V′.
+An automorphism of H is an isomorphism between H and itself.
+i.e., an automorphism of H is a one-to-one mapping between nodes
+of H such that there exists an edge from a node 𝑢 to a node 𝑣 in
+H if and only if there exists an edge from the node corresponding
+to 𝑢 to the node corresponding to 𝑣 in H. If denoting the set of
+automorphisms of H by 𝐴𝑢𝑡(H), the automorphism orbit of a node
+𝑢 ∈ V′ is the set {𝑦 ∈ V′ : ∃𝑔 ∈ 𝐴𝑢𝑡(H) s.t. 𝑦 = 𝑔(𝑢)} of
+nodes [36]. Formally, node roles are node automorphism orbits,
+and roughly, they are sets of symmetric positions of nodes within
+graphlets. Figure 2(b) (see the positions of black nodes) shows
+all 30 node roles in the 13 graphlets that we consider. We say a
+node 𝑣 “takes” node role 𝑖 in a graphlet instance if there exists
+an isomorphism of the graphlet instance and the 𝑖-th graph in
+Figure 2(b) that maps 𝑣 to the black node in the graph. We define the
+count of node role 𝑖 at a node 𝑣 as the number of graphlet instances
+where 𝑣 takes 𝑖, and 𝑚(𝑡)
+𝑖
+(𝑣) denotes the count at a snapshot G(𝑡).
+Edge Roles: Consider an induced subgraph H with an edge set
+E′. Based on the concepts defined above, we define the edge role
+of an edge 𝑢 → 𝑣 is the set {𝑥 → 𝑦 ∈ E′ : ∃𝑔 ∈ 𝐴𝑢𝑡(H) s.t. 𝑥 =
+𝑔(𝑢) ∧ 𝑦 = 𝑔(𝑣)} of edges. Roughly, edge roles are the sets of
+symmetric positions of edges within graphlets. Figure 2(c) (see the
+positions of edges from a red node to a blue node) shows all 30 edge
+roles in the 13 considered graphlets. We say an edge 𝑢 → 𝑣 “takes”
+edge role 𝑗 in a graphlet instance if there exists an isomorphism of
+the graphlet instance and the 𝑗-th graph in Figure 2(c) that maps 𝑢
+and 𝑣 to the red node and the blue node, respectively, in the graph.
+We define the count of edge role 𝑗 at an edge 𝑒 as the number of
+graphlet instances where 𝑒 takes 𝑗.
+2.2
+Datasets
+Throughout this paper, we use the nine real-world temporal graphs
+from the three domains, which are summarized in Table 2.
+Citation Graphs: Each node is a paper or a patent. Each directed
+edge from a node 𝑢 to a node 𝑣 means that 𝑢 cites 𝑣.
+Email/Message Graphs: Each node is a user. Each directed edge
+from a node 𝑢 to a node 𝑣 indicates that 𝑢 sends 𝑣 emails (messages).
+Online Q/A Graphs: Each node is a user. Each directed edge from
+a node 𝑢 to a node 𝑣 means that 𝑢 answers 𝑣’s questions.
+Algorithm 1: Counting the Instances of Each Graphlet in
+a Temporal Graph
+Input
+:Temporal Graph G = (V, E, T)
+Output:The count of the instances of each graphlet in G
+1 Initialize the count of the instances of each graphlet to zero
+2 Initialize E to an empty set
+3 for each edge 𝑒𝑖 = 𝑢 → 𝑣 in arrival order do
+4
+N ← union of the neighbors of 𝑢 and the neighbors of 𝑣
+(except for 𝑢 and 𝑣)
+5
+for each 𝑤 ∈ N do
+6
+if 𝑢, 𝑣 and 𝑤 form a graphlet instance then
+7
+decrement the count of the graphlet of the instance
+formed by 𝑢, 𝑣 and 𝑤
+8
+add 𝑢 → 𝑣 to E
+9
+for each 𝑤 ∈ N do
+10
+increment the count of the graphlet of the instance formed
+by 𝑢, 𝑣 and 𝑤
+11 return count of the instances of each graphlet instances
+3
+GRAPH LEVEL ANALYSIS
+In this section, we study the evolution of local structures in real-
+world graphs. We examine the dynamics in the distribution of
+graphlet instances and transitions between graphlets.
+3.1
+Global Level 1. Graphlets Over Time
+We track how the ratio of the instances of each graphlet changes as
+the considered real-world graphs evolve over time. Our tracking al-
+gorithm, which is described in Algorithm 1, is adapted from StreaM
+[38], which maintains the counts of the instances of the 4-node
+undirected graphlets in a fully dynamic graph stream, where edges
+are not just added but also deleted over time. The time complex-
+ity of Algorithm 1 is Θ(Σ𝑣∈V (𝑑(𝑣))2), as proven in Theorem 1. It
+should be noticed that, by Lemma 1, the time complexity is Θ(the
+number of instances of all graphlets in the last snapshot), which
+is the optimal time complexity achievable by any algorithm that
+counts graphlet instances by enumerating them.
+Theorem 1. The time complexity of Algorithm 1 is Θ(Σ𝑣∈V (𝑑(𝑣))2).
+Proof. Since the number of nodes forming each graphlet in-
+stance is a constant, finding the graphlet corresponding to a given
+instance and updating the corresponding count (lines 6-7 and 10)
+take 𝑂(1) time. Thus, the time complexity of processing each in-
+coming edge 𝑒𝑖 = 𝑢 → 𝑣 is that of computing the union of the neigh-
+bors of 𝑢 and 𝑣 (line 4), which is Θ(𝑑 (𝑡𝑖−1) (𝑢) +𝑑 (𝑡𝑖−1) (𝑣)). Hence,
+the total complexity is Θ(�
+𝑒𝑖=𝑢→𝑣∈𝐸 (𝑑 (𝑡𝑖−1) (𝑢) + 𝑑 (𝑡𝑖−1) (𝑣)) =
+Θ(�
+𝑣∈V (𝑑(𝑣))2).
+□
+Lemma 1. The number of instances of all graphlets in a snapshot
+G(𝑡) is Θ(Σ𝑣∈V (𝑡) (𝑑 (𝑡) (𝑣))2).
+Proof. Given a snapshot G(𝑡) = (V (𝑡), E (𝑡)), for each node
+𝑣 ∈ V (𝑡), if we count the instances of all graphlets that consist
+of 𝑣 and its two neighbors, then the count of such instances is
+Θ((𝑑 (𝑡) (𝑣))2) for each node 𝑣, and since 𝑑 (𝑡) (𝑣) ≥ 1 for every
+node 𝑣, the total count 𝐶 is Θ(Σ𝑣∈V (𝑡) (𝑑 (𝑡) (𝑣))2).
+
+Table 3: Ratios of graphlets over time. The colors in the plots are matched with the colors of the graphlets in Figure 2, and the
+evolution ratio means the fraction of edges added to graphs. The evolution patterns in real-world graphs vary depending on
+domains (Observation 1), and they are clearly distinguished from the evolution patterns in randomized graphs (Observation 2).
+Temporal graph G
+Randomized graph ˜G
+Citation
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+HepPh
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+HepTh
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Patent
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+HepPh
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+HepTh
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Patent
+Email/Message
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+EU
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Enron
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+College
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+EU
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Enron
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+College
+Online Q/A
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Math
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Ask
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Stack
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Math
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Ask
+0.0
+0.5
+1.0
+Evolution Ratio
+0.0
+0.5
+1.0
+Graphlet Ratio
+Stack
+Lower Bound: Since each graphlet instance, which consists of
+three nodes, is counted at most three times, 𝐶 is at most three times
+the number of instances of all graphlets in G(𝑡). In other words,
+the number of instances of all graphlets is at least 1/3 of 𝐶, and
+thus it is Ω(Σ𝑣∈V (𝑡) (𝑑 (𝑡) (𝑣))2).
+Upper Bound: In each graphlet instance, there exists at least one
+center node, who composes the graphlet together with its neighbors.
+Thus, each instance is counted at least once, and thus 𝐶 is at least
+the number of instances of all graphlets in G(𝑡). In other words,
+the number of instances of all graphlets is at most 𝐶, and thus it is
+𝑂(Σ𝑣∈V (��) (𝑑 (𝑡) (𝑣))2).
+□
+As seen in Table 3, the dynamics of the ratios depend on the
+domains of the graphs, as summarized in Observation 1.
+Observation 1. The dynamics in the distributions of graphlet in-
+stances in graphs from the same domain share some commonalities.
+• Instances of graphlet 4 are more dominant in the citation graphs
+than other graphs.
+• Graphlets with many edges (e.g., graphlets 8, 12, and 13) account for
+a larger fraction in email/message networks than in other networks.
+• The fraction of graphlet 1 increases over time only in the online
+Q/A graphs.
+However, the dynamics are not exactly the same within domains.
+For example, while graphlets 1, 2, and 4 are dominant compared to
+other graphlets in all citation graphs, the ratios among them vary
+greatly in different graphs.
+We also notice a consistent difference between the dynamics in
+real-world graphs and those in randomized graphs (see Section 2.1),
+as summarized in Observation 2.
+Observation 2. The ratios of graphlet instances change more lin-
+early in randomized graphs than in real-world graphs.
+Table 4: The non-linearity of the ratios of graphlet instances
+over time in real-world graphs and randomized graphs. We
+describe in Section 3.1 how the non-linearity is measured.
+The lower the non-linearity is, the more linear the change
+of the ratio of the corresponding graphlet instances is. Note
+that the ratios of graphlet instances change more linearly in
+randomized graphs than in real-world graphs.
+Dataset
+HepPh
+HepTh
+Patent
+EU
+Enron
+College
+Math
+Ask
+Stack
+real
+0.0027
+0.0080
+0.0093
+0.0107
+0.0042
+0.0095
+0.0028
+0.0038
+0.0047
+random
+0.0003
+0.0011
+0.0000
+0.0081
+0.0017
+0.0058
+0.0007
+0.0005
+0.0001
+In order to numerically support this observation, we measure the
+non-linearity [18, 22] of the ratios of graphlet instances over time.
+Specifically, we fit a linear regression model and a non-linear poly-
+nomial regression model to each time series in Table 3, and then
+we measure the average absolute difference between the predicted
+values of the two models as the non-linearity of the time series.1
+Lastly, we average the non-linearity of all time-series from each
+graph and report the results in Table 4. Note that non-linearity is
+significantly higher in real-world graphs than in corresponding
+randomized graphs. That is, the ratios of graphlet instances change
+more linearly in randomized graphs than in real-world graphs.
+3.2
+Global Level 2. Graphlet Transitions
+In a temporal graph, an instance of a graphlet may transition to
+an instance of another graphlet due to new edges added to it. In
+this subsection, we examine the counts of such transitions between
+graphlets to characterize the local dynamics in temporal graphs
+and also to make comparisons between them.
+1We use the linearity test implemented in Analyse-it (Ver. 5.65) and select a cubic
+model as the non-linearity polynomial model, as suggested in the program. For com-
+putational efficiency, we measure the absolute difference at 1,000 evolution ratios
+sampled uniformly at equal intervals.
+
+Table 5: Using graphlet transition graphs (GTGs) and characteristic profiles (CPs) from GTGs, we can accurately characterize
+the dynamics of local structures in real-world graphs. The colors of edges in GTGs indicate their normalized weights. Note
+that GTGs and CPs are particularly similar in real-world graphs from the same domains (Observation 3).
+Graphlet transition graphs (GTGs)
+Characteristic profiles (CPs)
+Citation
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+HepPh
+0.0
+0.3
+0.6
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+HepTh
+0.0
+0.3
+0.6
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+Patent
+0.0
+0.3
+0.6
+HepTh
+HepPh
+Patent
+Email/Message
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+EU
+0.0
+0.3
+0.6
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+Enron
+0.0
+0.3
+0.6
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+College
+0.0
+0.3
+0.6
+Enron
+EU
+College
+Online Q/A
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+Math
+0.0
+0.3
+0.6
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+Ask
+0.0
+0.3
+0.6
+3
+10
+2
+9
+7
+5
+8
+12
+4
+new
+1
+11
+6
+13
+Stack
+0.0
+0.3
+0.6
+Ask
+Math
+Stack
+Graphlet Transition Graph: We define graphlet transition graphs
+(GTGs) to encode transitions between graphlets.
+Definition 1 (Graphlet transition graph). A graphlet transi-
+tion graph (GTG) 𝐺 = (𝑉, 𝐸,𝑊 ) of a temporal graph G is a static
+directed weighted graph where the nodes are graphlets and each
+edge indicates that the source graphlet is transformed into the des-
+tination graphlet by an edge added to G. The weight of edges is
+the number of occurrences of the corresponding transitions. We use
+𝑊 = {𝑤1, · · · ,𝑤 |𝐸 |} to denote the edge weights.
+Since we focus on the 13 graphlets in Figure 2(a), a GTG consists
+of the 28 types of transitions between these graphlets. In Table 5,
+we visualize the GTGs from the real-world graphs. Algorithm 2 de-
+scribes the computation of the edge weights of a GTG. In a nutshell,
+for each edge in arrival order, we count the transitions caused by it.
+Its time complexity is formalized in Theorem 2.
+Theorem 2. The time complexity of Algorithm 2 is Θ(Σ𝑣∈V (𝑑(𝑣))2)
+= Θ(the number of instances of all graphlets in the last snapshot).
+Proof. We can prove the complexity of Θ(Σ𝑣∈V (𝑑(𝑣))2) simi-
+larly to Theorem 1, and by Lemma 1, it is Θ(the number of instances
+of all graphlets in the last snapshot).
+□
+Characteristic Profile (CP): We characterize the evolution of lo-
+cal structure in a graph G using the significance of edge weights in
+its GTG 𝐺 = (𝑉, 𝐸,𝑊 ). In order to measure the significance, we fol-
+low the steps in [31] for measuring the significance of each graphlet
+itself. To this end, we construct the graphlet transition graph ˜𝐺 of
+a randomized graph ˜G. Then, we measure the significance 𝑆𝑃𝑖 of
+each edge weight 𝑤𝑖 in 𝐺 as follows:
+𝑆𝑃𝑖 :=
+𝑤𝑖 − ˜𝑤𝑖
+𝑤𝑖 + ˜𝑤𝑖 + 𝜖 ,
+(1)
+Algorithm 2: Computing the Edge Weights of Graphlet
+Transition Graphs
+Input
+:Temporal graph G = (V, E, T)
+Output:Edge weights of the graphlet transition graph of G
+1 Initialize all edge weights to zero
+2 Initialize E to an empty set
+3 for each edge 𝑒𝑖 = 𝑢 → 𝑣 in arrival order do
+4
+for each 𝑤1 ∈ neighbors(𝑢) \ {𝑣} do
+5
+UPDATE(𝑢, 𝑣, 𝑤1)
+6
+for each 𝑤2 ∈ neighbors(𝑣) \{neighbors(𝑢) ∪ 𝑢} do
+7
+UPDATE(𝑢, 𝑣, 𝑤2)
+8
+add 𝑢 → 𝑣 to E
+9 return the edge weights
+10 Procedure UPDATE(𝑢, 𝑣, 𝑤)
+11
+if 𝑢, 𝑣, and 𝑤 form a graphlet instance then
+12
+prev ← graphlet of the instance (𝑢, 𝑣, 𝑤) without 𝑢 → 𝑣
+13
+next ← graphlet of the instance (𝑢, 𝑣, 𝑤) with 𝑢 → 𝑣
+14
+𝑖 ← index of the graphlet transition from prev to next
+15
+increase the weight of the edge 𝑖 (i.e., 𝑤𝑖) by 1
+where ˜𝑤𝑖 is the corresponding edge weight in ˜𝐺, and 𝜖 is a constant,
+which we fix to 4. For ˜𝑤𝑖, we generate 50 instances of randomized
+graphs and we use the average edge weights in them. Lastly, we
+normalize each significance as follows:
+𝐶𝑃𝑖 := 𝑆𝑃𝑖/
+√︃
+Σ|𝐸 |
+𝑖=1𝑆𝑃2
+𝑖 .
+(2)
+We characterize the evolution of local structures in G using the vec-
+tor of the normalized significances (i.e., [𝐶𝑃1, · · · ,𝐶𝑃|𝐸 |]), which
+we call characteristic profile (CP).
+Comparison between CPs: We plot the CPs of the considered
+real-world graphs in Table 5, and high levels of similarity are ob-
+served within domains. We numerically measure the similarity
+
+Vormalized
+0
+5
+10
+15
+20
+25
+GraphletTransitionIndexNormalized
+gnificance
+0
+5
+10
+15
+20
+25
+GraphletTransitionIndexNormalized
+0
+5
+10
+15
+20
+25
+GraphletTransitionIndexHigh
+Low
+Low
+High
+{2, 4}
+(a) 𝑑𝜃 = 2
+High
+Low
+Low
+High
+{2, 4, 7, 9}
+(b) 𝑑𝜃 = 4
+High
+Low
+Low
+High
+{2, 4, 7, 9, 11, 12}
+(c) 𝑑𝜃 = 8
+Figure 3: Example signals from the local structures of nodes
+regarding their future importance. The ratios of some node
+roles (e.g., node roles 2 and 4) at nodes monotonically in-
+crease with respect to the future in-degrees of the nodes.
+The ratios are rescaled so that their maximum values are
+the same.
+between CPs using the Pearson correlation coefficients, and the
+results are shown in Figure 1(a). The correlation coefficients are
+particularly high between graphs from the same domain, and specif-
+ically the domains can be classified with 97.2% accuracy if we use
+the best threshold of the correlation coefficient (0.58). The results
+demonstrate that CPs accurately characterize the evolution of local
+structures. Our observations are summarized in Observation 3.
+Observation 3. The evolution patterns of local structures are simi-
+lar in real-world graphs from the same domains.
+Comparison with Other Methods: We evaluate three other graph
+characterization methods, as we evaluate ours in the right above
+paragraph. In Figure 1(b), we provide the correlation coefficients
+between the CPs obtained from the count of the instances of each
+graphlet [31]. Note that the email/message graphs (blue) and the
+online Q/A graphs (green) are not distinguished clearly. Numeri-
+cally, with the best threshold of correlation coefficient (0.95), the
+classification accuracy is 83.3%.
+We also compute the similarity between the considered real-
+world graphs using Graphlet-orbit Transition (GoT) [4] and Orbit
+Temporal Agreement (OTA) [4], which are also based on transi-
+tions between graphlets (see Section 6 for details). Our way of
+characterization has the following major advantages over them:
+• (1) Speed: Empirically, GoT and OTA are up to 10× slower than
+our method, as shown in Appendix A. The time complexity of
+them is proportional to the sum of the counts of graphlet in-
+stances in all used snapshots, while the time complexity of Algo-
+rithm 2 is proportional only the to the count of graphlet instances
+in the last snapshot (Theorem 2).
+• (2) Space Efficiency: GoT and OTA run out of memory in the
+two largest graphs (Patent and Stackoverflow), as shown in Ap-
+pendix A, while our method does not. They need to store all
+graphlet instances in each considered snapshot for comparison
+with those in the next snapshot, while Algorithm 2 maintains only
+the latest snapshot without having to store graphlet instances.
+• (3) Characterization Accuracy: The best classification accura-
+cies computed using the considered real-world graphs (except
+for Patent and Stackoverflow for which GoT and OTA run out of
+memory) are 81.0% (GoT) and 85.7% (OTA), which is lower than
+our classification accuracy (97.2%). Detailed results are given in
+Appendix A. Note that GoT and OTA approximate the counts
+of transitions between graphlets based on a small number of
+snapshots, while Algorithm 2 exactly counts the transitions.
+Table 6: The absolute value of the Spearman’s rank correla-
+tion coefficients between node role ratios and future central-
+ities (averaged over all node roles and all datasets for each
+centrality measure) and each value of the threshold 𝑑𝜃. As
+the number of node neighbors increases (i.e., 𝑑𝜃 increases),
+the local-structural signals about future centralities become
+stronger (i.e., the absolute values increase).
+𝑑𝜃
+Degree
+Betweenness
+Closeness
+PageRank
+Edge Betweenness
+2
+0.640
+0.697
+0.682
+0.663
+0.546
+4
+0.721
+0.723
+0.712
+0.704
+0.558
+8
+0.816
+0.793
+0.759
+0.701
+0.599
+In summary, our way of characterizing temporal graphs
+using GTGs distinguishes the domains of temporal graphs
+most accurately with the accuracy of 97.2%. The accuracies of
+the other methods are 83.3%, 81.0%, and 85.7%.
+4
+NODE LEVEL ANALYSIS
+In this section, we study how local structures around nodes are re-
+lated to their future importance. Then, we enhance the predictability
+of future node centrality using the relations.
+4.1
+Patterns
+We characterize the local structures of nodes using node roles and
+examine their relation to the nodes’ future centrality.
+Local Structures of Nodes: Given a temporal graph G, we char-
+acterize the local structure of each node 𝑣 in their early stage by
+measuring the ratio of each node role at 𝑣 in the snapshot at time 𝑡
+when the in-degree of 𝑣 first reaches a threshold 𝑑𝜃. That is, each
+node 𝑣 is represented as a 30-dimensional vector whose 𝑖-th is
+𝑚(𝑡)
+𝑖
+(𝑣)/(�30
+𝑗=1 𝑚(𝑡)
+𝑗
+(𝑣)) (see Section 2.1 for 𝑚(𝑡)
+𝑖
+(𝑣)).
+Future Importance of Nodes: Given a temporal graph G, as fu-
+ture importance of each node, we measure its in-degree, node
+betweenness centrality [15], closeness centrality [9], and PageR-
+ank [34] in the last snapshot of G. Based on each centrality measure,
+we divide the nodes in G into six groups (Group 1: top 50-100%,
+Group 2: top 30-50%, Group 3: top 10-30%, Group 4: top 5-10%,
+Group 5: top 1-5%, and Group 6: top 0-1%).
+Finding Signals: For each group, we average the ratio vectors
+of the nodes in the group. Figure 3 shows some averaged ratios
+when in-degree is used as the centrality measure. Note that the
+ratios of node roles 2 and 4 monotonically grow as future centrality
+increases, regardless of 𝑑𝜃 values. That is, the ratios of node roles 2
+and 4 give a consistent signal regarding the nodes’ future in-degree.
+In Figure 4, we report the Spearman’s rank correlation coeffi-
+cient [43] between each averaged ratio and the future centralities
+of nodes (specifically, the above group numbers between 1 and
+6). We also report in Table 6 the absolute value of the coefficients
+(averaged over all node roles and all datasets) for each centrality
+measure and each value of the threshold 𝑑𝜃. Note that the average
+values are significantly greater than 0 and specifically around 0.7;
+and they increase as 𝑑𝜃 increases, as summarized in Observation 4.
+Observation 4. In real-world graphs, the local structures of nodes
+in their early stage provide a signal regarding their future impor-
+tance. The signals become stronger as nodes have more neighbors.
+
+Node Role
+Relative
+Ratio
+R
+12345
+6
+Centrality
+(Binned)Node Role
+Relative
+Ratio
+R
+12345
+6
+Centrality
+(Binned)Node Role
+Relative
+Ratio
+1
+2
+34
+5
+Centrality
+(Binned)5
+10
+15
+20
+25
+30
+Node Role
+HepPh
+HepTh
+Enron
+EU
+College
+Math
+Ask
+In-degree
+1.0
+0.5
+0.0
+0.5
+1.0
+5
+10
+15
+20
+25
+30
+Node Role
+HepPh
+HepTh
+Enron
+EU
+College
+Math
+Ask
+Betweenness
+1.0
+0.5
+0.0
+0.5
+1.0
+5
+10
+15
+20
+25
+30
+Node Role
+HepPh
+HepTh
+Enron
+EU
+College
+Math
+Ask
+Closeness
+1.0
+0.5
+0.0
+0.5
+1.0
+5
+10
+15
+20
+25
+30
+Node Role
+HepPh
+HepTh
+Enron
+EU
+College
+Math
+Ask
+Pagerank
+1.0
+0.5
+0.0
+0.5
+1.0
+Figure 4: The Spearman’s rank correlation coefficient between node role ratios (when nodes have in-degree four, i.e., 𝑑𝜃 = 4)
+and future node centralities. The darker a cell is, the larger the absolute value of the corresponding coefficient is. Note that
+the absolute values of most coefficients are significantly greater than 0.
+4.2
+Prediction
+Based on the observations above, we predict the future centrality
+of nodes using the counts of their roles at them in their early stage.
+Problem Formulation: We formulate the prediction problem as
+a classification problem, as described in Problem 1.
+Problem 1 (Node Centrality Prediction).
+• Given: the snapshot G(𝑡𝑣,𝑑𝜃 ) of the input graph when the in-
+degree of a node 𝑣 first reaches 𝑑𝜃,
+• Predict: whether the centrality of the node 𝑣 belongs to the top
+20% in the last snapshot of G.
+As the centrality measure, we use in-degree, betweenness centrality,
+closeness centrality, and PageRank. As 𝑑𝜃, we use 2, 4, or 8.
+Input Features: For each node 𝑣, we consider the snapshot G(𝑡)
+of the input graph G when the in-degree of 𝑣 first reaches 𝑑𝜃. That
+is, 𝑡 = 𝑡𝑣,𝑑𝜃 and G(𝑡) = G(𝑡𝑣,𝑑𝜃 ). Then, we extract the following
+sets of input features for 𝑣:
+• Local-NR: The count of each node role at 𝑣 in G(𝑡). That is,
+[𝑚(𝑡)
+1 (𝑣), 𝑚(𝑡)
+2 (𝑣), · · · ,𝑚(𝑡)
+30 (𝑣)] (see Section 2.1 for 𝑚(𝑡)
+𝑖
+(𝑣)).
+• Local-NPP [41]: In G(𝑡), we compute (1) the count of triangles
+at 𝑣, (2) the count of wedges centered at 𝑣, (3) the count of wedges
+ended at 𝑣.
+• Global-Basic: Counts of nodes and edges in the snapshot.
+• Global-NR: We compute the 30-dimensional vector whose 𝑖-th
+entry 𝑚(𝑡)
+𝑖
+(𝑣)/(�30
+𝑗=1 𝑚(𝑡)
+𝑗
+(𝑣)) is the ratio of each node role at 𝑣
+in G(𝑡). Then, we standardize (i.e., compute the 𝑧-score of) the
+role ratio vector using the mean and standard deviation from the
+role ratio vectors (in G(𝑡)) of all nodes with degree 𝑑𝜃 in G(𝑡).
+The features in Local-NR are also included.
+• Global-NPP [41]: In G(𝑡), we compute (1) the number of edges
+not incident to 𝑣 and (2) the number of non-adjacent node pairs
+where one is a neighbor of 𝑣 and the other is neither a neighbor
+of 𝑣 nor 𝑣 itself. The features in Local-NPP are also included.
+• ALL: All of Global-NR, Global-NPP, and Global-Basic.
+Note that we categorize the above sets into global and local
+depending on whether global information in G(𝑡) (i.e., the number
+of all nodes in G(𝑡)) is used or only local information at 𝑣 is used.
+Prediction Method: As the classifier, we use the random forest
+model from the Scikit-learn library. The model has 30 decision trees
+with a maximum depth of 10.
+Table 7: F1-score, accuracy, and AUROC on the task of pre-
+dicting future node importance when 𝑑𝜃 = 2 averaged over
+the 7 considered real-world graphs. Among local features,
+using Local-NR yields better performance than using Local-
+NPP in all settings. Using ALL leads to the best performance
+in most cases, indicating that the considered sets of features
+are complementary to each other. Detailed results on each
+dataset can be found in Appendix C.
+Target
+Degree
+Betweenness
+Measure
+F1-score
+Accuracy
+AUROC
+F1-score
+Accuracy
+AUROC
+Local-NR
+0.39
+0.69
+0.68
+0.59
+0.83
+0.82
+Local-NPP
+0.38
+0.68
+0.64
+0.58
+0.81
+0.79
+Global-NR
+0.57
+0.74
+0.78
+0.64
+0.84
+0.85
+Global-NPP
+0.57
+0.73
+0.77
+0.64
+0.84
+0.85
+Global-Basic
+0.50
+0.72
+0.73
+0.24
+0.73
+0.67
+ALL
+0.57
+0.74
+0.78
+0.65
+0.85
+0.86
+Target
+Closeness
+PageRank
+Measure
+F1-score
+Accuracy
+AUROC
+F1-score
+Accuracy
+AUROC
+Local-NR
+0.51
+0.76
+0.78
+0.42
+0.73
+0.73
+Local-NPP
+0.43
+0.70
+0.69
+0.37
+0.69
+0.67
+Global-NR
+0.68
+0.82
+0.87
+0.54
+0.75
+0.79
+Global-NPP
+0.66
+0.80
+0.85
+0.54
+0.74
+0.78
+Global-Basic
+0.59
+0.75
+0.79
+0.47
+0.71
+0.74
+ALL
+0.69
+0.83
+0.88
+0.56
+0.75
+0.79
+Evaluation Method: We use 80% of the nodes for training and
+the remaining 20% for testing. We evaluate the predictive perfor-
+mance in terms of F1-score, accuracy, and Area Under the ROC curve
+(AUROC). A higher value indicates better prediction performance.
+Result: Table 7 shows the predictive performance from each set
+of input features when 𝑑𝜃 = 2, and Table 8 shows how the per-
+formance depends on the in-degree threshold 𝑑𝜃. In the tables, we
+report the mean of each prediction performance over 10 runs in
+the 7 datasets in Section 2.2 except for the two largest ones (i.e.,
+Patent and Stackoverflow). From the results, we draw the following
+observations.
+Observation 5. Among local features, the counts of node roles at
+each node (Local-NR) are more informative than (Local-NPP) for
+future importance prediction.
+Observation 6. The considered sets of features are complementary
+to each other. Using them all (ALL) leads to the best predictive
+performance in most cases.
+Observation 7. As nodes have more neighbors, their future impor-
+tance can be predicted more accurately.
+
+Table 8:
+Average F1-score, accuracy, and AUROC on the
+task of predicting future node importance depending on 𝑑𝜃
+(i.e., in-degree of nodes when their input features are ex-
+tracted). The overall performance improves with respect to
+𝑑𝜃 in most cases. That is, as nodes have more neighbors,
+their future importance can be predicted more accurately.
+Detailed results on each dataset can be found in Appendix C.
+Target
+Degree
+Betweenness
+Measure
+F1-score
+Accuracy
+AUROC
+F1-score
+Accuracy
+AUROC
+ALL (𝑑𝜃 = 2)
+0.59
+0.74
+0.78
+0.65
+0.85
+0.86
+ALL (𝑑𝜃 = 4)
+0.69
+0.78
+0.79
+0.73
+0.83
+0.87
+ALL (𝑑𝜃 = 8)
+0.80
+0.81
+0.86
+0.82
+0.85
+0.90
+Target
+Closeness
+PageRank
+Measure
+F1-score
+Accuracy
+AUROC
+F1-score
+Accuracy
+AUROC
+ALL (𝑑𝜃 = 2)
+0.69
+0.83
+0.88
+0.55
+0.75
+0.79
+ALL (𝑑𝜃 = 4)
+0.78
+0.83
+0.89
+0.73
+0.77
+0.80
+ALL (𝑑𝜃 = 8)
+0.86
+0.88
+0.92
+0.85
+0.85
+0.83
+Figure 5: The Spearman’s rank correlation coefficient be-
+tween edge role ratios (when endpoints have in-degree 4 in
+total, i.e., 𝑑𝜃 = 4) and future edge centralities. The darker a
+cell is, the larger the absolute value of the corresponding
+coefficient is. Note that the absolute values of many coeffi-
+cients are significantly greater than 0, while they tend to be
+smaller than those in Figure 4.
+Feature Importance: Additionally, we measure the importance
+of each feature in the set ALL using Gini-importance [29], and we
+report the top five important features in Table 10 in Appendix B.
+Observation 8. Strong predictors vary depending on centrality
+measures. For example, for betweenness centrality, the counts of node
+roles as bridges (i.e., Local NR-4 and Global NR-4) are strong.
+5
+EDGE LEVEL ANALYSIS
+In this section, we investigate the signal of local structures of each
+edge regarding their future centrality, and based on the signal, we
+predict the future importance of edges.
+We generally follow the procedures in Section 4, except for the
+following differences: (a) we examine the ratios of edge roles at each
+edge 𝑢 → 𝑣 when the sum of the in-degrees of 𝑢 and 𝑣 becomes
+𝑑𝜃, (b) we use edge betweenness centrality [15] as the importance
+measure, (c) we formulate the problem of predicting future edge
+importance as described in Problem 2, (d) we extract feature sets
+Local-ER and Global-ER using the (relative) counts of edge roles
+at edges as we extract Local-NR and Global-NR, and (e) we union
+Global-ER and Global-Basic for ALL.
+Problem 2 (Edge Centrality Prediction).
+• Given: the snapshot G(𝑡𝑒,𝑑𝜃 ) of the input graph when the sum of
+the in-degrees of the endpoints of each edge first reaches 𝑑𝜃,
+• Predict: whether the centrality of each edge belongs to the top
+20% in the last snapshot of G.
+From Figure 5, Table 6, and Table 9, we draw the following
+observations.
+Table 9: F1-score, accuracy, and AUROC on the task of pre-
+dicting future edge importance averaged over the 7 con-
+sidered real-world graphs. Using edge role-based features
+(Local-ER and Global-ER) yields better performance than us-
+ing Global-Basic in most settings. The overall performance
+improves with respect to 𝑑𝜃. That is, as edges are better con-
+nected, their future importance is predicted more accurately.
+Detailed results on each dataset can be found in Appendix D
+Target
+Edge betweenness
+Measure
+F1-Score
+Accuracy
+AUROC
+Local-ER (𝑑𝜃 = 2)
+0.45
+0.78
+0.76
+Global-ER (𝑑𝜃 = 2)
+0.47
+0.81
+0.78
+Global-Basic (𝑑𝜃 = 2)
+0.42
+0.79
+0.73
+ALL (𝑑𝜃 = 2)
+0.50
+0.80
+0.75
+ALL (𝑑𝜃 = 2)
+0.50
+0.80
+0.75
+ALL (𝑑𝜃 = 4)
+0.53
+0.82
+0.84
+ALL (𝑑𝜃 = 8)
+0.52
+0.85
+0.85
+Observation 9. In real-world graphs, the signals from the local
+structures of edges in their early stage regarding their future im-
+portance are weaker, compared to the signals that from the local
+structures of nodes (see Figure 5).
+Observation 10. However, the signals become stronger as the edges
+are better connected, leading to better prediction performance (see
+Tables 6 and 9).
+Observation 11. The features from edge roles (Local-ER and
+Global-ER) are more informative than simple global statistics
+(Global-Basic) for future importance prediction (see Table 9).
+6
+RELATED WORK
+Previous studies on temporal graph analysis are largely categorized
+into (a) designing algorithms for streaming graphs [13, 23, 27, 30],
+(b) discovering temporal patterns in graphs [2, 3, 7, 11, 26], and
+(c) generating graphs with realistic dynamics [3, 8, 25]. This work
+belongs to the second category.
+Studies in this category have revealed (a) universal temporal
+patterns, such as densification [26], shrinking diameter [26], and
+power-laws between principle eigenvalues and edge counts [3]; and
+(b) domain-specific patterns in hyperlink networks [12], metabolic
+networks (e.g., biochemical reactions and protein interactions) [11],
+communication networks (e.g., phone calls and texts) [2, 16], and
+friendship networks [7].
+In particular, for the analysis of local structures, the concept of
+graphlets [36] (i.e., the sets of isomorphic small subgraphs with
+a predefined number of nodes) has been extended to temporal
+graphs. The extensions, which are called temporal network motifs,
+have multiple variants. Kovanen et al. [21] defined them as sets
+of temporal subgraphs with a fixed number of nodes that are (a)
+topologically equivalent, (b) temporally equivalent (specifically,
+relative orders of constituent edges are identical), (c) consecutive
+(specifically, constituent edges are consecutive for every node), and
+(d) temporally local (specifically, arrival times of consecutive edges
+are close enough). Hulovaty et al. [19] ignores (c); and Paranjape
+et al. [35] ignores (c) and relaxes (d) by restricting only the time
+difference between the first edge and the last edge. Note that all
+these notions focus on temporally local subgraphs, and thus they
+are suitable only for analyzing short-term dynamics.
+
+For long-term dynamics in local structures, David et al. [4] pro-
+posed Graph-orbit Transition (GoT) and Orbit Temporal Agreement
+(OTA), which characterize the dynamic of a temporal graph by ap-
+proximately counting the number of transitions between node roles.
+However, due to high computational overhead, only a small frac-
+tion of snapshots can be compared for estimating the counts of
+transitions, and as a result, their characterization powers are sig-
+nificantly weaker than our characterization method using GTGs
+(see Section 3.2). Recall that our method counts “every” transition
+between graphlets, and it is still significantly faster than GOT and
+OTA (see Section A in Appendix).
+For predicting the future in-degree of nodes, Yang et al. [41]
+proposed to use five features obtained from graphlets with three
+nodes (see Section 4.2 for descriptions). As shown empirically, our
+proposed features tend to provide better prediction performance
+than these five features, and more importantly, they are comple-
+mentary to each other. Faisal and Milenković [14] aimed to detect
+aging-related nodes, whose topological properties (e.g,. graphlet
+counts) change highly over time, in the gene expression process.
+On the algorithmic aspect, a great number of algorithms have
+been developed for the problem of counting the instances of each
+graphlet, which is also known as the subgraph counting problem.
+As suggested in a survey on subgraph counting [37], subgraph-
+counting algorithms are largely categorized into exact counting [1,
+32, 33, 38] and approximated counting [6, 40]. Those in the first cat-
+egory are further categorized into enumeration-based approaches
+[32, 38], matrix-based approaches [33], and decomposition-based
+approaches [1]. Algorithm 1 belongs to the first subcategory, and
+it achieves the optimal time complexity achievable by those in
+this subcategory, as discussed in the beginning of Section 3.1. It
+is adapted from StreaM [38], which maintains the counts of the
+instances of 4-node undirected graphlets in a fully dynamic graph
+stream (i.e., a stream of edge insertions and deletions).
+7
+CONCLUSION
+In this work, we examined the long-term evolution of local struc-
+tures captured by graphlets at the graph, node, and edge levels. We
+summarize our contribution as follows:
+• Patterns: We examined various patterns regarding the dynamics
+of local structures in temporal graphs. For example, the distribu-
+tions of graphlets over time in real-world graphs differ signifi-
+cantly from those in random graphs, and the transitions between
+graphlets are surprisingly similar in graphs from the same do-
+mains. Moreover, local structures at nodes and edges in their early
+stages provide strong signals regarding their future importance.
+• Tools: We introduced graphlet transition graphs, and we demon-
+strated that it is an effective tool for measuring the similarity
+between temporal graphs of different sizes.
+• Predictability: We enhanced the accuracy of predicting the fu-
+ture importance of nodes and edges by introducing new features
+based on node roles and edge roles. The features are also com-
+plementary to global graph statistics.
+Reproducibility: The code and the datasets are available at https:
+//github.com/deukryeol-yoon/graphlets-over-time.
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+Table 10: Results of feature importance analysis. We report the five strongest predictors and their Gini importance.
+Centrality
+Rank 1
+Rank 2
+Rank 3
+Rank 4
+Rank 5
+Degree
+# of edges
+0.07
+Global-NPP 2
+0.06
+# of nodes
+0.06
+Global-NR 3
+0.04
+Global-NR 10
+0.03
+Betweenness
+Local-NPP 2
+0.10
+Local-NR 4
+0.09
+Global-NR 4
+0.08
+Local-NR 9
+0.06
+Global-NR 3
+0.05
+Closeness
+Global-NR 5
+0.09
+Local-NR 5
+0.07
+# of edges
+0.07
+Global-NPP 2
+0.06
+# of nodes
+0.06
+PageRank
+Local-NR 1
+0.07
+# of edges
+0.06
+Global-NPP 2
+0.06
+# of nodes
+0.05
+Global-NR 1
+0.05
+Edge betweenness
+Global-ER 7
+0.11
+Global-ER 2
+0.09
+Global-ER 3
+0.09
+# of nodes
+0.07
+Local-ER 7
+0.07
+Hep
+pTPh
+He 
+h 
+Patent 
+Enron -
+EU -
+College -
+Math -
+Ask-
+Stack -
+I 
+-
+1• 
+| 
+- 1.0
+l卜 0.5 
+- 0.0
+(a) Ours
+(b) GoT
+HepPh 
+- 1.0
+HepTh 
+I- 0.8
+Enron -
+|� 0.6 
+EU -
+College -
+I'- 0.4
+Math-
+- 0.2
+Ask-
+- 0.0
+(c) OTA
+Figure 6: Similarity matrices from ours, GoT, and OTA. The
+domains of graphs (distinguished by colors) are classified
+more accurately by ours than by GoT or OTA.
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+network motifs and graphlets. CSUR 54, 2 (2021), 1–36.
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+A stream-based algorithm for counting motifs in dynamic graphs. In AlCoB.
+[39] Charalampos E Tsourakakis, Jakub Pachocki, and Michael Mitzenmacher. 2017.
+Scalable motif-aware graph clustering. In WWW.
+[40] Sebastian Wernicke. 2005. A faster algorithm for detecting network motifs. In
+WABI. Springer, 165–177.
+[41] Yang Yang, Yuxiao Dong, and Nitesh V Chawla. 2014. Predicting node degree
+centrality with the node prominence profile. Scientific reports 4, 1 (2014), 1–7.
+[42] Yanlei Yu, Zhiwu Lu, Jiajun Liu, Guoping Zhao, and Ji-rong Wen. 2019. Rum:
+Network representation learning using motifs. In ICDE.
+[43] Daniel Zwillinger and Stephen Kokoska. 1999. CRC standard probability and
+statistics tables and formulae. Crc Press.
+A
+COMPARISON WITH GOT AND OTA
+We provide additional details regarding the comparison between our
+characterization method based on graphlet transition graphs (GTGs)
+and regarding the comparison with Graphlet-orbit Transition (GoT)
+and Orbit Temporal Agreement (OTA).
+Detailed Setting: Our experiments were conducted on a desktop
+with a 3.8 GHz AMD Ryzen 3900x CPU and 128GB memory. We
+implemented our characterization method based on GTGs in Java,
+O.O.M
+O.O.M
+9.75x
+7.05x
+4.86x
+3.66x
+4.07x
+10.14x
+7.17x
+Ours
+GoT & OTA
+Ours
+GoT & OTA
+Figure 7:
+Running times of ours, GoT and OTA. Ours is
+consistently and significantly faster than both competitors,
+which run out of memory in the two largest datasets.
+and we used the official implementations for GoT and OTA provided
+by the authors, which were implemented in C++. In each dataset,
+we used 12 snapshots with the same intervals for GoT and OTA.
+Output Similar Matrix: Figure 6 shows the output similar matri-
+ces from our characterization method based on GTGs, GoT, and
+OTA. GoT and OTA run out of memory in the two largest datasets
+(Patent and Stackoverflow). Both GoT and OTA fail to distinguish
+email/message graphs (blue) and online Q/A graphs (green) clearly.
+Numerically, with the best thresholds of similarity, the classification
+accuracies are 81.0% (GoT) and 85.7% (OTA), while the accuracy
+is 97.2% in ours.
+Speed Comparison: As seen in Figure 7, ours is faster than GoT
+and OTA in all the graphs. Specifically, ours is 6.68× faster than
+the others on average.
+B
+FEATURE IMPORTANT ANALYSIS
+We measure the importance of each feature in the set ALL (see
+Section 4.2 of the main paper) using the Gini importance [29], and
+we report the top five important features in Table 10.
+C
+DETAILED RESULTS OF FUTURE NODE
+IMPORTANCE PREDICTION
+In Tables 11-16, we provide the average predictive performances
+and standard deviations over 10 runs on the task of predicting
+future node centrality in each real-world graph in terms of several
+evaluation metrics. The detailed experimental settings can be found
+in Section 4.2 of the main paper.
+D
+DETAILED RESULTS OF FUTURE EDGE
+IMPORTANCE PREDICTION
+In Tables 17-19, we provide the average predictive performances
+and standard deviations over 10 runs on the task of predicting
+future node centrality in each real-world graph in terms of several
+evaluation metrics. The detailed experimental settings can be found
+in Section 5 of the main paper.
+
+HepPh
+HepTh
+Enron
+EU
+College
+Math
+Ask三Table 11: F1-score on the task of predicting future node importance when 𝑑𝜃 = 2.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Degree
+Local-NR
+0.11±0.008
+0.20±0.014
+0.36±0.092
+0.68±0.007
+0.27±0.027
+0.38±0.017
+0.73±0.002
+0.39
+Local-NPP
+0.12±0.013
+0.19±0.016
+0.40±0.102
+0.60±0.011
+0.27±0.040
+0.36±0.016
+0.72±0.003
+0.38
+Global-NR
+0.52±0.013
+0.56±0.013
+0.51±0.077
+0.79±0.004
+0.37±0.025
+0.52±0.021
+0.70±0.004
+0.57
+Global-NPP
+0.51±0.010
+0.57±0.018
+0.51±0.063
+0.76±0.005
+0.40±0.030
+0.52±0.022
+0.70±0.006
+0.57
+Global-basic
+0.36±0.014
+0.38±0.024
+0.44±0.073
+0.77±0.008
+0.34±0.050
+0.51±0.022
+0.71±0.003
+0.50
+ALL
+0.53±0.010
+0.58±0.013
+0.52±0.043
+0.79±0.008
+0.38±0.041
+0.52±0.026
+0.70±0.005
+0.57
+Betweenness
+Local-NR
+0.59±0.011
+0.88±0.009
+0.34±0.063
+0.49±0.011
+0.34±0.033
+0.74±0.011
+0.73±0.007
+0.59
+Local-NPP
+0.58±0.010
+0.87±0.006
+0.35±0.076
+0.45±0.010
+0.36±0.073
+0.74±0.011
+0.73±0.007
+0.58
+Global-NR
+0.64±0.007
+0.90±0.005
+0.48±0.089
+0.62±0.014
+0.38±0.047
+0.75±0.008
+0.74±0.006
+0.64
+Global-NPP
+0.62±0.010
+0.89±0.006
+0.49±0.037
+0.58±0.019
+0.40±0.034
+0.75±0.010
+0.74±0.006
+0.64
+Global-basic
+0.01±0.003
+0.27±0.028
+0.40±0.079
+0.36±0.017
+0.25±0.038
+0.32±0.013
+0.10±0.013
+0.24
+ALL
+0.64±0.007
+0.90±0.007
+0.53±0.052
+0.62±0.016
+0.38±0.045
+0.75±0.010
+0.74±0.007
+0.65
+Closeness
+Local-NR
+0.49±0.010
+0.53±0.010
+0.28±0.077
+0.69±0.008
+0.24±0.034
+0.58±0.014
+0.75±0.005
+0.51
+Local-NPP
+0.37±0.015
+0.51±0.014
+0.31±0.067
+0.46±0.010
+0.25±0.038
+0.43±0.017
+0.66±0.006
+0.43
+Global-NR
+0.84±0.006
+0.75±0.007
+0.47±0.046
+0.83±0.008
+0.38±0.055
+0.69±0.024
+0.81±0.002
+0.68
+Global-NPP
+0.83±0.008
+0.74±0.007
+0.52±0.047
+0.76±0.008
+0.39±0.022
+0.64±0.019
+0.71±0.005
+0.66
+Global-basic
+0.82±0.004
+0.70±0.010
+0.44±0.074
+0.72±0.010
+0.33±0.010
+0.51±0.015
+0.60±0.004
+0.59
+ALL
+0.85±0.008
+0.76±0.008
+0.53±0.043
+0.83±0.007
+0.36±0.051
+0.69±0.022
+0.81±0.003
+0.69
+PageRank
+Local-NR
+0.44±0.013
+0.15±0.018
+0.42±0.069
+0.64±0.008
+0.25±0.038
+0.46±0.016
+0.58±0.009
+0.42
+Local-NPP
+0.41±0.012
+0.18±0.017
+0.43±0.086
+0.39±0.009
+0.25±0.040
+0.41±0.013
+0.53±0.008
+0.37
+Global-NR
+0.64±0.014
+0.41±0.015
+0.49±0.078
+0.74±0.006
+0.35±0.056
+0.53±0.019
+0.63±0.009
+0.54
+Global-NPP
+0.64±0.012
+0.43±0.015
+0.55±0.046
+0.65±0.011
+0.38±0.047
+0.52±0.017
+0.61±0.007
+0.54
+Global-basic
+0.63±0.010
+0.31±0.023
+0.41±0.054
+0.65±0.007
+0.28±0.035
+0.49±0.010
+0.54±0.008
+0.47
+ALL
+0.64±0.009
+0.44±0.013
+0.55±0.035
+0.74±0.006
+0.37±0.030
+0.53±0.020
+0.63±0.008
+0.56
+Table 12: Accuracy on the task of predicting future node importance when 𝑑𝜃 = 2.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Degree
+Local-NR
+0.71±0.008
+0.72±0.006
+0.80±0.024
+0.62±0.006
+0.77±0.019
+0.66±0.008
+0.58±0.003
+0.69
+Local-NPP
+0.71±0.010
+0.72±0.007
+0.79±0.031
+0.55±0.006
+0.75±0.020
+0.65±0.016
+0.58±0.002
+0.68
+Global-NR
+0.76±0.007
+0.77±0.007
+0.82±0.024
+0.77±0.004
+0.76±0.017
+0.68±0.014
+0.61±0.004
+0.74
+Global-NPP
+0.75±0.003
+0.77±0.007
+0.80±0.023
+0.75±0.004
+0.77±0.019
+0.68±0.012
+0.61±0.004
+0.73
+Global-basic
+0.73±0.006
+0.74±0.008
+0.77±0.028
+0.75±0.006
+0.74±0.019
+0.67±0.011
+0.60±0.003
+0.72
+ALL
+0.76±0.008
+0.78±0.006
+0.82±0.019
+0.77±0.005
+0.76±0.016
+0.68±0.014
+0.61±0.004
+0.74
+Betweenness
+Local-NR
+0.79±0.005
+0.93±0.005
+0.78±0.017
+0.82±0.006
+0.76±0.019
+0.86±0.005
+0.90±0.003
+0.83
+Local-NPP
+0.79±0.004
+0.92±0.004
+0.75±0.028
+0.81±0.005
+0.65±0.032
+0.86±0.005
+0.90±0.003
+0.81
+Global-NR
+0.81±0.004
+0.94±0.003
+0.81±0.017
+0.84±0.007
+0.75±0.016
+0.86±0.004
+0.90±0.002
+0.84
+Global-NPP
+0.80±0.005
+0.94±0.003
+0.79±0.021
+0.83±0.007
+0.76±0.020
+0.86±0.004
+0.90±0.003
+0.84
+Global-basic
+0.71±0.005
+0.73±0.010
+0.75±0.033
+0.77±0.007
+0.70±0.019
+0.67±0.008
+0.77±0.004
+0.73
+ALL
+0.81±0.003
+0.94±0.004
+0.82±0.018
+0.84±0.008
+0.75±0.019
+0.86±0.004
+0.90±0.003
+0.85
+Closeness
+Local-NR
+0.76±0.003
+0.76±0.005
+0.78±0.027
+0.75±0.005
+0.76±0.015
+0.72±0.006
+0.78±0.003
+0.76
+Local-NPP
+0.73±0.003
+0.75±0.006
+0.75±0.032
+0.63±0.007
+0.74±0.016
+0.65±0.010
+0.64±0.004
+0.70
+Global-NR
+0.91±0.003
+0.86±0.004
+0.80±0.018
+0.85±0.006
+0.75±0.019
+0.77±0.011
+0.82±0.002
+0.82
+Global-NPP
+0.90±0.005
+0.85±0.002
+0.81±0.013
+0.80±0.007
+0.76±0.011
+0.73±0.010
+0.73±0.003
+0.80
+Global-basic
+0.90±0.003
+0.82±0.004
+0.76±0.033
+0.77±0.008
+0.73±0.024
+0.66±0.009
+0.64±0.003
+0.75
+ALL
+0.91±0.004
+0.86±0.004
+0.82±0.020
+0.85±0.005
+0.75±0.017
+0.77±0.011
+0.82±0.002
+0.83
+PageRank
+Local-NR
+0.76±0.007
+0.72±0.009
+0.80±0.020
+0.74±0.006
+0.75±0.017
+0.66±0.011
+0.65±0.006
+0.73
+Local-NPP
+0.74±0.007
+0.72±0.007
+0.77±0.023
+0.65±0.018
+0.73±0.023
+0.63±0.005
+0.62±0.006
+0.69
+Global-NR
+0.81±0.005
+0.75±0.007
+0.81±0.022
+0.80±0.005
+0.75±0.015
+0.68±0.010
+0.67±0.006
+0.75
+Global-NPP
+0.81±0.005
+0.74±0.007
+0.82±0.023
+0.74±0.011
+0.76±0.019
+0.67±0.008
+0.66±0.006
+0.74
+Global-basic
+0.81±0.004
+0.73±0.008
+0.75±0.021
+0.74±0.006
+0.70±0.025
+0.65±0.007
+0.59±0.006
+0.71
+ALL
+0.81±0.005
+0.75±0.004
+0.81±0.017
+0.80±0.006
+0.75±0.014
+0.68±0.010
+0.67±0.005
+0.75
+
+Table 13: AUROC on the task of predicting future node importance when 𝑑𝜃 = 2.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Degree
+Local-NR
+0.69±0.006
+0.70±0.006
+0.80±0.037
+0.67±0.007
+0.69±0.020
+0.65±0.012
+0.58±0.006
+0.68
+Local-NPP
+0.64±0.005
+0.67±0.007
+0.75±0.032
+0.56±0.007
+0.64±0.025
+0.63±0.012
+0.56±0.004
+0.64
+Global-NR
+0.81±0.004
+0.83±0.005
+0.85±0.037
+0.86±0.005
+0.73±0.026
+0.71±0.013
+0.65±0.006
+0.78
+Global-NPP
+0.80±0.002
+0.83±0.005
+0.83±0.032
+0.83±0.005
+0.74±0.026
+0.71±0.012
+0.65±0.005
+0.77
+Global-basic
+0.74±0.004
+0.74±0.007
+0.82±0.035
+0.84±0.006
+0.68±0.037
+0.68±0.009
+0.62±0.005
+0.73
+ALL
+0.81±0.004
+0.84±0.006
+0.85±0.036
+0.86±0.005
+0.73±0.025
+0.71±0.014
+0.65±0.006
+0.78
+Betweenness
+Local-NR
+0.84±0.006
+0.98±0.002
+0.79±0.033
+0.80±0.007
+0.70±0.029
+0.83±0.010
+0.82±0.005
+0.82
+Local-NPP
+0.83±0.006
+0.98±0.003
+0.73±0.031
+0.68±0.008
+0.65±0.041
+0.82±0.010
+0.81±0.006
+0.79
+Global-NR
+0.87±0.005
+0.99±0.002
+0.81±0.030
+0.87±0.007
+0.71±0.032
+0.86±0.008
+0.86±0.005
+0.85
+Global-NPP
+0.85±0.007
+0.98±0.002
+0.81±0.032
+0.84±0.008
+0.72±0.033
+0.86±0.008
+0.86±0.005
+0.85
+Global-basic
+0.62±0.008
+0.74±0.013
+0.76±0.042
+0.74±0.005
+0.63±0.038
+0.63±0.014
+0.60±0.007
+0.67
+ALL
+0.87±0.006
+0.99±0.001
+0.83±0.024
+0.87±0.007
+0.71±0.033
+0.86±0.009
+0.86±0.005
+0.86
+Closeness
+Local-NR
+0.79±0.005
+0.79±0.006
+0.76±0.047
+0.84±0.006
+0.68±0.026
+0.76±0.009
+0.84±0.002
+0.78
+Local-NPP
+0.74±0.005
+0.78±0.008
+0.72±0.042
+0.65±0.009
+0.61±0.022
+0.63±0.007
+0.70±0.005
+0.69
+Global-NR
+0.97±0.002
+0.93±0.005
+0.82±0.035
+0.93±0.003
+0.73±0.024
+0.83±0.011
+0.89±0.002
+0.87
+Global-NPP
+0.96±0.003
+0.92±0.004
+0.83±0.028
+0.88±0.004
+0.73±0.026
+0.79±0.011
+0.81±0.003
+0.85
+Global-basic
+0.95±0.003
+0.89±0.005
+0.79±0.044
+0.85±0.007
+0.69±0.035
+0.68±0.012
+0.70±0.003
+0.79
+ALL
+0.97±0.002
+0.94±0.005
+0.82±0.027
+0.93±0.004
+0.75±0.028
+0.83±0.010
+0.90±0.001
+0.88
+PageRank
+Local-NR
+0.77±0.006
+0.65±0.010
+0.80±0.031
+0.81±0.004
+0.67±0.011
+0.69±0.015
+0.70±0.006
+0.73
+Local-NPP
+0.75±0.004
+0.63±0.008
+0.77±0.031
+0.63±0.010
+0.62±0.024
+0.64±0.009
+0.67±0.006
+0.67
+Global-NR
+0.87±0.005
+0.78±0.006
+0.84±0.035
+0.88±0.003
+0.72±0.029
+0.71±0.012
+0.73±0.005
+0.79
+Global-NPP
+0.86±0.005
+0.77±0.007
+0.85±0.028
+0.82±0.008
+0.72±0.018
+0.70±0.011
+0.71±0.006
+0.78
+Global-basic
+0.86±0.005
+0.73±0.010
+0.81±0.033
+0.81±0.008
+0.66±0.026
+0.66±0.011
+0.62±0.004
+0.74
+ALL
+0.87±0.005
+0.79±0.007
+0.85±0.035
+0.88±0.004
+0.72±0.031
+0.71±0.014
+0.73±0.005
+0.79
+Table 14: F1-score on the task of predicting future node importance depending on 𝑑𝜃 (i.e., in-degree of nodes when their input
+features are extracted).
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Degree
+ALL (𝑑𝜃 = 2)
+0.53±0.010
+0.58±0.013
+0.52±0.043
+0.19±0.005
+0.38±0.041
+0.52±0.026
+0.70±0.005
+0.59
+ALL (𝑑𝜃 = 4)
+0.67±0.007
+0.78±0.008
+0.62±0.062
+1.00*
+0.49±0.045
+0.89±0.006
+1.00*
+0.69
+ALL (𝑑𝜃 = 8)
+0.82±0.006
+0.93±0.006
+0.74±0.036
+1.00*
+0.71±0.022
+1.00*
+1.00*
+0.80
+Betweenness
+ALL (𝑑𝜃 = 2)
+0.64±0.007
+0.90±0.007
+0.53±0.052
+0.62±0.016
+0.38±0.045
+0.75±0.010
+0.74±0.007
+0.65
+ALL (𝑑𝜃 = 4)
+0.72±0.012
+0.94±0.007
+0.52±0.066
+0.75±0.006
+0.50±0.042
+0.84±0.009
+0.84±0.008
+0.73
+ALL (𝑑𝜃 = 8)
+0.77±0.008
+0.97±0.005
+0.65±0.045
+0.86±0.009
+0.69±0.057
+0.91±0.009
+0.89±0.007
+0.82
+Closeness
+ALL (𝑑𝜃 = 2)
+0.85±0.008
+0.76±0.008
+0.53±0.043
+0.83±0.007
+0.36±0.051
+0.69±0.022
+0.81±0.003
+0.69
+ALL (𝑑𝜃 = 4)
+0.87±0.008
+0.85±0.010
+0.55±0.072
+0.91±0.006
+0.54±0.032
+0.85±0.013
+0.91±0.004
+0.78
+ALL (𝑑𝜃 = 8)
+0.88±0.007
+0.90±0.009
+0.65±0.061
+0.97±0.004
+0.74±0.045
+0.95±0.007
+0.98±0.003
+0.86
+PageRank
+ALL (𝑑𝜃 = 2)
+0.64±0.009
+0.44±0.013
+0.52±0.035
+0.74±0.006
+0.37±0.030
+0.53±0.020
+0.63±0.006
+0.55
+ALL (𝑑𝜃 = 4)
+0.74±0.008
+0.71±0.010
+0.62±0.037
+0.87±0.006
+0.48±0.040
+0.79±0.008
+0.89±0.003
+0.73
+ALL (𝑑𝜃 = 8)
+0.83±0.007
+0.85±0.012
+0.68±0.049
+0.95±0.003
+0.72±0.033
+0.95±0.006
+0.98±0.003
+0.85
+* All nodes satisfying the condition on 𝑑𝜃 have the same class, belonging to top 20% in terms of the considered centrality measure.
+Table 15: Accuracy on the task of predicting future node importance depending on 𝑑𝜃.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Degree
+ALL (𝑑𝜃 = 2)
+0.76±0.008
+0.78±0.006
+0.82±0.019
+0.77±0.005
+0.76±0.016
+0.68±0.014
+0.61±0.004
+0.74
+ALL (𝑑𝜃 = 4)
+0.76±0.006
+0.80±0.009
+0.83±0.028
+1.00*
+0.70±0.046
+0.81±0.009
+1.00*
+0.78
+ALL (𝑑𝜃 = 8)
+0.79±0.006
+0.88±0.010
+0.86±0.021
+1.00*
+0.72±0.025
+1.00*
+1.00*
+0.81
+Betweenness
+ALL (𝑑𝜃 = 2)
+0.81±0.003
+0.94±0.004
+0.82±0.018
+0.84±0.008
+0.75±0.019
+0.86±0.004
+0.90±0.003
+0.85
+ALL (𝑑𝜃 = 4)
+0.81±0.008
+0.96±0.004
+0.80±0.023
+0.82±0.004
+0.70±0.023
+0.86±0.009
+0.89±0.006
+0.83
+ALL (𝑑𝜃 = 8)
+0.81±0.004
+0.98±0.004
+0.82±0.019
+0.85±0.009
+0.70±0.049
+0.88±0.010
+0.88±0.006
+0.85
+Closeness
+ALL (𝑑𝜃 = 2)
+0.91±0.004
+0.86±0.004
+0.82±0.020
+0.85±0.005
+0.75±0.017
+0.77±0.011
+0.82±0.002
+0.83
+ALL (𝑑𝜃 = 4)
+0.91±0.005
+0.88±0.007
+0.80±0.022
+0.89±0.007
+0.70±0.021
+0.80±0.015
+0.86±0.006
+0.83
+ALL (𝑑𝜃 = 8)
+0.91±0.006
+0.89±0.009
+0.82±0.020
+0.94±0.006
+0.73±0.046
+0.91±0.013
+0.95±0.006
+0.88
+PageRank
+ALL (𝑑𝜃 = 2)
+0.81±0.005
+0.75±0.004
+0.81±0.017
+0.80±0.006
+0.75±0.014
+0.68±0.010
+0.67±0.006
+0.75
+ALL (𝑑𝜃 = 4)
+0.81±0.006
+0.75±0.007
+0.83±0.017
+0.83±0.007
+0.68±0.025
+0.68±0.011
+0.81±0.003
+0.77
+ALL (𝑑𝜃 = 8)
+0.83±0.005
+0.81±0.012
+0.82±0.023
+0.92±0.004
+0.72±0.025
+0.91±0.011
+0.96±0.003
+0.85
+* All nodes satisfying the condition on 𝑑𝜃 have the same class, belonging to top 20% in terms of the considered centrality measure.
+
+Table 16: AUROC on the task of predicting future node importance depending on 𝑑𝜃.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Degree
+ALL (𝑑𝜃 = 2)
+0.81±0.004
+0.84±0.005
+0.85±0.035
+0.86±0.005
+0.73±0.025
+0.71±0.014
+0.65±0.005
+0.78
+ALL (𝑑𝜃 = 4)
+0.83±0.005
+0.87±0.006
+0.85±0.036
+1.00*
+0.72±0.027
+0.68±0.018
+1.00*
+0.79
+ALL (𝑑𝜃 = 8)
+0.87±0.007
+0.90±0.013
+0.88±0.027
+1.00*
+0.78±0.031
+1.00*
+1.00*
+0.86
+Betweenness
+ALL (𝑑𝜃 = 2)
+0.87±0.005
+0.99±0.001
+0.83±0.024
+0.87±0.007
+0.71±0.033
+0.86±0.009
+0.86±0.005
+0.86
+ALL (𝑑𝜃 = 4)
+0.89±0.006
+0.99±0.001
+0.81±0.040
+0.89±0.004
+0.73±0.026
+0.90±0.007
+0.91±0.004
+0.87
+ALL (𝑑𝜃 = 8)
+0.90±0.003
+1.00±0.001
+0.84±0.026
+0.93±0.006
+0.77±0.044
+0.94±0.009
+0.94±0.006
+0.90
+Closeness
+ALL (𝑑𝜃 = 2)
+0.97±0.002
+0.94±0.005
+0.84±0.033
+0.93±0.004
+0.73±0.028
+0.83±0.010
+0.90±0.002
+0.88
+ALL (𝑑𝜃 = 4)
+0.97±0.002
+0.95±0.004
+0.82±0.027
+0.95±0.004
+0.75±0.030
+0.88±0.012
+0.93±0.004
+0.89
+ALL (𝑑𝜃 = 8)
+0.97±0.003
+0.96±0.006
+0.88±0.024
+0.98±0.004
+0.79±0.043
+0.92±0.016
+0.95±0.013
+0.92
+PageRank
+ALL (𝑑𝜃 = 2)
+0.87±0.005
+0.79±0.008
+0.85±0.035
+0.88±0.004
+0.72±0.031
+0.71±0.014
+0.73±0.005
+0.79
+ALL (𝑑𝜃 = 4)
+0.89±0.006
+0.83±0.008
+0.87±0.018
+0.90±0.006
+0.71±0.028
+0.69±0.009
+0.70±0.007
+0.80
+ALL (𝑑𝜃 = 8)
+0.91±0.005
+0.87±0.013
+0.87±0.034
+0.95±0.009
+0.79±0.018
+0.73±0.040
+0.71±0.049
+0.83
+* All nodes satisfying the condition on 𝑑𝜃 have the same class, belonging to top 20% in terms of the considered centrality measure.
+Table 17: F1-score on the task of predicting future edge importance.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Edge Betweenness
+Local-ER (𝑑𝜃 = 2)
+0.68 ± 0.004
+0.59 ± 0.011
+0.14 ± 0.094
+0.74 ± 0.013
+0.41 ± 0.042
+0.21 ± 0.038
+0.40 ± 0.013
+0.45
+Global-ER (𝑑𝜃 = 2)
+0.69 ± 0.015
+0.63 ± 0.041
+0.18 ± 0.122
+0.79 ± 0.051
+0.38 ± 0.060
+0.23 ± 0.058
+0.39 ± 0.022
+0.47
+Global-Basic (𝑑𝜃 = 2)
+0.69 ± 0.013
+0.51 ± 0.168
+0.22 ± 0.132
+0.75 ± 0.072
+0.37 ± 0.064
+0.15 ± 0.116
+0.26 ± 0.180
+0.42
+ALL (𝑑𝜃 = 2)
+0.71 ± 0.005
+0.68 ± 0.009
+0.25 ± 0.186
+0.84 ± 0.005
+0.40 ± 0.062
+0.23 ± 0.060
+0.36 ± 0.018
+0.50
+ALL (𝑑𝜃 = 2)
+0.71 ± 0.005
+0.68 ± 0.009
+0.25 ± 0.186
+0.84 ± 0.005
+0.40 ± 0.062
+0.23 ± 0.060
+0.36 ± 0.018
+0.50
+ALL (𝑑𝜃 = 4)
+0.71 ± 0.007
+0.72 ± 0.009
+0.33 ± 0.104
+0.77 ± 0.006
+0.43 ± 0.086
+0.29 ± 0.071
+0.46 ± 0.014
+0.53
+ALL (𝑑𝜃 = 8)
+0.69 ± 0.004
+0.75 ± 0.009
+0.17 ± 0.079
+0.72 ± 0.011
+0.39 ± 0.052
+0.31 ± 0.055
+0.53 ± 0.023
+0.52
+Table 18: Accuracy on the task of predicting future edge importance.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Edge Betweenness
+Local-ER (𝑑𝜃 = 2)
+0.66 ± 0.003
+0.72 ± 0.005
+0.87 ± 0.041
+0.75 ± 0.012
+0.70 ± 0.030
+0.91 ± 0.008
+0.91 ± 0.003
+0.78
+Global-ER (𝑑𝜃 = 2)
+0.68 ± 0.020
+0.75 ± 0.023
+0.88 ± 0.041
+0.81 ± 0.058
+0.70 ± 0.035
+0.91 ± 0.009
+0.91 ± 0.003
+0.81
+Global-Basic (𝑑𝜃 = 2)
+0.66 ± 0.036
+0.72 ± 0.046
+0.87 ± 0.040
+0.79 ± 0.052
+0.67 ± 0.053
+0.91 ± 0.009
+0.91 ± 0.008
+0.79
+ALL (𝑑𝜃 = 2)
+0.67 ± 0.037
+0.73 ± 0.047
+0.87 ± 0.042
+0.81 ± 0.055
+0.68 ± 0.052
+0.91 ± 0.009
+0.91 ± 0.007
+0.80
+ALL (𝑑𝜃 = 2)
+0.67 ± 0.037
+0.73 ± 0.005
+0.87 ± 0.042
+0.81 ± 0.055
+0.68 ± 0.052
+0.91 ± 0.009
+0.91 ± 0.007
+0.80
+ALL (𝑑𝜃 = 4)
+0.73 ± 0.006
+0.79 ± 0.005
+0.86 ± 0.050
+0.78 ± 0.024
+0.78 ± 0.024
+0.93 ± 0.007
+0.90 ± 0.004
+0.82
+ALL (𝑑𝜃 = 8)
+0.75 ± 0.003
+0.81 ± 0.005
+0.88 ± 0.046
+0.82 ± 0.017
+0.82 ± 0.017
+0.94 ± 0.005
+0.91 ± 0.002
+0.85
+Table 19: AUROC on the task of predicting future edge importance.
+Centrality
+Feature
+Citation Networks
+Email/Message Networks
+Online Q/A Networks
+Average
+HepPh
+Hepth
+Email-EU
+Email-Enron
+Message-College
+Mathoverflow
+Askubuntu
+Edge Betweenness
+Local-ER (𝑑𝜃 = 2)
+0.71 ± 0.003
+0.77 ± 0.007
+0.64 ± 0.080
+0.82 ± 0.009
+0.68 ± 0.035
+0.85 ± 0.013
+0.86 ± 0.006
+0.76
+Global-ER (𝑑𝜃 = 2)
+0.74 ± 0.027
+0.80 ± 0.033
+0.63 ± 0.092
+0.88 ± 0.058
+0.68 ± 0.032
+0.85 ± 0.015
+0.86 ± 0.007
+0.78
+Global-Basic (𝑑𝜃 = 2)
+0.71 ± 0.053
+0.77 ± 0.055
+0.64 ± 0.093
+0.87 ± 0.049
+0.65 ± 0.058
+0.73 ± 0.164
+0.76 ± 0.150
+0.73
+ALL (𝑑𝜃 = 2)
+0.72 ± 0.054
+0.79 ± 0.057
+0.64 ± 0.099
+0.89 ± 0.051
+0.66 ± 0.056
+0.76 ± 0.150
+0.78 ± 0.139
+0.75
+ALL (𝑑𝜃 = 2)
+0.72 ± 0.054
+0.79 ± 0.057
+0.64 ± 0.099
+0.89 ± 0.051
+0.66 ± 0.056
+0.76 ± 0.150
+0.78 ± 0.139
+0.75
+ALL (𝑑𝜃 = 4)
+0.80 ± 0.004
+0.87 ± 0.004
+0.75 ± 0.050
+0.94 ± 0.002
+0.74 ± 0.026
+0.89 ± 0.016
+0.89 ± 0.009
+0.84
+ALL (𝑑𝜃 = 8)
+0.82 ± 0.005
+0.89 ± 0.005
+0.70 ± 0.046
+0.93 ± 0.002
+0.79 ± 0.021
+0.90 ± 0.011
+0.90 ± 0.008
+0.85
+

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+arXiv:2301.11423v1  [math.CO]  26 Jan 2023
+Improved Permutation Arrays for Kendall Tau Metric∗
+Sergey Bereg
+William Bumpass
+Mohammadreza Haghpanah
+Brian Malouf
+I. Hal Sudborough
+Abstract
+Permutation arrays under the Kendall-τ metric have been considered for error-correcting
+codes. Given n and d ∈ [1,
+�n
+2
+�
+], the task is to find a large permutation array of permutations
+on n symbols with pairwise Kendall-τ distance at least d. Let P(n, d) denote the maximum
+size of any permutation array of permutations on n symbols with pairwise Kendall-τ distance d.
+Using new recursive techniques, new automorphisms, and programs that combine randomness
+and greedy strategies, we obtain several improved lower bounds for P(n, d).
+1
+Introduction
+In [1, 2, 5, 8, 10, 11], permutation arrays under the Kendall-τ metric were studied. This comple-
+mented many studies of permutation arrays under other metrics, such as the Hamming metric [3]
+[4] [6], Chebyshev metric [9] and several others [7]. The use of the Kendall-τ metric was motivated
+by applications of error correcting codes and rank modulation in flash memories [8].
+Let σ and π be two permutations (or strings) over an alphabet Σ ⊆ [1...n] = {1, 2, ..., n}. The
+Kendall-τ distance between σ and π, denoted by d(σ, π), is the minimum number of adjacent trans-
+positions (bubble sort operations) required to transform σ into π. For an array (set) A of permuta-
+tions (strings), the pairwise Kendall-τ distance of A, denoted by d(A), is min{ d(σ, π) | σ, π ∈ A }.
+An array A of permutations on [1...n] with d(A) = d will be called a (n, d)-PA. Let P(n, d) denote
+the maximum cardinality of any (n, d)-PA A.
+Vijayakumaran [10] showed several lower bounds for P(5, d) and P(6, d) using integer linear
+programming. Buzaglo and Etzion [5] showed many new bounds, including that P(7, 3) ≥ 588 using
+two permutation representatives and a set of permutations generated by specific automorphism
+operations. We also show results using automorphisms, namely those given in Table 1. Details of
+these automorphisms are shown in Section 4.
+We also used other programs to compute good lower bounds:
+1. Programs which find a maximum size clique in a graph.
+2. Programs which combine randomness with a Greedy approach.
+That is, the first constructs a graph with a node for each permutation on n symbols and an edge
+connecting two nodes whose permutations are at Kendall-τ distance at least d. The set of nodes
+(permutations) in a maximum size clique in this graph is a (n, d)-PA. The second initially chooses
+randomly a specified size set of permutations at pairwise Kendall-τ distance d, and then proceeds
+through all remaining permutations in lexicographic order and adds them to the set if they have
+Kendall-τ distance at least d.
+1
+
+In Tables 1 and 2 are given sporadic results obtained by these techniques. Blank positions in
+our tables signify other papers have the best lower bounds known e.g. [5], [10]. All other lower
+bounds we give are larger than previous lower bounds, except for the two noted in Table 1.
+n:d
+3
+4
+5
+6
+7
+8
+9
+6
+102(∗)
+7
+588(∗)
+336
+126
+84
+42
+8
+3,752
+2,240
+672
+448
+168
+9
+1,008
+288
+Table 1: Improved lower bounds on P(n, d) by automorphisms. (The bounds for P(6, 3) and P(7, 3)
+are from [10] and [5], respectively.)
+n:d
+3
+4
+5
+6
+7
+8
+9
+8
+115
+57
+9
+26,831
+15,492
+3,882
+2,497
+608
+10
+233,421
+133,251
+29,113
+18,344
+5,629
+3,832
+1,489
+11
+247,014
+153,260
+42,013
+28,008
+9,747
+12
+73,068
+n:d
+10
+11
+12
+13
+14
+15
+7
+13
+8
+7
+4
+8
+43
+26
+21
+15
+12
+8
+9
+195
+100
+77
+46
+37
+24
+10
+1,066
+491
+370
+195
+152
+89
+11
+6,890
+2,861
+2,108
+1,005
+768
+409
+12
+50,649
+19,227
+13,935
+6,087
+4,564
+2,239
+Table 2: Improved lower bounds by random Greedy.
+In [2] Barg and Mazumdar described their Theorem 4.5, which is given below:
+Theorem 1. [2] Let m = ((n − 2)t+1 − 1)/(n − 3), where n − 2 is a prime power. Then
+P(n, 2t + 1) ≥
+n!
+t(t + 1)m.
+This was improved by Wang, Zhang, Yang, and Ge in [11].
+Theorem 2. [11] Let m = ((n − 2)t+1 − 1)/(n − 3), where n − 2 is a prime power. Then
+P(n, 2t + 1) ≥
+n!
+(2t + 1)m.
+For example, by choosing t = 1 and n = 11, one obtains, by Theorem 2, P(11, 3) ≥ 1,330,560.
+Theorem 2 applies only when n is two greater than a power of a prime. To compute good lower
+bounds for P(n, d) when n is not two greater than a power of a prime, one needs other techniques.
+The lower bounds given by Theorem 2 are close to corresponding upper bounds when the Kendall-τ
+2
+
+distance is small, but not so close when the Kendall-τ distance is close to n. Our Theorems 6 and
+7, described below, give better lower bounds when the Kendall-τ distance is close to n.
+The following theorem from [8] allows one to obtain good lower bounds for even Kendall-τ
+distances.
+Theorem 3. [8] For all n ≥ 1 and even d ≥ 2, we have P(n, d) ≥ 1
+2P(n, d − 1).
+Theorem 4. [8] For all n, d ≥ 1 we have P(n + 1, d) ≤ (n + 1) · P(n, d).
+Using Theorems 4 and 2 we have P(14, 11) ≥ P(15, 11)/15 ≥ 15!/(11 · 402234 · 15) ≈ 19, 703.2
+Theorem 5. [8] For all n, d > 1 we have P(n + 1, d) ≥ ⌈n+1
+d ⌉P(n, d).
+For example, to compute a lower bound for P(14, 11) one can use, iteratively, Theorem 5 to
+obtain P(14, 11) ≥ ⌈14
+11⌉·⌈13
+11⌉·P(12, 11) = 4·P(12, 11). By computation (using the random greedy
+algorithm) we have P(12, 11) ≥ 19, 277, so P(14, 11) ≥ 76, 908. We next give generalizations of
+Theorem 5 that yield improvements.
+Let Sn,m be the set of permutations on [1...n] with the restriction that the first n−m symbols are
+in sorted order, for a given m < n. A set A ⊆ Sn,m with Kendall-τ distance d is called a (n, m, d)-PA
+or (n, m, d)-array. Let P(n, m, d) denote the maximum cardinality of any (n, m, d)-array A.
+Theorem 6. For any m < n and d, P(n, d) ≥ P(n, m, d) · P(n − m, d).
+Proof. Let A be a (n, m, d)-array and B be a (n − m, d)-array. For each permutation π in A and
+each permutation τ in B, form the permutation (π, τ) by substituting the n − m symbols in the
+order given by τ for the first n − m symbols, given in order, in π.
+It is easily seen that d((π, τ), (ρ, σ)) ≥ d, if either π ̸= ρ or σ ̸= τ. That is, for π, ρ ∈ A, if
+π ̸= ρ, then d(π, ρ) ≥ d. Clearly, changing the order of the other n − m symbols, which appear in
+order in permutations in A, does not make the distance smaller. A symmetric argument applies
+when σ, τ are different permutations in the (n − m, d)-array B.
+In [8] Theorem 5 was proved using the set {1, d + 1, 2d + 1, . . . , ⌈n+1
+d ⌉d + 1}, which corresponds
+to a (n + 1, 1, d)-array. In general, a (n, m, d)-array can be much larger than one obtained by the
+iterative use of Theorem 5. For example, for all n, we give (n, 2, 3)-arrays with n(n+1)
+6
+permutations,
+when n−1 is not divisible by 3. Also, for n = 14 we computed a (14, 2, 11)-array with 5 permutations
+τ1, . . . , τ5 shown in Table 3. Thus, using Theorem 6 we obtain P(14, 11) ≥ 5·P(12, 11) ≥ 5·19, 277 =
+96, 135 which is a better lower bound than obtained by Theorem 5.
+One can also improve on Theorem 6. For each permutation, say τ in a (n, m, d)-array A, one
+can generally find a larger set of permutations than in the best (n−m, d)-array. Let Pτ(n, d) denote
+the maximum cardinality of any (n, d) PA with the highest m symbols in the same positions as in
+τ, but where the other n − m symbols can be in any order. We also denote it by P(n, d; i1, . . . , im),
+where i1, . . . , im are the fixed positions of symbols n − m + 1, . . . , n, not necessarily in that order.
+Theorem 7. For any (n, m, d)-array A, P(n, d) ≥ �
+τ∈A Pτ(n, d).
+Proof. Let A be a (n, m, d)-array and, for each permutation π ∈ A, let τ be a permutation in an
+(n, d)-PA with the highest m symbols in the same position as in π. Form the new permutation
+(π, τ) by substituting the n − m symbols in the order given by τ for the first n − m symbols, given
+in order, in π.
+It is easily seen, as in the proof of Theorem 6, that d((π, τ), (ρ, σ)) ≥ d, if either π ̸= ρ or
+σ ̸= τ.
+3
+
+1
+2
+3 4 5 6
+7
+8 9 10 11 12 13 14 Pτi(14, 11)
+τ1
+0
+0
+0 0 0 0 13 14 0
+0
+0
+0
+0
+0
+47,851
+τ2
+0
+0 14 0 0 0
+0
+0 0
+0
+0
+0
+0 13
+36,250
+τ3
+0 13
+0 0 0 0
+0
+0 0
+0
+0
+0
+0 14
+19,227
+τ4 13 14
+0 0 0 0
+0
+0 0
+0
+0
+0
+0
+0
+19,227
+τ5
+0
+0
+0 0 0 0
+0
+0 0
+0
+0
+0 14 13
+19,227
+Table 3: (14, 2, 11)-array with 5 permutations τ1, . . . , τ5.
+Since the first 12 symbols in all τi
+are sorted, they are replaced by zeros.
+The last column contains lower bounds for
+Pτi(14, 11), i = 1, . . . , 5.
+For example, we saw the result P(14, 11) ≥ 96, 125 using Theorem 6, with a (14, 2, 11)-array with
+five permutations τi, i = 1, . . . , 5. We computed lower bounds for Pτi(14, 11), see the last column
+in Table 3. By Theorem 7, we obtain the improved lower bound of P(14, 11) ≥ �5
+i=1 Pτi(14, 11) ≥
+141, 782.
+2
+Bounds for P(n, m, d)
+There are
+n!
+(n−m)! permutations in Sn,m for finding P(n, m, d). When m is small, this is relatively
+small compared to the n! permutations to explore for finding P(n, d). Also, P(n, m, d) generalizes
+P(n, d) as P(n, d) = P(n, n, d). Finding exact values or bounds for P(n, m, d) is an interesting
+problem in its own right. Clearly, P(n, 1, d) = ⌈n/d⌉. In general, by Theorem 5
+P(n, m, d) ≥
+�n
+d
+�
+·
+�n − 1
+d
+�
+· · · · ·
+�n − m + 1
+d
+�
+.
+(1)
+We denote by ε the identity permutation (1, 2, . . . , n).
+Proposition 8. P(n, m, d) ≥ 2 if d ≤ mn−m(m+1)/2. The bound for d is tight for all n > m ≥ 1.
+Proof. Let π = (n, n−1, . . . , n−m+1, 1, 2 . . . , n−m). The bubble sort for π uses n−1 transpositions
+for symbol n, n − 2 transpositions for symbol n − 1, etc. Then d(ε, π) = (n − 1) + (n − 2) + · · · +
+(n − m) = nm − (1 + 2 + · · · + m) = mn − m(m + 1)/2.
+The bound is tight since for any permutation σ ̸= π, d(ε, σ) < mn − m(m + 1)/2.
+We improve the bound in Equation 1 for m = 2.
+Theorem 9. For any d ≥ 1,
+(a) P(n, 2, d) ≥ 3 if d ≤ n + ⌊n/3⌋ − 2.
+(b) P(n, 2, d) ≥ 5 if d ≤ n − 2.
+Proof. (a) Let τ1 = (n − 1, n, 1, 2, . . . , n − 2), τ2 = (1, . . . , x − 1, n − 1, x, . . . , n − 2, n) and τ3 =
+(1, . . . , x, n, x + 1, . . . , n − 1) where x = ⌊n/3⌋, see an example in Table 4. Transformation of τ1
+to τ2 requires n − 1 transpositions for symbol n − 1 and x − 1 transpositions for symbol n. Then
+d(τ1, τ2) = n+x−2 ≥ d. Similarly d(τ1, τ3) = (n−2)+x ≥ d, and d(τ2, τ3) = (n−x)+(n−x−2) =
+2n − 2x − 2 ≥ n + x − 2 ≥ d.
+4
+
+1 2 3 4 5 6 7 8 9
+τ1 8 9 1 2 3 4 5 6 7
+τ2 1 2 9 3 4 5 6 7 8
+τ3 1 2 3 8 4 5 6 7 9
+Table 4: P(9, 2, 10) ≥ 3.
+(b) Suppose n = 2k. Consider 5 permutations τi, i = 1, . . . , 5 where symbols n − 1 and n are
+placed at positions 1 and 2 for τ1, n − 1 and n for τ2, k and k + 1 for τ3, 1 and n for τ4, n and 1
+for τ5, see an example in Table 5. We show that d(τi, τj) ≥ n − 2 if 1 ≤ i < j ≤ 5. For all pairs
+i, j ∈ {1, 2, 4, 5} with i < j, transformation of τi to τj requires n − 2 transpositions for only one
+of two symbols n − 1 or n. Transformation of τ3 to any τi, i = 1, 2, 4 requires k − 1 transpositions
+for each symbol n − 1 and n. Transformation of τ3 to any τ5 requires k − 1 transpositions for each
+symbol n − 1 and n after transposition of n − 1 and n.
+1
+2
+3
+4
+5
+6
+7
+8
+9 10 11 12
+τ1 11 12
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+τ2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0 11 12
+τ3
+0
+0
+0
+0
+0 11 12
+0
+0
+0
+0
+0
+τ4 11
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0 12
+τ5 12
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0 11
+Table 5: P(12, 2, 10) ≥ 5. The first 10 symbols in all τi are in the sorted order and replaced by
+zeros.
+1
+2
+3
+4
+5
+6
+7
+8
+9 10 11 12 13
+τ1 12 13
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+τ2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0 12 13
+τ3
+0
+0
+0
+0
+0 13 12
+0
+0
+0
+0
+0
+0
+τ4 12
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0 13
+τ5 13
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0 12
+Table 6: An example for P(13, 2, 11) ≥ 5.
+Similarly, a (n, 2, n − 2)-array can be constructed for n = 2k + 1 where symbols n and n − 1 are
+placed at positions k and k + 1 for τ3, see an example in Table 6.
+We have constructed a program for computing P(n, m, d) for various values of n, m, and d. For
+each of the
+� n
+m
+�
+positions for m symbols out of n, and each of the possible m! orders of the m
+symbols, the program uses the random/Greedy strategy described earlier. That is, it chooses a
+specified number of random choices first and then tries adding all remaining possible permutations
+in increasing order. When m is small, the program finds solutions quickly. It allows one to compute
+P(15, 12), for example, without examining all 15! permutations of 15 symbols. That is, by Theorem
+6 one can first compute, for example, P(15, 3, 12), which as shown in Table 9 is at least 12, and
+then compute P(12, 12).
+As shown in Table 11 these are useful for obtaining improved lower bounds for P(n, d) when
+5
+
+the Kendall-τ distance d is close to n. We give lower bounds for P(n, m, d), for 8 ≤ d ≤ 15 and
+10 ≤ n ≤ 20 in Tables 7, 8, 9, and 10.
+n:m
+2
+3
+4
+5
+6
+10
+5
+14
+37
+113
+335
+11
+5
+16
+55
+186
+645
+12
+6
+21
+73
+285
+1145
+13
+6
+26
+99
+428
+1920
+14
+8
+31
+130
+625
+3117
+15
+8
+37
+172
+884
+4872
+16
+10
+45
+219
+1233
+7367
+17
+10
+52
+278
+1676
+10828
+18
+13
+61
+344
+2227
+15567
+19
+13
+71
+426
+2939
+21862
+20
+15
+80
+517
+3805
+30196
+n:m
+2
+3
+4
+5
+6
+10
+3
+9
+24
+63
+162
+11
+5
+15
+34
+99
+301
+12
+5
+16
+46
+149
+523
+13
+6
+18
+59
+219
+861
+14
+6
+22
+78
+315
+1383
+15
+7
+26
+100
+445
+2119
+16
+8
+31
+128
+610
+3165
+17
+8
+36
+162
+824
+4613
+18
+10
+42
+201
+1097
+6589
+19
+10
+49
+244
+1427
+9179
+20
+12
+55
+292
+1827
+12581
+Table 7: Lower bounds for P(n, m, 8) (left) and P(n, m, 9) (right).
+n : m
+2
+3
+4
+5
+6
+10
+3
+7
+19
+48
+125
+11
+5
+10
+27
+76
+226
+12
+5
+13
+37
+116
+394
+13
+6
+16
+50
+167
+644
+14
+6
+18
+64
+241
+1011
+15
+6
+21
+83
+342
+1570
+16
+6
+25
+103
+467
+2337
+17
+8
+30
+129
+629
+2239
+18
+8
+35
+158
+829
+3185
+19
+10
+40
+192
+1084
+4405
+20
+10
+46
+233
+4184
+6017
+n : m
+2
+3
+4
+5
+6
+10
+3
+6
+13
+27
+73
+11
+3
+7
+16
+41
+128
+12
+3
+10
+22
+61
+214
+13
+5
+11
+31
+96
+344
+14
+5
+13
+37
+120
+539
+15
+5
+17
+55
+163
+810
+16
+6
+20
+70
+220
+1193
+17
+6
+23
+86
+366
+1716
+18
+7
+26
+106
+472
+2413
+19
+8
+31
+127
+618
+3362
+20
+8
+35
+151
+789
+4571
+Table 8: Lower bounds for P(n, m, 10) (left) and P(n, m, 11) (right).
+3
+Improved Lower Bounds by Theorems 5, 6, and 7.
+Each of the improved lower bounds given in Table 11 is explained in this section. Many of the
+computations described took weeks on Apple MacBook Air computers with an M1 or M2 processor.
+• By Theorem 7, P(12, 5) ≥ P(12, 5; 2) + P(12, 5; 7) + P(12, 5; 12) ≥ 318, 641 + 334, 200 +
+246, 968 = 899, 809.
+• By Theorem 7, P(12, 7) ≥ P(12, 7; 3) + P(12, 7; 10) ≥ 2 · 64, 649 = 129, 298.
+• By Theorem 7, P(12, 8) ≥ P(12, 8; 3) + P(12, 8; 11) ≥ 44, 042 + 41049 = 85, 091.
+• By Theorem 7, P(13, 9) ≥ P(13, 9; 3) + P(13, 9; 12) ≥ 124, 047 + 112, 717 = 236, 764
+6
+
+n : m
+2
+3
+4
+5
+6
+10
+2
+6
+13
+26
+58
+11
+3
+7
+17
+40
+101
+12
+3
+9
+23
+59
+168
+13
+3
+10
+30
+84
+273
+14
+5
+13
+37
+117
+420
+15
+5
+16
+45
+159
+622
+16
+5
+17
+58
+216
+919
+17
+6
+20
+72
+287
+1323
+18
+6
+22
+87
+375
+1859
+19
+6
+25
+103
+485
+2580
+20
+8
+30
+125
+620
+3503
+n : m
+2
+3
+4
+5
+6
+10
+2
+4
+10
+20
+37
+11
+2
+6
+13
+28
+63
+12
+3
+7
+16
+40
+103
+13
+3
+9
+22
+56
+163
+14
+3
+10
+27
+79
+247
+15
+5
+12
+35
+106
+370
+16
+5
+15
+44
+141
+533
+17
+5
+16
+52
+181
+757
+18
+6
+18
+63
+242
+1058
+19
+6
+20
+73
+308
+1447
+20
+6
+23
+90
+390
+1965
+Table 9: Lower bounds for P(n, m, 12) (left) and P(n, m, 13) (right).
+n : m
+2
+3
+4
+5
+6
+10
+2
+4
+10
+16
+30
+11
+2
+4
+11
+23
+51
+12
+3
+6
+15
+34
+85
+13
+3
+7
+18
+48
+133
+14
+3
+9
+24
+65
+203
+15
+3
+10
+30
+88
+298
+16
+5
+13
+38
+118
+431
+17
+5
+15
+46
+153
+609
+18
+5
+16
+54
+197
+844
+19
+6
+18
+63
+254
+1163
+20
+6
+20
+75
+323
+1568
+n : m
+2
+3
+4
+5
+6
+10
+2
+4
+6
+12
+19
+11
+2
+4
+10
+20
+31
+12
+2
+5
+12
+21
+48
+13
+3
+6
+15
+30
+72
+14
+3
+7
+16
+40
+107
+15
+3
+9
+23
+52
+154
+16
+3
+10
+29
+84
+221
+17
+5
+12
+35
+109
+385
+18
+5
+14
+41
+138
+530
+19
+5
+16
+41
+174
+720
+20
+5
+17
+46
+220
+961
+Table 10: Lower bounds for P(n, m, 14) (left) and P(n, m, 15) (right).
+• By Theorem 7, P(14, 9) ≥ P(14, 9; 2, 6, 8)+P(14, 9; 2, 3, 5)+P(14, 9; 1, 5, 6)+P(14, 9; 1, 6, 8)+
+P(14, 9; 4, 7, 8)+P(14, 9; 4, 9, 10)+P(12, 9; 8)+P(12, 9; 9)+P(13, 9; 4, 5)+P(12, 9; 3)+P(14, 9; 3, 5, 7)+
+P(14, 9; 3, 5, 14)+P(14, 9; 2, 4, 14)+P(14, 9; 2, 7, 9)+P(13, 9; 4, 5)+P(12, 9; 7)+2∗P(12, 9; 4)+
+P(13, 9; 3, 5)+P(13, 9; 2, 3)+P(12, 9; 3)+P(13, 9; 3, 4)+ ≥ 51, 871+26, 347+19, 878+31, 130+
+39, 622 + 42, 132 + 18, 649 + 18, 397 = 19, 914 + 17, 294 + 48, 029 + 28, 367 + 25, 367 + 52, 958 +
+19, 915 + 18, 807 + 36, 794 + 28, 348 + 16, 073 + 17, 294 + 18, 542 = 575, 728
+• By Theorem 7, P(13, 10) ≥ P(13, 10; 2) + P(13, 10; 12) ≥ 2 ∗ 79, 104 = 158, 208.
+• By Theorem 7, P(14, 10) ≥ P(14, 10; 5, 14)+P(14, 10; 8, 14)+P(14, 10; 6, 7)+P(14, 10; 1, 11)+
+P(14, 10; 1, 12)+P(14, 10; 1, 2) ≥ 94, 643+95, 052+102, 965+93, 157+89, 021+50, 649 ≥ 525, 427
+• By Theorem 7, P(13, 11) ≥ P(13, 11; 2) + P(13, 11; 13) ≥ 31, 809 + 19, 227 = 51, 046.
+• By Theorem 7, P(14, 11) ≥ P(14, 11; 7, 8)+P(14, 11; 14, 3)+P(14, 11; 13, 14)+P(14, 11; 1, 2)+
+P(14, 11; 1, 14) ≥ 47, 851 + 36, 250 + 3 ∗ 19, 227 = 141, 782.
+• By Theorem 7, P(15, 11) ≥ P(15, 11; 1, 7, 9) + P(15, 11; 9, 10, 15) + P(15, 11; 11, 14, 15) +
+7
+
+n:d
+5
+7
+8
+9
+10
+11
+12
+13
+14
+15
+12 899,809 129,298 85,091
+13
+236,764 158,208
+51,046
+29,859
+14,158
+10,756
+5,527
+14
+595,728 525,427
+141,782
+100,813
+52,565
+41,673
+15,674
+15
+1,049,633
+524,817
+105,130 83,346
+37,104
+16
+2,099,266 1,049,634 267,828 173,432 74,208
+17
+244,051
+Table 11: Improved lower bounds using Theorems 5, 6 and 7. Blanks indicate other methods have
+the best lower bounds known e.g. [11] or, for n=12, the best lower bounds are in Table
+2.
+P(15, 11; 8, 9, 11)+P(15, 11; 6, 10, 15)+P(15, 11; 5, 7, 13)+P(15, 11; 5, 6, 15)+P(15, 11; 4, 12, 14)+
+P(15, 11; 4, 5, 6)+P(15, 11; 3, 14, 15)+P(15, 11; 2, 7, 11)+P(15, 11; 1, 13, 15)+P(15, 11; 1, 2, 3)+
+P(15, 11; 1, 2, 15)+P(15, 11; 1, 2, 15)+P(15, 11; 1, 2, 13)+P(15, 11; 1, 9, 11) ≥ 70, 509+47, 069+
+36, 430+93, 986+85, 010+138, 475+47, 027+107, 707+45, 837+145, 804+3∗19227+31, 861+
+69, 377 ≥ 1, 049, 633
+• By Theorem 4, P(16, 11) ≥ 2 ∗ P(15, 11) ≥ 2 ∗ 1, 049, 633 = 2, 099, 266
+• By Theorem 7, P(13, 12) ≥ P(13, 12; 7) ≥ 29, 859.
+• By Theorem 7, P(14, 12) ≥ P(14, 12; 7, 8)+P(14, 12; 13, 14)+P(14, 12; 14, 2)+P(14, 12; 1, 14)+
+P(14, 12; 1, 2) ≥ 35, 709 + 13, 935 + 23, 299 + 19, 227 + 19, 227 = 100, 813.
+• By Theorem 3, P(15, 12) ≥ 1
+2P(15, 11) ≥ 524, 817.
+• By Theorem 4, P(16, 12) ≥ 2 ∗ P(15, 12) ≥ 1, 049, 634
+• By Theorem 7, P(13, 13) ≥ P(13, 13; 7) ≥ 14, 158.
+• By Theorem 7 P(14, 13) ≥ P(14, 13; 7, 13) + P(14, 13; 6, 14) + P(14, 13; 3, 4) ≥ 23, 388 +
+14, 073 + 15, 104 ≥ 52, 565.
+• By Theorem 5, P(15, 13) ≥ 2 ∗ P(14, 13) ≥ 2 ∗ 52, 565 = 105, 130.
+• By Theorem 6, P(16, 13) ≥ P(12, 13) ∗ P(16, 4, 13) ≥ 6, 087 ∗ 44 = 267, 828.
+• By Theorem 7, P(13, 14) ≥ P(13, 14; 7) ≥ 10, 756.
+• By Theorem 7, P(14, 14) ≥ P(14, 14; 1, 3) + P(14, 14; 4, 14) + P(14, 14; 6, 11) ≥ 8, 036 +
+10, 060 + 23, 577 = 41, 673
+• By Theorem 5, P(15, 14) ≥ 2 ∗ P(14, 14) ≥ 2 ∗ 41, 673 = 83, 346.
+• By Theorem 6, P(16, 14) ≥ P(12, 14) ∗ P(16, 4, 14) ≥ 4, 564 ∗ 38 = 173, 432.
+• By Theorem 7, P(13, 15) ≥ P(13, 15; 7) ≥ 5, 527.
+• By Theorem 7, P(14, 15) ≥ P(14, 15; 6, 14) + P(14, 15; 14, 6) + P(14, 15; 2, 3) ≥ 5, 493 +
+5, 493 + 4, 688 = 15, 674.
+• By Theorem 7, P(15, 15) ≥ P(15, 15; 3, 4, 7, 8) + 3 ∗ P(11, 15) + P(15, 15; 4, 5, 6, 7) + 3 ∗
+P(14, 15; 6, 7, 8) + P(13, 15 : 2, 10) + P(15, 15; 2, 3, 4, 13) + P(3, 5, 7, 11) + P(15, 15, 2, 4, 10, 11) +
+2P ∗ (13, 15; 2, 3) + P(12, 15; 3) + 4 ∗ P(12, 15; 2) + P(14, 15; 3, 4, 5) ≥ 4, 279 + 3 ∗ 409 + 1, 787 +
+3 ∗ 1, 848 + 1, 738 + 1, 964 + 7, 798 + 5, 773 + 2 ∗ 879 + 895 + 4 ∗ 743 + 1, 369 ≥ 37, 104.
+8
+
+• By Theorem 5, P(16, 15) ≥ 2 ∗ P(16, 15) ≥ 74, 208.
+• By Theorem 6, P(17, 15) ≥ P(12, 15) ∗ P(17, 5, 15) ≥ 2, 239 ∗ 109 = 244, 051.
+4
+Automorphism Lower Bounds
+It is known that for a permutation π(x) : Fq → Fq, where Fq denotes a finite field of order q, the
+operations of multiplying by a non-zero constant a, adding a constant c, and adding to the argument
+a constant b, each yield another permutation on Fq. That is, aπ(x + b) + c, for all non-zero a and
+all b, c ∈ Fq, is again a permutation. We use this to search for sets of permutations at specified
+Kendall-τ distance d. That is, the search can be done for a set of representative permutations and
+expanded into a full set of permutations using operations on the representatives. Our program
+verifies that the full set of permutations has the stipulated Kendall-τ distance.
+Example.
+Use the operation π(x) + c on the following 17 representatives.
+This gives 102
+permutations for P(6, 3).
+0 1 2 3 5 4
+0 1 2 4 5 3
+0 1 3 5 4 2
+0 1 5 4 2 3
+0 2 3 4 1 5
+0 2 4 5 1 3
+0 2 5 3 4 1
+0 3 1 4 2 5
+0 3 2 5 1 4
+0 3 4 2 5 1
+0 3 5 4 1 2
+0 4 1 5 3 2
+0 4 2 1 3 5
+0 4 5 3 2 1
+0 5 2 1 3 4
+0 5 3 1 2 4
+0 5 4 2 1 3
+Example. Use the operations aπ(x) + c on the following 14 representatives. This gives 1, 008
+permutations for P(9, 7).
+0 1 2 4 8 3 7 5 6
+0 1 2 7 8 5 3 4 6
+0 1 3 4 7 2 8 6 5
+0 1 3 8 2 6 7 4 5
+0 1 3 8 4 6 5 7 2
+0 1 4 5 6 7 3 8 2
+0 1 4 5 8 2 7 6 3
+0 1 6 2 3 4 7 8 5
+0 1 6 2 8 7 5 4 3
+0 1 6 4 5 2 3 8 7
+0 1 6 7 3 4 8 5 2
+0 1 7 2 4 6 8 5 3
+0 1 7 4 8 3 5 2 6
+0 1 8 5 7 4 6 3 2
+Example.
+Use the operations aπ(x) + c on the following 8 representatives.
+This gives 576
+permutations for P(9, 8).
+0 1 2 3 8 4 6 5 7
+0 1 2 5 8 6 3 7 4
+0 1 4 5 2 8 6 7 3
+0 1 5 3 2 4 6 8 7
+0 1 5 6 4 8 3 7 2
+0 1 6 4 7 2 5 8 3
+0 1 6 7 3 2 8 5 4
+0 1 8 3 6 5 7 2 4
+Example. Use the operations aπ(x) + c on the following four representatives. This gives 288
+permutations for P(9, 9).
+0 1 2 6 5 8 7 4 3
+0 1 3 8 4 5 2 6 7
+0 1 4 6 5 3 7 2 8
+0 1 5 2 4 7 3 6 8
+Example.
+Use the operations π(x) + c on the following 12 representatives.
+This gives 84
+permutations for P(7, 6).
+0 1 3 6 5 4 2
+0 1 4 2 3 6 5
+0 1 6 2 5 4 3
+0 2 3 4 1 5 6
+0 2 3 6 5 1 4
+0 3 4 6 1 2 5
+0 3 5 4 1 2 6
+0 4 5 6 3 1 2
+0 5 2 4 3 6 1
+0 5 3 6 1 2 4
+0 6 3 5 4 2 1
+0 6 4 2 1 3 5
+Example. Use the operation aπ(x) + c on 8 permutations. This gives 448 permutations for
+P(8, 6).
+9
+
+0 1 7 4 5 6 2 3
+0 2 1 5 3 4 6 7
+0 2 6 4 7 3 1 5
+0 3 7 5 4 2 1 6
+0 5 4 6 7 1 2 3
+0 7 3 1 2 6 5 4
+0 7 5 4 3 6 1 2
+0 7 6 4 2 1 3 5
+Example. Use the operation aπ(x) + c on 67 permutation representatives. This gives 3,752
+permutations for P(8, 3).
+0 1 2 3 4 5 6 7
+0 1 2 5 3 6 7 4
+0 1 3 5 7 2 6 4
+0 1 5 4 3 6 2 7
+0 1 6 2 7 3 4 5
+0 1 6 3 4 2 7 5
+0 1 6 7 4 5 2 3
+0 1 7 3 2 5 6 4
+0 1 7 5 3 2 4 6
+0 1 7 5 6 3 4 2
+0 2 3 5 1 4 7 6
+0 2 3 5 7 6 4 1
+0 2 3 6 5 4 7 1
+0 2 4 1 6 5 7 3
+0 2 4 5 6 3 1 7
+0 2 5 1 7 4 3 6
+0 2 5 3 4 6 7 1
+0 2 5 4 3 1 6 7
+0 2 5 6 4 1 7 3
+0 2 6 4 3 5 1 7
+0 2 6 4 7 1 5 3
+0 3 1 5 4 7 2 6
+0 3 2 4 1 7 6 5
+0 3 2 5 4 7 1 6
+0 3 2 6 1 4 5 7
+0 3 6 2 4 5 1 7
+0 3 7 4 5 6 2 1
+0 3 7 5 4 2 1 6
+0 4 1 6 2 3 5 7
+0 4 2 7 3 1 5 6
+0 4 2 7 5 6 1 3
+0 4 5 6 2 1 3 7
+0 4 6 1 7 2 3 5
+0 4 6 2 5 3 7 1
+0 4 6 2 7 1 5 3
+0 4 7 5 2 3 1 6
+0 4 7 6 3 5 2 1
+0 5 1 6 7 4 3 2
+0 5 1 7 3 6 2 4
+0 5 2 1 6 3 7 4
+0 5 2 3 6 4 1 7
+0 5 2 6 4 3 7 1
+0 5 3 1 4 6 2 7
+0 5 3 2 6 1 7 4
+0 5 3 4 1 2 7 6
+0 5 3 7 6 1 4 2
+0 5 4 6 2 7 1 3
+0 5 4 6 3 1 2 7
+0 5 6 3 1 2 7 4
+0 5 6 3 7 4 1 2
+0 5 7 6 4 3 1 2
+0 6 1 5 2 3 4 7
+0 6 2 4 3 7 5 1
+0 6 3 1 7 4 5 2
+0 6 3 7 2 4 5 1
+0 6 4 3 5 7 1 2
+0 6 5 1 7 3 2 4
+0 6 7 1 3 5 4 2
+0 6 7 5 3 2 1 4
+0 7 1 2 3 4 5 6
+0 7 1 3 5 4 6 2
+0 7 1 4 3 6 2 5
+0 7 3 4 2 1 5 6
+0 7 3 6 1 4 2 5
+0 7 4 6 3 1 2 5
+0 7 4 6 5 2 3 1
+0 7 5 1 2 3 6 4
+Example.
+Use the operation aπ(x) + c on 12 permutation representatives.
+This gives 672
+permutations for P(8, 5).
+0 2 3 6 5 4 7 1
+0 2 4 3 1 5 6 7
+0 3 2 1 6 4 7 5
+0 3 5 1 6 2 7 4
+0 5 7 2 4 6 1 3
+0 6 3 4 5 2 1 7
+0 6 3 7 1 5 2 4
+0 6 5 4 7 3 1 2
+0 7 1 5 4 6 2 3
+0 7 3 6 4 2 1 5
+0 7 4 1 2 6 5 3
+0 7 5 6 4 1 3 2
+Example. Use the operation aπ(x) + c on 40 permutation representatives. This gives 2,242
+permutations for P(8, 4).
+0 1 4 5 7 6 3 2
+0 1 7 3 2 5 6 4
+0 2 1 3 7 4 5 6
+0 2 1 5 7 4 6 3
+0 2 1 6 7 5 4 3
+0 2 3 6 1 5 4 7
+0 2 4 3 5 6 1 7
+0 2 5 3 7 4 6 1
+0 2 7 1 4 5 3 6
+0 2 7 3 1 4 6 5
+0 2 7 3 6 5 1 4
+0 2 7 6 1 4 5 3
+0 3 2 1 5 7 4 6
+0 3 5 6 4 7 1 2
+0 3 5 7 6 1 2 4
+0 3 6 2 5 1 7 4
+0 4 1 6 2 3 5 7
+0 4 1 7 6 2 3 5
+0 4 2 1 5 6 3 7
+0 4 2 5 7 6 3 1
+0 4 2 7 1 5 6 3
+0 4 3 1 7 5 6 2
+0 4 3 5 6 1 7 2
+0 5 2 1 6 3 7 4
+0 5 3 2 6 1 7 4
+0 5 3 2 7 1 4 6
+0 5 4 2 1 3 6 7
+0 5 4 7 6 2 3 1
+0 5 6 2 1 7 4 3
+0 5 6 4 1 3 2 7
+0 5 7 1 6 4 2 3
+0 5 7 3 2 4 6 1
+0 6 1 2 4 3 5 7
+0 6 7 2 4 3 1 5
+0 7 1 2 6 3 5 4
+0 7 2 5 1 4 6 3
+0 7 2 5 3 6 4 1
+0 7 4 3 1 5 2 6
+0 7 5 1 4 2 3 6
+0 7 5 6 2 1 4 3
+Example. Use the operation aπ(x) + c on 3 permutation representatives. This gives 168 permu-
+tations for P(8, 7).
+0 5 3 1 4 6 2 7
+0 6 1 3 2 5 7 4
+0 7 3 1 2 6 5 4
+Example. Use the operation π(x) + c on the following 48 permutations. This gives 336 permu-
+tations for P(7, 4).
+10
+
+0 1 2 4 3 6 5
+0 1 2 5 4 6 3
+0 1 3 2 6 5 4
+0 1 3 4 2 5 6
+0 1 4 5 6 2 3
+0 1 5 3 6 2 4
+0 1 6 2 3 4 5
+0 1 6 5 2 4 3
+0 2 1 3 5 4 6
+0 2 3 4 1 6 5
+0 2 3 6 5 4 1
+0 2 4 1 5 3 6
+0 2 4 6 5 1 3
+0 2 5 1 6 3 4
+0 2 5 3 4 1 6
+0 2 5 6 4 3 1
+0 2 6 1 4 3 5
+0 3 1 5 2 4 6
+0 3 1 6 5 4 2
+0 3 2 5 1 6 4
+0 3 2 6 1 4 5
+0 3 4 1 6 2 5
+0 3 4 2 5 1 6
+0 3 4 5 6 1 2
+0 3 6 5 4 2 1
+0 4 2 1 6 3 5
+0 4 2 5 3 6 1
+0 4 3 1 5 2 6
+0 4 3 6 2 5 1
+0 4 5 1 2 3 6
+0 4 6 1 3 2 5
+0 4 6 5 1 2 3
+0 5 1 2 3 4 6
+0 5 2 4 1 6 3
+0 5 3 1 4 6 2
+0 5 3 6 2 1 4
+0 5 4 1 6 3 2
+0 5 4 3 6 2 1
+0 5 6 1 4 2 3
+0 6 1 4 2 5 3
+0 6 1 5 3 4 2
+0 6 2 5 3 1 4
+0 6 3 1 4 2 5
+0 6 3 2 4 5 1
+0 6 3 5 1 2 4
+0 6 4 2 3 1 5
+0 6 4 5 3 2 1
+0 6 5 2 4 1 3
+Example. Use the operation π(x) + c on the following 18 permutations. This gives 126 permu-
+tations for P(7, 5).
+0 1 4 2 5 3 6
+0 1 4 6 3 2 5
+0 1 5 2 6 4 3
+0 2 1 3 5 4 6
+0 2 4 5 6 3 1
+0 2 6 4 1 3 5
+0 3 1 5 6 4 2
+0 3 2 4 5 1 6
+0 3 2 6 1 4 5
+0 3 5 4 6 2 1
+0 4 3 1 5 2 6
+0 4 3 6 2 1 5
+0 4 5 1 6 3 2
+0 5 1 3 4 2 6
+0 5 3 2 6 1 4
+0 6 1 2 5 3 4
+0 6 5 2 4 1 3
+0 6 5 3 4 1 2
+5
+Patterns for P(n, m, d)
+In this section, let us, for convenience, describe general patterns for strings (permutations) in
+P(n, 2, d) and P(n, 3, d), by replacing the symbols [1 . . . n−2] ([1 . . . n−3], respectively), which are
+in order, by blank symbols, i.e. ’-’.
+For example, for P(5, 2, 3), we have the set
+{ 4 5 - - - ,
+- 5 4 - - ,
+- - 4 5 - ,
+- - - 5 4 ,
+4 - - - 5 , 5 - - - 4 }.
+It is easy to verify that the Kendall-τ distance between any two strings in this set is at least 3.
+This set agrees with that found by our program, namely P(5, 2, 3) ≥ 6.
+Also, for P(10, 2, 3), we have the set
+{ 9 10 - - - - - - - - ,
+- 10 9 - - - - - - -,
+- - 9 10 - - - - - - ,
+- - - 10 9 - - - - -,
+- - - - 9 10 - - - -,
+- - - - - 10 9 - - -,
+- - - - - - 9 10 - -,
+- - - - - - - 10 9 -,
+- - - - - - - - 9 10,
+9 - - - 10 - - - - -,
+10 - - - 9 - - - - -,
+- - 9 - - - 10 - - -,
+- - 10 - - - 9 - - -
+- - - 9 - - - - 10 -,
+- - - 10 - - - - 9 -,
+- - - - - 9 - - - 10,
+- - - - - 10 - - - - 9,
+9 - - - - - 10 - - -,
+10 - - - - - 9 - - -,
+- 9 - - - - - - - 10,
+- 10 - - - - - - - 9 }.
+It is easy to verify that the Kendall-τ distance between any two strings in this set is at least 3.
+This set agrees with that found by our program, namely P(10, 2, 3) ≥ 21.
+These examples show that sets of strings that form a (n, 2, 3)-array contain easily recognized
+patterns. It is an interesting open question if such patterns can be determined for other choices of
+n, m, and d.
+Along these lines, for d = 3, consider π1(a, b, c) = . . . , n − 1, . . . , n, . . . and π2(a, b, c) =
+. . . , n, . . . , n − 1, . . ., where a, b, c denote the number of symbols in the 3 gaps represented by the
+“. . .”. We will use π1(a, b, c) for a = 0, 2, 4, . . . and b = 0, 3, 6, . . ., and π2(a, b, c) for a = 1, 3, 5, . . .
+11
+
+and b = 0, 3, 6, . . ., for each choice of a and b for which the resulting string has length at most n.
+Using π1(a, b, c) and π2(a, b, c), it can be observed that P(n, 2, 3) ≥ n(n+1)
+6
+, for n ̸≡ 1 mod 3
+and P(n, 2, 3) ≥ (n+2)(n−1)
+6
+for n ≡ 1 mod 3. Similarly, for Kendall-τ distance 4 and for n = 2k +1,
+use π1(a, b, c) for a = 0, 2, 4, . . . and b = 0, 4, 8, . . .; π2(a, b, c) for a = 0, 2, 4, . . . and b = 3, 7, 11, . . ..
+Using these patterns, it can be observed that P(4k+1, 2, 4) ≥ 2k2+k for k ≥ 1 and P(4k+3, 2, 4) ≥
+2k2 + 3k + 1 for k ≥ 0.
+6
+Conclusions and Open Questions
+Theorems 6 and 7 improve many lower bounds.
+All of the bounds shown in Tables 1, 2, and
+11 are improvements on previous results. The techniques described can be used to obtain other
+improvements, with sufficient time. Many of our computations required weeks.
+Our work on good patterns for (n, m, d)-arrays is continuing. We conjecture that (n, m, d)-
+arrays can be used to compute improved lower bounds for P(n, d), for all n, and for d close to
+n.
+References
+[1] A. Abdollahi, J. Bagherian, F. Jafari, M. Khatami, F. Parvaresh, and R. Sobhani.
+New
+bounds on the size of permutation codes with minimum Kendall τ-distance of three. arXiv,
+abs/2206.10193, 2022.
+[2] A. Barg and A. Mazumdar. Codes in permutations and error correction for rank modulation.
+IEEE Transactions on Information Theory, 56(7):3158–3165, 2010.
+[3] S. Bereg, A. Levy, and I. H. Sudborough.
+Constructing permutation arrays from groups.
+Designs, Codes and Cryptography, 86(5):1095–1111, 2018.
+[4] S. Bereg, Z. Miller, L. G. Mojica, L. Morales, and I. H. Sudborough. New lower bounds for
+permutation arrays using contraction. Designs, Codes and Cryptography, 87:2105–2128, 2019.
+[5] S. Buzaglo and T. Etzion. Bounds on the size of permutation codes with the Kendall tau
+metric. IEEE Trans. on Inform. Theory, 61(6):3241–3250, 2015.
+[6] W. Chu, C. J. Colbourn, and P. Dukes. Constructions for permutation codes in powerline
+communications. Designs, Codes and Cryptography, 2004.
+[7] M. M. Deza and T. Huang. Metrics on permutations, a survey. J. Comb. Inf. System Sci.,
+23:173–185, 1998.
+[8] A. Jiang, M. Schwartz, and J. Bruck.
+Correcting charge-constrained errors in the rank-
+modulation scheme. IEEE Transactions on Information Theory, 56(5):2112–2120, 2010.
+[9] T. Kløve, T.-T. Lin, S.-C. Tsai, and W.-G. Tzeng. Permutation arrays under the Chebyshev
+distance. IEEE Trans. on Info. Theory, 56(6):2611 – 2617, 2010.
+12
+
+[10] S. Vijayakumaran. Largest permutation codes with Kendall τ-metric in S4 and S5. IEEE
+Communications Letters, 20(10):1912–1915, 2016.
+[11] X. Wang, Y. Zhang, Y. Yang, and G. Ge. New bounds of permutation codes under Hamming
+metric and Kendall’s τ-metric. Des. Codes Cryptography, 85(3):533–545, 2017.
+13
+

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+Optical fingerprints of the electronic band reconstruction in van der Waals magnetic
+materials
+M. Corasaniti†,1 R. Yang†,1 Y. Liu‡,2 C. Petrovic,2 and L. Degiorgi∗1
+1Laboratorium f¨ur Festk¨orperphysik, ETH - Z¨urich, 8093 Z¨urich, Switzerland
+2Condensed Matter Physics and Materials Science Department,
+Brookhaven National Laboratory, Upton NY 11973, USA
+(Dated: January 13, 2023)
+We report a broadband study of the charge dynamics in the van der Waals (vdW) magnetic
+materials 2H-MxTaS2 (M = Mn and Co), which span the onset of both long-range antiferromagnetic
+(AFM) and ferromagnetic (FM) order, depending on the intercalation M and its concentration x.
+We discover a spectral weight (SW) shift from high to low energy scales for FM compositions, while
+reversely SW is removed from low towards high spectral energies for AFM compounds. This maps
+the related reconstruction of the electronic band structure along the crossover from the FM to AFM
+order, which restores an occupation balance in the density of states between spin majority and
+minority bands of the intercalated 3d elements.
+Understanding the physical mechanism as well as
+functionalities of van der Waals (vdW) heterostructures
+and electronic/spintronic devices is at present a cen-
+tral topic of the ongoing solid state physics research
+activities [1].
+In this context, Fe3GeTe2 and MnSe2
+are prototype examples [2, 3] of two-dimensional vdW
+magnets.
+The discovery of such a long-range mag-
+netism lately boosted the several decades long investiga-
+tion of the (mostly non-magnetic) vdW transition-metal
+dichalcogenides (TMDCs) with appropriate magnetic in-
+tercalations.
+This enlarges the fundamental studies of
+low-dimensional magnetism and provides a platform for
+questing the nature of the critical behaviour of the spin
+interactions, ranging from Heisenberg and XY to Ising
+type [1].
+The TMDC materials of general formula TX2 (e.g.,
+with T = Nb, Ta and X = Se, S) consist of layered-
+like sandwiches held together by relatively weak forces
+across the vdW gaps, within which intercalation may oc-
+cur. The latter is notably accompanied by charge transfer
+between the intercalated species and the host layer but
+is not supposed to change the local bonding within the
+sandwich layers, so that the electronic properties of the
+host TMDC materials could be treated within the rigid-
+band model and consequently be fine-tuned depending
+on the effective d-bands filling. Such a controllable de-
+gree of band filling seems to be a feature quite unique to
+low-dimensional structures [4, 5].
+In this work, we focus our attention on the 2H-MxTaS2
+compounds with M = Co and Mn. While the possibil-
+ity to intercalate TMDCs with magnetic 3d elements M,
+exhibiting diverse magnetic properties, is known since
+the eighties, their physical properties have been only ad-
+dressed recently [6–8].
+Any intercalation with atoms,
+that tend to preserve local moment when embedded in a
+metallic host, induces long range magnetic order [9, 10].
+This is different from the simple charge transfer that oc-
+curs upon intercalation of non-magnetic atom such as Pd
+or Li, and which commonly perturbs the charge-density-
+wave (CDW) and enhances superconductivity [11, 12].
+The pristine 2H-TaS2 compound is in fact a familiar
+CDW material [13] and the peculiarities with respect to
+its broken-symmetry ground state are amply documented
+in the literature. The (magnetic) Co and Mn intercala-
+tion (Fig. S1 in Supplemental Material (SM) [14]), be-
+sides removing the CDW transition, expands the vdW
+gap of the 2H-MxTaS2 crystals along the c-axis (i.e., or-
+thogonal to the layers) and should result in an electronic
+doping via hybridization with atoms around the vdW
+gap [8]. This induces a ferromagnetic (FM) state with
+an easy-plane anisotropy in 2H-MnxTaS2 [7, 15–19]. For
+Co-intercalation, ferromagnetism with strong uniaxial
+anisotropy in 2H-Co0.22TaS2 [6] and a three-dimensional
+antiferromagnetic (AFM) state in 2H-Co0.34TaS2 [20]
+have been firmly established. The evolving FM to AFM
+transition promoted by the Co-intercalation and the re-
+lated sign-change of the ordinary Hall coefficient hint
+to an electronic multi-band occurrence and its recon-
+struction based on the magnetic ground state [8]. The
+temperature-dependent electrical resistivity (ρ(T)) dis-
+plays an overall metallic behaviour for all samples (Fig.
+S2 in SM [14]). A clear kink in ρ(T), a weak anomaly
+in thermal conductivity, as well as a slope change in
+thermopower were yet observed at the magnetic transi-
+tions for 2H-Mn0.28TaS2 (TC ∼ 82 K) and 2H-Co0.34TaS2
+(TN ∼ 36 K), albeit weaker for crystals with lower con-
+centration x [8].
+From a spectroscopic point of view, these materials
+are still not comprehensively scrutinised. Here, we inves-
+tigate the temperature (T) dependence of the (in-plane)
+absorption spectrum [21] over a broad spectral range of
+both Co-and Mn-intercalated 2H-MxTaS2. 2H-CoxTaS2
+harbour a distinct and reverse spectral weight (SW)
+reshuffling with regard to the FM and AFM transition at
+TC and TN, respectively. The overall optical response of
+the FM Co-intercalated materials (i.e., x ≤ 0.22) copies
+with the data collected on the Mn compositions, which all
+refer to a FM state. We supply arguments, which favour
+a reconstruction of the electronic band structure upon
+crossing over from the FM to AFM order with increas-
+Typeset by REVTEX
+arXiv:2301.04898v1  [cond-mat.mtrl-sci]  12 Jan 2023
+
+2
+ing Co-concentration, bearing testimony to the progres-
+sive lifting of the occupation imbalance in the density of
+states between spin majority and minority bands of the
+intercalated 3d elements.
+We launch first the survey about the T dependence of
+the real part (σ1(ω)) of the optical conductivity, shown in
+Figs. 1(a-c) for three selected Mn- and Co-concentration
+in the energy interval spanning the far- (FIR), mid-
+(MIR) and near- (NIR) infrared up to the visible spec-
+tral ranges at 5 or 10 and 300 K. We refer to SM in
+Ref. 14 for further details and additional data for other
+concentrations. The selected compositions are particu-
+larly pertinent, since they encompass both the FM and
+AFM magnetic phase transitions for the Co-intercalation
+and address a representative Mn compound towards its
+FM one (Table I in SM [14]). There is a metallic Drude-
+like component, which gets narrow as well as robust upon
+lowering T (i.e., it generally gains SW, see below). It
+merges into a broad and T-dependent MIR absorption
+between 1000 and 3000 cm−1 [14].
+In order to focus the discussion on the impact of
+the magnetic phase transition on the electronic proper-
+ties, we propose their phenomenological Drude-Lorentz
+fit [14], which is singled out for the spectra at 5 or 10
+and 300 K in Figs.
+1(a-c).
+The chosen layout of the
+collected data allows to emphasise the SW redistribu-
+tion and its evolution as a function of T among the
+Lorentz (Li, i = 1 to 9) harmonic oscillators (HO). In
+general, SW of the optical conductivity corresponds to
+its integral SW(T) = Z0
+π2
+� ω2
+ω1 σ1(ω′; T)dω′, expressed in
+units of cm−2 (Z0 = 376.73 Ω, being the impedance of
+free space) [21]. ωi (i = 1 and 2) define the energy in-
+terval, relevant for the SW estimation. Ahead, we al-
+ternatively propose to identify specific energy intervals
+via the phenomenological fit components, for which the
+related SW corresponds to the square of the (Drude)
+plasma frequency or of the (Lorentz) HO strength (i.e.,
+ω2
+p,Di or Ω2
+j in Eq.
+(S1) in SM [14]).
+As elaborated
+in SM [14], the metallic part of σ1(ω) necessitates of two
+Drude terms for the Co-intercalated materials, thus spot-
+ting the multiband nature of their electronic structure,
+while the Mn-intercalated compositions feature a single
+Drude component. The resulting global Drude SW (i.e.,
+SW = �
+i ω2
+p,Di, i ranging over the number of contem-
+plated Drude terms) will be anyhow at the centre of our
+attention.
+The main findings of our work are summarised in Fig.
+1(d-f). First, in all compounds the T dependence of the
+plasma frequencies (Fig. S8 in SM [14]) is such that the
+total Drude SW either barely changes or moderately in-
+creases upon lowering T. For the Co-compositions, this
+also pairs with an additional SW accumulating into the
+high frequency tail of the purely metallic response (rep-
+resented by HO L1, Figs. 1(b-c) and S7(b) in SM [14]).
+Such an enhancement results from a shift of SW from
+higher energy scales. For the Mn-compounds, the Drude
+tail is given by the combination of HO L1 and L2 (Fig.
+1(a) and S7(a) in SM [14]), which equally suffer a SW
+reordering among them and in favour of the Drude term.
+The SW reshuffling affecting the Drude term and even-
+tually its high-energy tail turns out though to be rather
+residual, compared to the SW redistribution at higher
+energies (as e.g. emphasised by the inset in Fig. 1(d)
+as well as in Fig. S7(c) in SM [14]). Second and even
+more relevant for our discussion, there is an opposite
+and strong redistribution of SW between HO L2 and L3
+depending on whether a FM or AFM transition (Figs.
+1(b-c)) takes place for the Co-intercalated compounds.
+For the Co-concentration x = 0.22, HO L3 losses SW,
+which merges almost totally in HO L2 upon crossing TC
+(Fig. 1(e)). The trend in the SW removal and reallo-
+cation across the FM transition as observed in the Co-
+intercalated composition is similarly confirmed by the ob-
+servations in the Mn-intercalated ones (e.g., for x = 0.09
+and 0.19 in Fig. 1(d) and Fig. S7(c) in SM [14], respec-
+tively). On the contrary, for the AFM Co-concentration
+x = 0.34 we principally encounter a progressive and grad-
+ual shift of SW from HO L2 towards HO L3 upon ap-
+proaching TN from high T (Fig. 1(f)). An equivalent
+incidence is observed for the x = 0.26 Co-intercalation
+(Fig. S7(d) in SM [14]), so that the SW reshuffling for
+the AFM transition is fully exploited and leans towards a
+constant behaviour pattern below TN. Summarising, we
+overall discern a mostly incremental SW redistribution
+at FIR and MIR-NIR energy intervals upon approaching
+the magnetic phase transition from high T, which then
+tends to saturate into the magnetic state (i.e., at T < TN
+or TC). Additionally, a change of slope in ∆SW(T) is ob-
+served in all compositions at T ranging between ∼ 100
+and 250 K. This is noted by T ∗ (black arrows in Figs.
+1(d-f) and S7(c-d) in SM [14]).
+The resulting kink in
+∆SW(T) is smooth in the Mn-intercalated materials and
+rather abrupt and sudden in the Co-ones (at least for x
+= 0.22 (Fig. 1(e)) and 0.26 (Fig. S7(d) in SM [14]) and
+somehow stepwise for x = 0.34 (Fig. 1(f)). These ob-
+servations fairly agree with similar findings (as kink or
+upturn around T ∗) in the measured T dependence of the
+in-plane thermopower and total thermal conductivity as
+well as Hall resistivity [8]. The origin of these peculiari-
+ties and their link to the advanced multiband nature of
+these materials need to be better understood, as offered
+here from the optical perspective.
+The SW distribution and its evolution upon crossing
+TC or TN seems to be a common property for both Co
+or Mn intercalations and is exclusively driven by the tar-
+geted, final magnetic state (i.e., independent of the el-
+ement choice). Moreover, the encountered shift of SW
+occurs at FIR-MIR energy scales up to the NIR spectral
+range, while at visible and ultra-violet frequencies SW is
+constant at any T (grey shaded areas in Figs. 1(a-c) and
+Figs. S7(a-b) in SM [14]). This also means that the full
+recovery of SW is achieved at about 1 eV. Nonetheless,
+the opposite SW allocation discovered upon lowering T
+through either a FM or AFM transition (see rounded ar-
+rows in Figs. 1(a-c) and Figs. S7(a-b) in SM [14]) calls
+for a yet different reconstruction of the related electronic
+
+3
+Temperature (K)
+0
+100
+200
+300–6
+–4
+–2
+0
+2
+4
+6
+ΔSW(T) [×107 (cm)-2]
+Co0.34TaS2
+Drude
+L1
+L2
+L3
+(f)
+–12
+–8
+–4
+0
+4
+8
+12
+ΔSW(T) [×107 (cm)-2]
+Co0.22TaS2
+Drude
+L1
+L2
+L3
+(e)
+101
+102
+103
+104
+0
+3
+6
+Frequency (cm-1)
+σ1(ω) [×103 (Ωcm)-1]
+Co0.34TaS2 (TN = 36 K)
+10 K
+Fit 10 K
+300 K
+Fit 300 K
+L1
+L2
+L3
+Drude
+L4-9
+(c)
+0
+2
+4
+6
+σ1(ω) [×103 (Ωcm)-1]
+Co0.22TaS2 (TC = 26 K)
+10 K
+Fit 10 K
+300 K
+Fit 300 K
+(b)
+Drude
+L1
+L2
+L3
+L4-9
+–2
+–1
+0
+1
+2
+ΔSW(T) [×107 (cm)-2]
+Mn0.09TaS2
+Drude
+L1
+L2
+L3-L4
+L5-L6
+(d)
+0
+100 200 300
+–0.04
+0
+0.04
+10–2
+10–1
+100
+0
+1
+σ1(ω) [×103 (Ωcm)-1]
+Mn0.09TaS2 (TC = 11 K)
+5 K
+Fit 5 K
+300 K
+Fit 300 K
+(a)
+Energy (eV)
+Drude L1
+L2
+L3
+L4
+L5
+L6
+L7-9
+Figure 1. (a-c) In-plane σ1(ω) below 4×104 cm−1 (1 eV = 8.06548×103 cm−1, please note the logarithmic energy scale) at
+5 or 10 and 300 K together with their respective total Drude-Lorentz fit (thick dashed line) after Eq. S1 in SM [14], and
+(d-f) T dependence of the SW relative variation with respect to 300 K, i.e., ∆SW(T) = SW(T) − SW(300 K) for selected
+fit components (see legend in panels (a-c)) of 2H-MxTaS2 (M = Mn and Co): (a,d) x = 0.09 (FM) Mn-concentration, and
+(b,e) x = 0.22 (FM) as well as (c,f) x = 0.34 (AFM) Co-concentration. Panels (a-c) explicitly show all fit components: the
+total Drude and Lorentz (Li, i = 1 to 9) HOs [14]. The coloured shaded areas emphasise SW encountered by each component
+(reddish and blueish colours refer to 300 and 5 or 10 K, respectively, while the grey shaded area corresponds to SW being
+T-independent). The rounded arrows in panels (a-c) highlight the direction in energy of the SW reshuffling upon lowering T,
+which is stronger with thicker arrows. The inset in panel (d) is a blow-up of ∆SW for the total Drude term and HOs L1 and
+L2. The vertical dashed and dotted lines in panels (d-f) mark TC and TN (see Table I in SM [14]), respectively. The error bars
+in ∆SW(T) correspond to the direct propagation of the error in the HOs strength, estimated numerically within the non-linear
+least-squares fit technique. The vertical black arrows in panels (d-f) indicate T ∗, as the onset of the faster ∆SW(T) variation
+upon lowering T. Additional data with similar analysis are available in Fig. S7 in SM [14].
+band structure, upon which we wish to argue for the rest
+of our paper.
+
+4
+EF
+E
+DOS
+DOS
+FM
+AFM
+MIR-NIR
+energy scales
+FIR
+energy scales
+Figure 2. Proposal for DOS particularly emphasising the in-
+terband transitions grouping around the characteristic FIR
+energy scales of 0.05-0.2 eV (i.e., between the blue bands)
+and MIR-NIR ones of 0.3-0.5 eV (i.e., between the red bands).
+The trend in the related transition probability (i.e., SW re-
+distribution of the proposed interband excitations) is indi-
+cated by the arrows thickness (violet vertical arrows for the
+FM state and green vertical arrows for the AFM state). The
+colour code of the arrows is in accord with the convention
+used for the SW reshuffling in the AFM and FM state of
+Figs. 1(a-c) and Figs. S7(a-b) in SM [14].
+Figure 2 schematically depicts the density-of-states
+(DOS), factual for the alleged progression of the inter-
+band transition probability upon lowering T across the
+FM and AFM transitions. The charge dynamics and the
+evolution of its stored SW convey the presence of in-
+terband transitions grouping within two distinct spectral
+intervals, i.e., at FIR resonance energies between 50 and
+200 meV (i.e., involving the blue bands in Fig.
+2) as
+well as at MIR-NIR ones between 300 and 500 meV (i.e.,
+involving the red bands in Fig.
+2).
+These ranges are
+described by the combination of HOs L3-L4 and L5-L6
+for the Mn-intercalated material and L2 and L3 for the
+Co-intercalated compositions (Figs. 1(a-c)), respectively.
+The thickness of the coloured (vertical) arrows in Fig. 2
+mimics the strength (i.e., SW) of those two possible in-
+terband transitions, which alike discriminate between the
+two magnetic states. Below TN, the (convoluted) MIR-
+NIR transition is stronger than the lower FIR one, while
+the opposite seems to apply below TC.
+The characteristic ingredients pertinent to the recon-
+struction of the electronic band structure, depicted in
+Fig. 2 and then driving the SW redistribution observed
+in σ1(ω) (Fig. 1 and Fig. S7 in SM [14]), embrace several
+findings and achieved knowledge on related, sister mate-
+rials. First of all and from a general perspective, the FM
+state in both Mn- and Co-intercalated 2H-TaS2 has been
+tentatively reconciled within a Ruderman-Kittel-Kasuya-
+Yosida interaction scenario [6–8], in which the local spins
+of intercalated Mn and Co ions align ferromagnetically
+through the itinerant Ta 5d electrons, as initially pro-
+posed for the related Co-intercalated 2H-NbS2 material
+[22].
+Further, it is speculated that Co atoms tend to
+hybridise more strongly with the electronic states associ-
+ated with covalently bonded structural subunit (i.e., Ta
+and S) when intercalated in the vdW gap. By enlarg-
+ing the Co concentration, this leads to a suppression of
+the spontaneous magnetic moment and to a stronger ten-
+dency for AFM exchange coupling parameters [8, 23].
+Along this line of thoughts, the FM to AFM crossover,
+hither studied upon changing the intercalation and con-
+tingently the concentration of the intercalated Co atom,
+can be also achieved after two alternative ways: either
+by differentiating the element-intercalation at given con-
+centration or by applying pressure on a selected inter-
+calated compound. It is experimentally known that the
+structurally equivalent 2H-NbS2 compound [24], inter-
+calated with 3d elements Cr, Mn and Fe for the con-
+centration 1/3, exhibits a variety of magnetic states,
+(roughly) classified as FM for the Cr and similarly Mn
+materials [20, 25–27] and as AFM for the Fe composi-
+tion [20, 28, 29].
+The first-principles electronic band
+structure calculations that have been performed us-
+ing the fully relativistic Korringa-Kohn-Rostoker Green
+function method [30] have pointed out the complex-
+ity of their magnetic ordering.
+For Cr1/3NbS2 and
+Mn1/3NbS2, the in-plane magnetocrystalline anisotropy
+and Dzyaloshinskii-Moriya interactions give rise to a heli-
+magnetic structure along the c-axis, following the experi-
+mental observations [25, 26]. On the other hand, the neg-
+ative exchange interactions in the Fe1/3NbS2 compound
+result in a noncollinear frustrated magnetic structure if
+the magnetocrystalline anisotropy is not taken into ac-
+count. However, a strong magnetocrystalline anisotropy
+along the c-axis does lead to a magnetic state referred to
+as an ordering of the third kind, which was indeed deter-
+mined experimentally [20, 28, 29]. This may be pinned
+down to the diverse DOS with respect to the magnetic
+state in these series of compounds. For all compounds,
+DOS is rather large for the majority-spin states at the
+Fermi energy (EF ). In the case of minority-spin states of
+Cr1/3NbS2 and Mn1/3NbS2, one can however observe a
+pseudogap between the occupied and unoccupied states
+[23, 30]. Such an imbalance then leads to the FM ground
+state.
+In the case of Fe1/3NbS2, EF is located at the
+DOS maximum corresponding to the Fe minority-spin d
+states. DOS is thus finite for both spin directions, which
+facilitates an AFM ground state [30]. This latter trend
+is reflected in the isotropic exchange coupling parameter
+for the three compositions, which is predominantly posi-
+tive for the Cr and Mn compounds (i.e., promoting FM)
+but negative (i.e., leading to AFM) for Fe-Fe interactions
+at short distance [30].
+In turn, one can induce the crossover from the FM to
+
+5
+AFM state in Mn1/4NbS2 upon applying pressure [31].
+This possibility is instrumental, in order to better jus-
+tify and support our schematic proposal in Fig. 2. Fo-
+cusing the attention on the spin-resolved DOS on Mn
+sites, which is foremost contributed by the d-orbitals, the
+majority-spin states are almost occupied around both the
+Γ and K points in the Brillouin zone at ambient pressure,
+while the minority-spin states, essentially around the K
+point of the Brillouin zone, are unoccupied. This favors
+FM and further implies a dominating hole-type charac-
+ter of the electric carriers and a positive slope of the Hall
+resistivity [31]. Interband transitions within the FIR en-
+ergy interval (so roughly peaked at 100 meV) are fore-
+seen from the electronic band structure at both Γ and
+K points of the Brillouin zone [31] and should play the
+most prominent role in σ1(ω). Figure 2 catches a glimpse
+of such a trim for our FM compositions (Figs.
+1(a-b)
+and S7(a) in SM [14]). Conversely, the pressure increase
+results in the broadening of the energy bands, which
+premises an occupation of the bottom of the minority-
+spin states (having mainly dx2−y2 and dxy character)
+and a draining of the top of minority-spin states (pri-
+marily of dxzand dyz character) at the K point of the
+Brillouin zone in Mn1/4NbS2 [31]. This is accompanied
+by a decrease of the exchange splitting of the majority-
+and minority-spin d-states of Mn, which potentially turns
+negative for the first neighbour interaction, as the prereq-
+uisite for an AFM alignment of the magnetic moments
+in the absence of any other interactions. In fact, such
+a setting is translated into an electron-like character of
+the ordinary Hall effect [31]. By inspecting the result-
+ing electronic band structure in the AFM state [31], we
+recognise the most cogent consequence for the excitation
+spectrum: namely, low energy FIR interband transitions
+are less probable, while the probability for high energy
+ones at MIR and NIR frequencies (i.e., settled around
+400 meV) increases sensitively, as evinced in σ1(ω) (Figs.
+1(c) and S7(b) in SM [14]) and as also sketched in Fig.
+2 for our case. Therefore, we claim that the band recon-
+struction in Mn1/4NbS2 upon applying pressure is imple-
+mentable to 2H-CoxTaS2 with respect to the FM-AFM
+crossover as a function of x, as backed up by the same
+sign-change of the ordinary Hall coefficient upon varying
+the magnetic ground state [8, 31].
+In conclusion, the charge dynamics of 2H-MxTaS2 (M
+= Mn and Co) allows to determine a distinct SW re-
+distribution and to shed light on the relevant energy
+scales shaping the reconstruction of the electronic band
+structure upon crossing over from the FM to AFM order
+with varying intercalation (i.e., element and/or its con-
+centration). Our spectroscopic findings seem to be con-
+sistent with dedicated first-principles calculations upon
+tuning element-intercalation, pressure and/or magnetic
+field on similar intercalated materials. The applicability
+of the proposed comparison is validated by the fact, that
+the quoted calculations do tackle the impact of (similar)
+magnetic transitions on the electronic band structure of
+equivalent materials, seemingly reflected in the charge
+dynamics of our compositions, too.
+Summing up, the significance of this work consists
+in the first instance in the systematic investigation of
+2H-TaS2 with different intercalations and across distinct
+magnetic transitions, hence widening out a previous,
+more restricted attempt [32] and possibly challenging the
+so far broadly accepted implementation of the rigid-band
+model.
+Moreover, since the continuous change of the
+dominant magnetic exchange controls the FM to AFM
+switching and being its impact mapped onto the easily
+accessible FIR-MIR-NIR spectral range, one may exploit
+our straightforward (broadband) experimental tool also
+as a function of alternative tuneable variables than
+chemical-intercalation, which would thoroughly flash on
+the overall consistency of the emerging physical picture.
+Finally, the present work demonstrates a feasible route
+towards understanding magnetism in low dimensions
+and may help in revealing robust properties, relevant
+for the development of low-power spin-logic circuits
+from layered materials [6]. The possibility of integration
+of FM is generally of interest for spintronics, so that
+novel fabrication of low-dimensional heterostructures
+might be envisaged and motivated, as well; the few-layer
+graphene/2H-TaS2 heterostructures with robust spin-
+helical state [33] is already a promising development
+and the recent discovery of one-dimensional vdW (yet
+non magnetic) heterostructures [34] and multi-walled
+2H-TaS2 nanotubes [35] may open new avenues. Since
+exotic phenomena such as skyrmions and magnetic
+solitons [36, 37] were recently discovered in 2H-TaS2-
+based vdW magnets, our results may therefore help to
+efficiently tune their properties for further applications.
+ACKNOWLEDGEMENTS
+Work at Brookhaven National Laboratory was sup-
+ported by the U.S. Department of Energy, Office of
+Basic Energy Science, Division of Materials Science
+and Engineering, under Contract No.
+DE-SC0012704
+(materials synthesis).
+† Authors M.C. and R.Y. contributed equally to the
+work.
+‡ Present address: Los Alamos National Laboratory,
+Los Alamos, New Mexico 87545
+∗ Correspondence and requests for materials should be
+addressed to: L. Degiorgi, Laboratorium f¨ur Festk¨orper-
+physik, ETH - Z¨urich, 8093 Z¨urich, Switzerland; email:
+degiorgi@solid.phys.ethz.ch.
+
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+and N. L. Wang, Phys. Rev. B 76, 045103 (2007).
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+C. Felser, M. Dressel, and A. V. Pronin, Phys. Rev. B
+103, 115206 (2021).
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+B 102, 161109 (2020).
+

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+1 
+ 
+Choosing statistical models to assess biological interaction as a 
+departure from additivity of effects   
+ 
+David M. Thompson, Yan Daniel Zhao 
+ 
+Abstract 
+ 
+Vanderweele and Knol define biological interaction as an instance wherein “two exposures 
+physically interact to bring about the outcome.”  A hallmark of biological interaction is that the 
+total effect, produced when factors act together, differs from the sum of effects when the factors 
+operate independently.  
+ 
+Epidemiologists construct statistical models to assess biological interaction. The form of the 
+statistical model determines whether it is suited to detecting departures from additivity of effects 
+or for detecting departures from multiplicativity of effects. A consensus exists that biological 
+interaction should be assessed as a departure from additivity of effects. 
+ 
+This paper compares three statistical models’ assessment of a data example that appears in 
+several epidemiology textbooks to illustrate biological interaction in a binomial outcome. A 
+linear binomial model quantifies departure from additivity in the data example in terms of 
+differences in probabilities. It generates directly interpretable estimates and 95% confidence 
+intervals for parameters including the interaction contrast (IC). Log binomial and logistic 
+regression models detect no departure from multiplicativity in the data example. However, their 
+estimates contribute to calculation of a “Relative Excess Risk Due to Interaction” (RERI), a 
+measure of departure from additivity on a relative risk scale.   
+ 
+The linear binomial model directly produces interpretable assessments of departures from 
+additivity of effects and deserves wider use in research and in the teaching of epidemiology. 
+Strategies exist to address the model’s limitations.   
+ 
+ 
+Key Words: additivity and multiplicativity of effects; biological interaction; statistical 
+interaction; generalized linear models; interaction contrast (IC); Relative Excess Risk Due to 
+Interaction (RERI) 
+ 
+__________________________________________________________________________ 
+ 
+ 
+ 
+
+2 
+ 
+Key Messages 
+• A consensus exists in epidemiology, and is reflected in the STROBE statement, that 
+biological interaction should be assessed as a departure from additivity of effects. 
+• The log binomial and logistic regression models are widely used in epidemiology to 
+assess biological interaction, even though their statistical forms suit them for detecting 
+departures from multiplicativity of effects. 
+• The log binomial and logistic regression models can quantify departures from additivity, 
+on a relative risk scale, by estimating statistics like the Relative Excess Risk Due to 
+Interaction (RERI).  However, the RERI is not directly interpretable as excess risk, and 
+inference (estimation and hypothesis testing) for the RERI are complicated.  
+• The linear model binomial model, which can be estimated using available software for 
+generalized linear models, directly estimates the interaction contrast (IC), which is 
+interpretable as excess risk. 
+• The linear binomial model deserves wider use in research and in the teaching of 
+epidemiology.  
+ 
+ 
+ 
+
+3 
+ 
+Biological interaction and statistical interaction 
+Hypotheses related to biological interaction are often of interest in studies of clinical or 
+population health.  Vanderweele and Knol [1 (p. 54)] define biological interaction as an instance 
+in which “two exposures physically interact to bring about the outcome.”  Rothman [2 (p. 171)] 
+states that “biologic interaction between two causes occurs whenever the effect of one is 
+dependent on the presence of the other.”  
+ 
+Investigators construct statistical models to detect interaction and effect modification. Rothman 
+[2 (p.169)] points out that “in statistics, the term ‘interaction’ is used to refer to departure from 
+the underlying form of a statistical model.”  A model’s form can suit it for detecting departures 
+from additivity of effects or for detecting departures from multiplicativity of effects. Because a 
+statistical model’s form affects the interpretation of statistical interaction, Rothman [2 (p.170)] 
+prefers the term “effect measure modification” to interaction. 
+ 
+Rothman links “biological independence” with an additivity of effects and connects “biological 
+interaction” with a departure from an additivity of effects. “Why is it,” Rothman asks, “that 
+biological interaction should be evaluated as departures from additivity of effect” [2 (p.178)]?  
+By 2007, the STROBE statement regarded the response to Rothman’s rhetorical question to 
+reflect a “consensus that the additive scale, which uses absolute risks, is more appropriate [than 
+the multiplicative scale] for public health and clinical decision making” [3 (p.817)].   The authors 
+of the STROBE statement remind investigators that “in many circumstances, the absolute risk 
+associated with an exposure is of greater interest than the relative risk” and ask them to “consider 
+
+4 
+ 
+translating estimates of relative risk into absolute risk for a meaningful time period” [3 (p.825)].  
+Vanderweele and Knol [1 (p. 37)] remark, more pointedly, that “one reason why additive 
+interaction is important to assess (rather than only relying on multiplicative interaction measures) 
+is that it is the more relevant public health measure.” 
+ 
+Additivity and multiplicativity of effects 
+ 
+This paper aligns with this consensus but avoids using the term “additive interaction.” Instead, it 
+links the concept to statistical models that assess evidence of a departure from additivity of 
+effects.  One such model, the “binomial model for the risk difference” [4], directly quantifies 
+departures from additivity of effects in terms of differences in probabilities, including the 
+interaction contrast (IC).  This model is also called the “binomial regression model” [5, 6].  
+Richardson et al. [7], who employ it as a final step in a marginal structural model, call it the 
+“linear binomial model,” the term we will use.  
+ 
+In the linear binomial model, detection of statistical interaction constitutes direct evidence of a 
+departure from additivity of effects. The log binomial and logistic regression models can also 
+assess additivity indirectly, when their estimates of relative risks or odds ratios are recombined to 
+calculate statistics like the “Relative Excess Risk due to Interaction” (RERI).  
+ 
+The paper also avoids using the term “multiplicative interaction” but links that concept to 
+statistical models that assess evidence of departures from multiplicativity of effects.  Log 
+binomial models estimate effects in terms of relative risks, also called risk ratios, prevalence 
+
+5 
+ 
+ratios [4,7] or prevalence proportion ratios.  Logistic regression models estimate effects in terms 
+of odds and odds ratios.  In the log binomial and logistic models, which employ log 
+transformations of probabilities or of their corresponding odds, detection of statistical interaction 
+constitutes direct evidence of a departure from multiplicativity among effects.  
+ 
+Statistical models for binomial outcomes 
+ 
+The linear binomial, log binomial and logistic regression models are all examples of generalized 
+linear models.  Each treats the outcome as arising from a binomial distribution.  Each features a 
+linear predictor structured as a sum of terms.  In this regard, all generalized linear models might 
+be considered “additive.”  Accordingly, this paper does not refer to “additive or multiplicative 
+models” but refers instead to statistical models that assess additivity or multiplicativity of effects.   
+ 
+All three models link a binomial outcome to a linear predictor. They are distinguished by the link 
+functions they employ.  The linear binomial model uses the identity link, the log binomial model 
+uses the log link, and the logistic regression model uses the logit link.  Thus, the linear binomial 
+model operates directly on probabilities, while the others apply log transformations of the 
+probabilities or of their corresponding odds.  Because each model estimates a different effect 
+measure, they differ in their ability to detect statistical interaction in a collection of data.   
+ 
+After reviewing the definition of additivity of effects, we compare the three statistical models 
+using a widely cited example of biological interaction [8].  The linear binomial model detects 
+statistical interaction in these data.  The log binomial and logistic regression models, which 
+
+6 
+ 
+assess multiplicativity of relative risks or of odds ratios, find no evidence of statistical 
+interaction.  The absence of statistical interaction in these models does not point to an absence of 
+biological interaction, but to a lack of departure from multiplicativity of effects.   
+ 
+We conclude by summarizing the three models’ advantages and limitations for assessing 
+additivity of effects.  The RERI is commonly used in epidemiologic research to quantify 
+departures from additivity despite complications in its estimation, testing and interpretation.  In 
+comparison, the linear binomial model produces readily interpretable estimates of effects, 
+including the interaction contrast.   
+ 
+Defining additivity of effects  
+Consider a comparison of the probability or “risk” of an outcome Y among individuals who are 
+exposed or not exposed to one or both of two “risk factors,” X and Z.  Then, pxz is a probability 
+whose subscripts signify the probability or risk of the outcome Y at “levels” of X and Z (Table 
+1). 
+Table 1. Probabilities of an outcome (Y) at levels of two exposure or risk factors (X and Z) 
+____________________________________________________________________ 
+____________________________________________________________________ 
+                                                                    Z=1                                       Z=0 
+                                                     (“exposed to factor Z”)    (“not exposed to factor Z”) 
+_____________________________________________________________________ 
+X=1 (“exposed to factor X”)                      p11                                         p10 
+X=0 (“not exposed to factor X”)                p01                                         p00 
+_____________________________________________________________________ 
+ 
+ 
+ 
+
+7 
+ 
+Rothman [2 (p.178)] states that the following equation “establishes additivity as the definition of  
+biological independence.” 
+𝑝11 − 𝑝00 = (𝑝10 − 𝑝00) + (𝑝01 − 𝑝00) 
+ 
+ 
+(Equation 1) 
+ 
+According to Rothman’s equation, two exposures (X and Z) are biologically independent, and 
+their effects are additive, when the effect on Y of their joint and simultaneous effects (p11 − p00) 
+is equal to the sum of the separate and independent effects of X (p10 − p00) and of Z (p01 −
+p00). A departure from additivity of effect, which Rothman considers evidence of biological 
+interaction, is present when the exposures’ joint and simultaneous effect differs from the sum of 
+their separate effects.  
+ 
+Additivity can be defined equivalently as a homogeneity of effects.  The terms of Equation 1 can 
+be reordered to obtain 
+ 
+𝑝11 − 𝑝01 = 𝑝10 − 𝑝00 ,  
+ 
+ 
+(Equation 2) 
+ 
+p11 − p10 = p01 − p00. 
+ 
+ 
+(Equation 3) 
+ 
+Equation 2 states that the effect of X on Y is the same whether Z = 1 (p11 − p01) or Z = 0 
+(p10 − p00).  Homogeneity of effects is reciprocal. Equation 3 states that the effect of Z on Y is 
+the same at all levels of X, that is, whether X=1 (𝑝11 − 𝑝10) or X=0 (𝑝01 − 𝑝00).   When the 
+effects of X and Z are additive, the association between Y and X is homogenous at levels of Z, 
+and the association between Y and Z is homogenous at levels of X.   
+
+8 
+ 
+ 
+Assessing additivity of effects using probabilities (the interaction contrast) or 
+ratios (the RERI)  
+ 
+Departures from an additivity of effects (or from biological independence), whether defined as 
+an inequality between joint and independent effects, or as a heterogeneity among effects, can be 
+formally assessed through the interaction contrast, whose terms are probabilities, and the RERI, 
+whose terms are relative risks. 
+ 
+The terms in equation (1) can be ordered to produce the interaction contrast [9]:   
+ 
+𝑝11 − 𝑝10 − 𝑝01 + 𝑝00 = 0 
+ 
+ 
+(Equation 4) 
+ 
+Reordering the terms in Equation 4 and dividing each by p00 yields:    
+ 
+p11/p00 − p01/p00 − p10/p00 + 1 = 0.  
+ 
+Recognizing that these ratios of probabilities are relative risks (RR), we obtain: 
+ 
+RR11 − RR01 − RR10 + 1 = 0. 
+ 
+(Equation 5) 
+ 
+Rothman [10] names the quantity on the left side of equation 5 the “Relative Excess Risk due to 
+Interaction” (RERI).  Rothman and Greenland [9] call it the “interaction contrast ratio” (ICR).  
+
+9 
+ 
+Hosmer and Lemeshow [11] define it as “the proportion of disease among those with both 
+exposures that is attributable to their interaction.”   
+ 
+The algebraic equivalence between equations 1 (for the IC) and 5 (for the RERI) validates the 
+assessment of additivity of effects on either probability or relative risk scales.  The IC and the 
+RERI formally test the hypothesis that the effects on Y of X and Z are additive or, equivalently, 
+that no interaction exists between X and Z.  The STROBE statement [3 (p.825)] illustrates how 
+to use the RERI to assess departures from additivity of effects.   
+ 
+Data example: lung cancer mortality among workers with different exposures 
+to asbestos and smoking 
+Hammond et al. [8] compared the risk of a dichotomous outcome, mortality from lung cancer, 
+among 17,800 asbestos workers and among 73,763 workers who were not exposed to asbestos.  
+They also recorded smoking status, so participants displayed combinations of exposure to 
+cigarette smoking and to asbestos (Table 2).  Hammond’s study is widely used in epidemiology 
+textbooks [2 (pp.168-180),12] to illustrate biological interaction. 
+ 
+Supplementary File 1 illustrates the creation of a dataset that closely approximates the properties 
+of the published data.  So that the dataset’s risk probabilities (reported as lung cancer deaths per 
+100,000) reflect the published ones, we assumed a smoking prevalence of 0.28 for both the 
+asbestos workers and for the comparison group of unexposed workers.   
+ 
+
+10 
+ 
+Table 2. Lung cancer deaths (per 100,000 workers) among those with exposure to asbestos 
+and/or cigarette smoking 
+________________________________________________________________________ 
+                                                                                 Asbestos Exposure 
+                                       _____________________________________________________ 
+Cigarette smoking         Asbestos Workers (n= 17800)     Comparison Group (n=73763) 
+_____________________________________________________________________ 
+Smokers                                      p11=601.9                             p10= 121.1      
+Non-smokers                              p01=  54.6                              p00=  11.3      
+________________________________________________________________________ 
+ 
+ 
+The data example illustrates a departure from additivity of effects 
+ 
+If the effects of asbestos exposure and cigarette smoking are additive, the expected effect of 
+experiencing both exposures would equal the sum of the exposures’ separate effects (Equation 
+1).  Following the notation introduced in Table 1 to define pxz, where X denotes cigarette 
+smoking (1 = smokers and 0 = nonsmokers) and Z denotes asbestos exposure (1=exposed and 0= 
+not exposed), the estimated risk probabilities are: 
+ 
+𝑝̂11 − 𝑝̂00 = 601.9 − 11.3 = 590.6 excess deaths per 100,000 people, attributable to 
+joint effects of both exposures. 
+ 
+𝑝̂10 − 𝑝̂00 = 121.0 − 11.3 = 109.7 excess deaths per 100,000 attributable to smoking by 
+itself. 
+ 
+
+11 
+ 
+𝑝̂01 − 𝑝̂00 = 54.6 − 11.3 = 43.3 excess deaths per 100,000 people, attributable to 
+asbestos exposure by itself. 
+ 
+The number of lung cancer deaths attributable to dual exposure appears to exceed the sum of the 
+exposures’ separate effects.  The interaction contrast for the data example: 𝑝11 − 𝑝10 − 𝑝01 +
+𝑝00 indicates that the risk of lung cancer death in those who experience both exposures exceeds, 
+by about 437.6 deaths per 100,000, the sum of the separate risks from smoking or from asbestos 
+exposure.  Calculated for the data example, the RERI, which quantifies additivity of effects on 
+the relative risk scale, RR11 − RR01 − RR10 + 1 = [601.9/11.3] - [54.6/11.3] - [121.0/11.3] +1 = 
+38.7.   
+ 
+The linear binomial model directly estimates the interaction contrast in the data 
+example 
+The linear binomial model [4,7] estimates the interaction contrast directly in terms of 
+probabilities and differences in probabilities:  
+ 
+P(Y = 1) = β0 + β1X + β2Z + β3XZ    
+ 
+ 
+(Equation 6) 
+ 
+Recalling that X and Z take values of 1 for “exposure” and 0 for “no exposure”, then 
+𝑝̂00 = 𝛽0 
+𝑝̂10 − 𝑝̂00 = (𝛽0 + 𝛽1) − 𝛽0 = 𝛽1 
+𝑝̂01 − 𝑝̂00 = (𝛽0 + 𝛽2) − 𝛽0 = 𝛽2 
+𝑝̂11 − 𝑝̂00 = (𝛽0 + 𝛽1 + 𝛽2 + 𝛽3) − 𝛽0 = 𝛽1 + 𝛽2 + 𝛽3 
+
+12 
+ 
+ 
+Substituting these expressions into Equation 1, which defines additivity of effects,  
+𝑝11 − 𝑝00 = (𝑝10 − 𝑝00) + (𝑝01 − 𝑝00) 
+ 
+ 
+𝛽1 + 𝛽2 + 𝛽3 = 𝛽1 + 𝛽2 
+ 
+In the linear binomial model, effects are additive if 𝛽3, the regression coefficient associated with 
+the product or interaction term, is equal to zero. 
+ 
+Substituting the expressions into Equation 4 illustrates that the model’s estimate for β3 directly 
+estimates the interaction contrast: 
+ 
+p11 − p10 − p01 + p00 = (β0 + β1 + β2 + β3) − (β0 + β1) − (β0 + β2) + β0 = β3 
+ 
+Thus, the linear binomial model’s estimates for the interaction contrast and for the X*Z 
+interaction are equivalent.  Both provide direct tests of additivity; evidence against the 
+hypothesis that β3 =0 is evidence of a departure from additivity.   
+ 
+Supplementary File 2 illustrates the construction of the linear binomial model using SAS PROC 
+GENMOD [4,7]. The model’s point estimates for the number of deaths per 100,000 workers, 
+which are presented in Table 3, are equal to those reported in Table 2.  Table 3 also reports the 
+model’s estimates (and 95% CI) for regression coefficients.  These coefficients include estimates 
+for the effect on lung cancer mortality of smoking among those not exposed to asbestos (β1), and 
+of asbestos exposure in non-smokers (β2).   
+
+13 
+ 
+ 
+Table 3. Absolute risks (and risk differences) for death from lung cancer (per 100,000 workers) 
+for those with exposure to asbestos and/or cigarette smoking, estimated by linear binomial model 
+________________________________________________________________________ 
+              Smoking  Asbestos    Estimate      Deaths per 100,000            95% CI on estimate 
+                                                                                                                ________________ 
+                                                                                                                 Lower         Upper 
+________________________________________________________________________ 
+ p11          1 (yes)     1 (yes)       0.006019              601.926                 387.183      816.669 
+ p10          1 (yes)     0 (no)        0.001210              121.048                   73.627      168.469 
+ p01          0 (no)       1 (yes)       0.000546                54.619                    14.169        95.070 
+ p00          0 (no)       0 (no)        0.000113                11.298                      2.258        20.337 
+ 
+β1            smk (𝑝̂10 − 𝑝̂00)         0.001098              109.750                    61.475      158.025 
+β2        asbestos  (𝑝̂01 − 𝑝̂00)     0.000433                43.322                      1.873        84.770 
+β3            smk*asbestos             0.004376              437.557                 213.768       661.345  
+IC            p11-p10-p01+p00             0.004376              437.557                 213.768       661.345  
+________________________________________________________________________ 
+ 
+ 
+The linear binomial model produces identical inference for β3 , which estimates the statistical 
+interaction between smoking and asbestos exposure, and for the IC (estimate: 437.6 deaths per 
+100,000; 95% CI: 213.8, 661.3; P=0.00012702).  The consistency between the p values 
+generated for these statistics verifies that they offer equivalent tests of the null hypothesis that 
+the effects of smoking and asbestos exposure are additive.   
+ 
+Figure 1, which depicts the estimates and confidence intervals generated by the linear binomial 
+model, illustrates the heterogeneity of the effects of smoking on lung cancer mortality in groups 
+defined by asbestos exposure.  The syntax that produced Table 3 and Figure 1 is contained in 
+Supplementary File 3. 
+ 
+
+14 
+ 
+Figure 1.  Biological interaction, between asbestos exposure and smoking, illustrated as a non-
+additivity or heterogeneity of effects 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Log binomial and logistic regression models detect no departure from multiplicativity 
+of effects in the data example. 
+ 
+In contrast to the linear binomial model, models that employ logarithmic transformations of 
+probabilities (log binomial models) or their corresponding odds (logistic regression models) 
+assess departures from multiplicativity of effects.  Multiplicativity of effects is defined in a 
+manner analogous to the definition of additivity of effects.  The effects of two factors (X and Z) 
+on an outcome (Y) are multiplicative if their joint effects are equal to the product of their 
+separate and independent effects.  When effects are multiplicative, relative risks will conform to 
+ 
+
+800-
+AsbestosWorkers(n=17800)
+--ComparisonGroup(n=73763)
+100,000
+600
+I cancer deaths per
+400
+200
+Lung
+0
+Non-smokers
+Smokers15 
+ 
+the relationship: RRXZ = RRX × RRZ, and odds ratios will conform to the relationship: ORXZ =
+ORX × ORZ. A log binomial model estimates and tests the multiplicativity of relative risks.   
+ 
+ln[P(Y = 1)] = β0 + β1X + β2Z + β3XZ,  
+ 
+P(Y = 1) = exp (β0 + β1X + β2Z + β3XZ). 
+ 
+it follows that: RRXZ = exp(β1X + β2Z + β3XZ); RRX = exp(β1X);  RRZ = exp(β2Z). 
+ 
+If there is no departure from multiplicativity among relative risks, then: 
+ 
+RRXZ = RRX RRz 
+ 
+exp(β1X + β2Z + β3XZ) = exp(β1X)exp(β2Z) =  exp(β1X + β2Z). 
+ 
+These equalities hold only if β3, the regression coefficient associated with the product term XZ, 
+is equal to zero.  Similarly, the logistic regression model, ln[P(Y = 1) P(Y = 0)
+⁄
+] = β0 + β1X +
+β2Z + β3XZ, assesses multiplicativity of effects expressed as odds or odds ratios.  In either 
+model, estimates or hypothesis tests that suggest that β3 does not equal zero constitute evidence 
+of a departure from multiplicativity of effects. 
+ 
+Applied to the data example, the log binomial model finds no evidence of statistical interaction 
+between smoking and asbestos exposure (P=0.9637); measured as relative risks, the factors’ 
+
+16 
+ 
+effects are multiplicative and homogenous.   Similarly, a logistic regression model finds no 
+statistical interaction between smoking and asbestos exposure (P=0.9581) to suggest a departure 
+from multiplicativity of effects measured as odds ratios.  Figures 2 and 3 depict the estimates 
+generated by the log binomial and logistic regression models.  The models’ construction, using 
+SAS PROC GENMOD, is detailed in Supplementary File 4 along with the syntax that produced 
+Figures 2 and 3. 
+ 
+Figure 2. Predicted log probabilities illustrate a lack of departure from multiplicativity of effects 
+in the log binomial model. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3. Predicted log odds illustrate a lack of departure from multiplicativity of effects in the 
+logistic regression model. 
+ 
+
+-5
+AsbestosWorkers(n=17800)
+-
+ComparisonGroup(n=73763)
+-8 
+-9
+-10
+Non-smokers
+Smokers17 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+The models’ differences in detecting statistical interaction do not confound the question of 
+whether the data exemplify biological interaction.  Rather, they illustrate the importance of (1) 
+identifying an effect measure (either a difference or a ratio between probabilities or risks) that 
+reflects the hypothesized form of the interaction and then (2) constructing a statistical model that 
+directly estimates that effect measure.  
+  
+Choosing among statistical models 
+ 
+Choosing log binomial or logistic regression models that generate estimates of the 
+RERI  
+ 
+Neither the log binomial model nor the logistic regression model detects statistical interaction in 
+the data example. The models’ form suits them for detecting departures from multiplicativity of 
+ 
+
+-5 
+AsbestosWorkers(n=17800)
+0--
+ComparisonGroup(n=73763
+-8 
+-9
+-10 
+Non-smokers
+Smokers18 
+ 
+effects.  Nevertheless, they are widely used in epidemiology to assess departures from additivity 
+of effects through ratio measures like the RERI [3]. 
+ 
+Although widely used, the RERI has disadvantages.  Because it is constructed from ratios, the 
+RERI is not interpretable as the number of excess deaths attributable to exposure to both 
+smoking and asbestos.  The RERI of 38.7, calculated for the data example, lacks the ease of 
+interpretation of the linear binomial model’s estimate of the IC of 437.6 excess deaths per 
+100,000 (Table 3.)   A second disadvantage relates to difficulties in obtaining standard errors 
+with which to construct confidence intervals for or to test hypotheses related to the RERI.  An 
+influential approach, introduced by Hosmer and Lemeshow [11], estimates the RERI using 
+logistic regression and obtains standard errors for its estimates using the delta method.  SAS 
+syntax for the approach is provided by Andersson et al. [13] and by Richardson and Kaufman 
+[14], who construct a “linear odds ratio model” using SAS PROC NLMIXED.  As an alternative 
+approach, Richardson and Kaufmann [14] recommend bootstrapping for obtaining confidence 
+intervals. An empirical 95% confidence interval on the RERI, calculated for the data example 
+from 500 bootstrap samples, is 15.9, 132.6.  However, because the bounds for the RERI’s 
+confidence interval are ratios, they present the same challenges to interpretation as the estimate 
+itself. 
+ 
+ 
+Choosing the linear binomial model that directly estimates the interaction contrast 
+ 
+
+19 
+ 
+Logistic regression is widely used in epidemiology to study binomial outcomes, even though its 
+form is suited for detecting departures from multiplicativity of effects. A major reason for the 
+model’s popularity and durability is that its use of the logit link, which is the canonical link for a 
+binomial response, affords desirable statistical properties. Among these is logistic regression’s 
+reliability in converging on parameter estimates.  Models that use other link functions can 
+encounter problems with convergence. Zou [15] and Spiegelman and Herzmark [4] discuss 
+problems with convergence in the log binomial model and advocate use of a modified Poisson 
+model to address the problem when it arises.   
+ 
+The linear binomial model, which uses the non-canonical identity link, can also fail to converge 
+on estimates.  This limitation interferes with the model’s wider acceptance, despite its ability to 
+directly assess additivity of effects by estimating the interaction contrast.  To address non-
+convergence in the linear binomial model, Spiegelman and Herzmark [4] advocate modifying the 
+model, retaining the identity link but assuming that the outcome follows a Poisson distribution.  
+Although the approach ensures convergence, imposing the Poisson assumption causes the model 
+to misspecify the binomial outcome’s variance.  This intentional misspecification of the 
+outcome’s distribution reduces the efficiency of the model’s standard errors and of the 
+hypothesis tests and confidence intervals that are based on them.  Accordingly, Spiegelman and 
+Herzmark [4] recommend calculating standard errors that are robust despite misspecification.   
+Richardson et al. [7] also recommend the calculation of robust standard errors but, because they 
+apply it to weighted data, do not advocate otherwise modifying the linear binomial model.  
+Supplementary File 2 shows how to incorporate these various recommendations using SAS 
+PROC GENMOD.   
+
+20 
+ 
+ 
+Cheung [5] addresses non-convergence in the linear binomial model by proposing a modified 
+least squares (MLS) model that also uses the identity link.  Cheung’s approach also calculates 
+robust standard errors.  Cheung’s approach differs in that it uses ordinary least squares (OLS) 
+instead of maximum likelihood estimation (MLE).  In doing so, it avoids specifying the 
+outcome’s assumed distribution. This strategy cures the problem of non-convergence but cannot 
+guarantee that estimated probabilities will be in the logical range from 0 to 1.    
+ 
+ 
+Conclusions 
+ 
+Biological interaction is often hypothesized to manifest itself as a non-additivity of effects that 
+are quantified as differences in risks or probabilities.  Applied to a data example widely used in 
+epidemiology education to illustrate biological interaction, a linear binomial model detects 
+statistical interaction while logistic and log binomial models do not.   
+ 
+The result affirms the consensus that biological interaction should generally be assessed as a 
+departure from an additivity of effects.  Statistics like the RERI are widely used in epidemiology 
+to assess additivity on a relative risk scale. In contrast, the linear binomial model produces 
+estimates of differences in probabilities, including the interaction contrast, that are directly 
+interpretable as excess risks. 
+ 
+
+21 
+ 
+Widely available software for generalized linear models permit researchers to construct the linear 
+binomial model and to obtain estimates and confidence intervals for the interaction contrast and 
+other effects.  The model deserves wider use in research and judicious use in the teaching of 
+epidemiology.  The linear binomial model can encounter problems with convergence, but 
+strategies exist to address this limitation. 
+ 
+ 
+Funding 
+ 
+Dr. Zhao’s work was partially supported by funding provided by National Institutes of Health, 
+National Institute of General Medical Sciences [Grant 1 U54GM104938, PI Judith James]. 
+ 
+ 
+Acknowledgement: 
+ 
+The authors thank Dr. Tabitha Garwe for important comments on the manuscript. 
+ 
+ 
+Conflict of Interest: None declared. 
+ 
+ 
+References: 
+ 
+[1] VanderWeele TJ, Knol MJ. A tutorial on interaction. Epidemiologic Methods 2014;3:33-72. 
+ 
+[2] Rothman KJ. Epidemiology: an introduction. New York: Oxford University Press, 2002. 
+ 
+[3] Vandenbroucke JP, von Elm E, Altman DG, et al. Strengthening the reporting of 
+observational studies in epidemiology (STROBE): explanation and elaboration. Epidemiology 
+2007;18(6):805‐835.  
+ 
+[4] Spiegelman D, Hertzmark E. Easy SAS calculations for risk or prevalence ratios and 
+differences. Am J Epidemiol 2005;162(3):199-200. 
+ 
+[5] Cheung YB. A modified least-squares regression approach to the estimation of risk 
+difference. Am J Epidemiol 2007;166(11):1337-44.  
+ 
+[6] Bieler GS, Brown GG, Williams RL, Brogan DJ. Estimating model-adjusted risks, risk 
+differences, and risk ratios from complex survey data.  Am J Epidemiol 2010; 171(5):618-623. 
+ 
+[7] Richardson DB, Kinlaw AC, MacLehose RF, Cole SR. Standardized binomial models for 
+risk or prevalence ratios and differences. Int J Epidemiol 2015;44(5):1660-72.  
+
+22 
+ 
+ 
+[8] Hammond EC, Selikoff IJ, Seidman H. Asbestos exposure, cigarette smoking and death rates. 
+Ann N Y Acad Sci 1979; 330:473-90.   
+ 
+[9] Rothman KJ, Greenland S. Modern epidemiology. Philadelphia: Lippincott Williams and 
+Wilkins, 1998. 
+ 
+[10] Rothman KJ. Modern epidemiology (1st Ed.). Boston:  Little, Brown and Company, 1986. 
+ 
+[11] Hosmer DW, Lemeshow S. Confidence interval estimation of interaction. Epidemiology 
+1992; 3(5):452-456. 
+ 
+[12] Szklo M, Nieto FJ. Epidemiology: beyond the basics (3rd Ed.). Sudbury, MA: Jones and 
+Bartlett, 2004. 
+ 
+[13] Andersson T, Alfredsson L, Källberg H, Zdravkovic S, Ahlbom, A. Calculating measures of 
+biological interaction. Eur J Epidemiol 2005; 20(7):575-579.  
+ 
+[14] Richardson DB, Kaufman JS. Estimation of the relative excess risk due to interaction and 
+associated confidence bounds, Am J Epidemiol 2009; 16(6):756–760.  
+ 
+[15] Zou G. A modified Poisson regression approach to prospective studies with binary data. Am 
+J Epidemiol 2004;159(7):702-706.  
+ 
+ 
+ 
+Contact information 
+ 
+David M. Thompson, Department of Biostatistics and Epidemiology, University of Oklahoma 
+Health Sciences Center, Oklahoma City, OK 73104 (e-mail: dave-thompson@ouhsc.edu).  
+ 
+Yan Daniel Zhao, Department of Biostatistics and Epidemiology, University of Oklahoma 
+Health Sciences Center, Oklahoma City, OK 73104 (e-mail: daniel-zhao@ouhsc.edu).  
+ 
+ 
+
+23 
+ 
+Supplementary File 1. Data example  
+ 
+SAS syntax that creates a dataset that approximates the one published in Hammond et al. [8].  
+Outcome is lung cancer deaths per 100,000 workers.  The prevalence of smoking is assumed to 
+be 0.28 for both the asbestos workers and for the comparison group of unexposed workers.   
+ 
+/*Data example*/ 
+data one; 
+  array asbn (2) (73763 17800); /*n in published study*/ 
+  array rate (4) (11.3 122.6 58.4  601.6);  
+                     /*lung ca deaths per 100k in published study*/ 
+  smokeprev=0.28;    /*assumed prevalence of smoking*/ 
+  do asbestos=0 to 1; 
+    do smk=0 to 1; 
+ 
+  do lungcadeath=0 to 1; 
+ 
+     mult=rate [2*asbestos + smk +1] / 100000; 
+ 
+     count1=asbn[asbestos+1] * 
+ 
+ 
+       (abs((1-smk)-smokeprev)) * 
+               (abs((1-lungcadeath)-mult)); 
+ 
+     count=round(count1,1); 
+ 
+     output; 
+        end; 
+    end; 
+  end; 
+keep asbestos smk lungcadeath count; 
+run; 
+proc sort data=one (keep=asbestos smk lungcadeath count) out=two; 
+  by descending asbestos descending smk descending lungcadeath; 
+run; 
+ 
+/*version of dataset with individual observations*/ 
+data long; 
+  set two; 
+  do i=1 to count; 
+    id+1; 
+    output; 
+  end; 
+run; 
+ 
+proc format; 
+  value smkf 1="Smokers" 0="Non-smokers"; 
+  value gpf   1="Asbestos Workers (n= 17800)" 
+              0="Comparison Group (n=73763)"; 
+  value death 1="Deaths due to lung CA"  
+              0="Alive or dead due to other causes"; 
+run; 
+ 
+/* Table 2. Lung cancer deaths (per 100,000 workers) among those with exposure 
+to asbestos and/or cigarette smoking*/ 
+proc freq data=two order=data; 
+  weight count; 
+  tables asbestos*smk*lungcadeath / nocol nopct outpct out=three; 
+  format smk smkf. asbestos gpf. lungcadeath death.; 
+run; 
+ 
+
+24 
+ 
+data four; 
+  set three; 
+  perhunthou=pct_row*1000; 
+run; 
+ 
+proc report nowd data=four; 
+  where lungcadeath=1; 
+  columns smk asbestos, perhunthou; 
+  define smk / group "Cigarette smoking" format=smkf. order=data; 
+  define asbestos / across "Asbestos Exposure" format=gpf. order=data; 
+  define perhunthou / analysis '' format=6.2; 
+run; 
+ 
+ 
+
+25 
+ 
+Supplementary File 2. SAS syntax for linear binomial model 
+ 
+The syntax below illustrates the construction of the linear binomial model [4,7] using SAS 
+PROC GENMOD.  A MODEL statement identifies the independent variables included in 
+comprise the linear predictor: smk (smoking status); asbestos (asbestos exposure status); and 
+smk*asbestos, the interaction between smoking status and asbestos exposure.  Options in the 
+MODEL statement specify that the outcome (lung cancer) follows a binomial distribution and 
+link it directly (through an identity link) to the linear predictor.  The LSMEANS statement 
+estimates the number of deaths per 100,000 workers for each combination of exposures.  The 
+ESTIMATE statement lists the four coefficients (1 − 1 − 1   1) that define the interaction 
+contrast (IC): 
+ 
+(1) 𝑝11 + (−1)𝑝10 + (−1)𝑝01 + (1)𝑝00 = 0 
+ 
+/*linear binomial model*/  
+proc genmod data=long descending; 
+  class smk (ref=first) asbestos (ref=first); 
+  model lungcadeath = smk asbestos smk*asbestos 
+          / link=identity dist=bin type3 wald;   
+  lsmeans smk*asbestos / cl; 
+  ods output lsmeans=lsmeans estimates=estimates parameterestimates=betas                
+modelanova=type3; 
+  estimate "IC" smk*asbestos 1 -1 -1 1; 
+run; 
+ 
+/*Syntax that includes a REPEATED statement, which initiates GEE estimation 
+of robust standard errors, advocated by Richardson et al.[7].*/ 
+proc genmod data=long descending; 
+  class smk (ref=first) asbestos (ref=first) id; 
+  model lungcadeath = smk asbestos smk*asbestos 
+          / link=identity dist=bin type3 wald; 
+  repeated subject=id / type=ind; 
+  lsmeans smk*asbestos / cl; 
+  estimate "IC" smk*asbestos 1 -1 -1 1; 
+run; 
+ 
+/*modification of linear binomial model advocated by Spiegelman and Herzmark 
+[4] for instances when convergence fails*/  
+proc genmod data=long descending; 
+  class smk (ref=first) asbestos (ref=first) id; 
+  model lungcadeath = smk asbestos smk*asbestos 
+          / link=identity dist=poisson type3 wald ;   
+  repeated subject=id / type=ind; 
+  lsmeans smk*asbestos / cl; 
+  estimate "IC" smk*asbestos 1 -1 -1 1; 
+run; 
+ 
+ 
+
+26 
+ 
+Supplementary File 3. SAS syntax for Table 3 and Figure 1 
+ 
+The syntax below uses data sets output from the linear binomial model (Supplementary Box S2) 
+to create Table 3 and Figure 1.   
+ 
+/* Table 3. Absolute risks (and risk differences) for death from lung cancer 
+(per 100,000 workers) for those with exposure to asbestos and/or cigarette 
+smoking, estimated by linear binomial model*/ 
+data mortality; 
+  set lsmeans; 
+  mortality=estimate*100000; 
+  ucl=upper*100000; 
+  lcl=lower*100000; 
+run; 
+proc print noobs data=mortality; 
+  var smk asbestos estimate mortality lcl ucl; 
+run; 
+ 
+/*estimate for interaction contrast (IC)*/ 
+data ic; 
+  set estimates; 
+  ic=meanestimate*100000; 
+  ic_lcl=meanlowercl*100000; 
+  ic_ucl=meanuppercl*100000; 
+run; 
+proc print noobs data=ic; 
+  var label meanestimate  ic ic_lcl ic_ucl probchisq; 
+  format meanestimate 9.6 probchisq 12.8 ; 
+run; 
+ 
+/*Estimates of regression coefficients, which are interpretable as excess 
+deaths*/ 
+data beta2; 
+  set betas (where=(df=1)); 
+  excessdeaths=estimate*100000; 
+  ucl=upperwaldcl*100000; 
+  lcl=lowerwaldcl*100000; 
+run; 
+proc print noobs data=beta2; 
+  var parameter estimate excessdeaths lcl ucl probchisq; 
+  format estimate 9.6 probchisq 12.8; 
+run; 
+ 
+/* Figure 1.  Biological interaction, between asbestos exposure and smoking, 
+illustrated as a non-additivity or heterogeneity of effects*/ 
+proc template; 
+  define style styles.mystyle; 
+  parent=styles.default; 
+    class graphbackground / color=white; 
+    style GraphData1 from GraphData1 / 
+          contrastcolor=black linestyle=1; 
+    style GraphData2 from GraphData2 / 
+          contrastcolor=black linestyle=2; 
+  end; 
+run; 
+
+27 
+ 
+ods html style=styles.mystyle; 
+proc sgplot data=mortality; 
+  series y=mortality x=smk / group=asbestos name="one" 
+            groupdisplay=cluster clusterwidth=0.05 
+            markers markerattrs=(symbol=squarefilled size=10); 
+  highlow x=smk high=ucl low=lcl / group=asbestos 
+            groupdisplay=cluster clusterwidth=0.05 
+            type=line lineattrs=(pattern=1) lowcap=serif highcap=serif; 
+  xaxis values=(0 1) label=" " valueattrs=(size=14 weight=bold); 
+  yaxis label="Lung cancer deaths per 100,000" 
+        labelattrs=(size=14 weight=bold) 
+        valueattrs=(size=14 weight=bold); 
+  format smk smkf. asbestos gpf.; 
+  keylegend "one" / title="" location=inside down=2 position=topleft 
+             valueattrs=(size=12 weight=bold) ; 
+run; 
+ods html close; 
+ 
+ 
+
+28 
+ 
+Supplementary File 4. Log binomial and logistic regression models 
+ 
+Log binomial model and depiction of its estimates in Figure 2.  
+ 
+proc genmod data=long descending; 
+  class smk (ref=first) asbestos (ref=first)  ; 
+  model lungcadeath = smk asbestos smk*asbestos 
+          / link=log dist=bin type3 wald lrci; 
+  lsmeans smk*asbestos / cl; 
+  ods output lsmeans=lsmeans ; 
+run; 
+ 
+ods html style=styles.mystyle; 
+proc sgplot data=lsmeans; 
+  series y=estimate x=smk / group=asbestos name="one" 
+            groupdisplay=cluster clusterwidth=0.05 
+            markers markerattrs=(symbol=squarefilled size=10); 
+  highlow x=smk high=upper low=lower / group=asbestos 
+            groupdisplay=cluster clusterwidth=0.05 
+            type=line lineattrs=(pattern=1) lowcap=serif highcap=serif; 
+  xaxis values=(0 1) label=" " valueattrs=(size=14 weight=bold); 
+  yaxis label="ln[p(Death from lung cancer)]" 
+        labelattrs=(size=14 weight=bold) 
+        valueattrs=(size=14 weight=bold); 
+  format smk smkf. asbestos gpf.; 
+  keylegend "one" / title="" location=inside down=2 position=topleft 
+             valueattrs=(size=12 weight=bold) ; 
+run; 
+ods html close; 
+ 
+Logistic regression model and depiction of its estimates in Figure 3.  
+ 
+proc genmod data=long descending; 
+  class smk (ref=first) asbestos (ref=first)  ; 
+  model lungcadeath = smk asbestos smk*asbestos 
+          / link=logit dist=bin type3 wald lrci; 
+  lsmeans smk*asbestos / cl; 
+  ods output lsmeans=lsmeans ; 
+run; 
+ 
+ods html style=styles.mystyle; 
+proc sgplot data=lsmeans; 
+  series y=estimate x=smk / group=asbestos name="one" 
+            groupdisplay=cluster clusterwidth=0.05 
+            markers markerattrs=(symbol=squarefilled size=10); 
+  highlow x=smk high=upper low=lower / group=asbestos 
+            groupdisplay=cluster clusterwidth=0.05 
+            type=line lineattrs=(pattern=1) lowcap=serif highcap=serif; 
+  xaxis values=(0 1) label=" " valueattrs=(size=14 weight=bold); 
+  yaxis label="Log odds of death from lung cancer" 
+        labelattrs=(size=14 weight=bold) 
+        valueattrs=(size=14 weight=bold); 
+  format smk smkf. asbestos gpf.; 
+  keylegend "one" / title="" location=inside down=2 position=topleft 
+             valueattrs=(size=12 weight=bold) ; 
+
+29 
+ 
+run; 
+ods html close; 
+ 
+ 
+ 
+ 
+ 
+ 
+

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@@ -0,0 +1,953 @@
+Quantum information entropy of heavy mesons in the presence of a point-like
+defect
+C. A. S. Almeida1a, C. O. Edet2b,c,d, F. C. E. Lima3a, N. Ali4c,e, and M. Asjad5f
+aUniversidade Federal do Cear´a (UFC), Departamento de F´ısica, Campus do Pici, Fortaleza-CE, 60455-760, Brazil.
+bInstitute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia.
+cFaculty of Electronic Engineering Technology, Universiti Malaysia Perlis, Malaysia.
+dDepartment of Physics, Cross River University of Technology, Calabar, Nigeria.
+eAdvanced Communication Engineering (ACE) Centre of Excellence, Universiti Malaysia Perlis, 01000 Kangar, Perlis, Malaysia.
+fDepartment of Mathematics, Khalifa University, Abu Dhabi 127788, United Arab Emirates.
+Abstract
+Using Schr¨odinger’s formalism, we investigate the quantum eigenstates of the heavy mesons trapped by a point-like
+defect and by Cornell’s potential. One implements this defect to the model considering a spherical metric profile
+coupled to it. Furthermore, the Nikiforov-Uvarov method is applied to theory to study the quantum eigenstates
+of the heavy mesons. To calculate the quantum information entropy (QIE), one considers the wave functions that
+describe the charmonium and bottomonium states. To explore the QIE, we use the well-known Shannon’s entropy
+formulated at the position and reciprocal space.
+The analysis of the QIE gives us relevant information about
+how the quantum information change with the variation of the point-like defect. Consequently, considering the
+Bialynicki-Birula and Mycielski (BBM) relation, we show how this defect influences the quarkonium position and
+momentum uncertainty measures.
+Keywords: Schr¨odinger equation; Nikiforov-Uvarov method; Cornell potential; Heavy mesons.
+1. Introduction
+The growing interest in non-relativistic quantum mechanical systems is notorious [1, 2, 3, 4]. This interest is
+because the results from quantum mechanics give us a good prediction of some phenomenological data [5, 6, 7].
+Furthermore, these systems are the first step toward understanding more complex models [8]. Thus, considering
+non-relativistic quantum mechanical systems, we found several studies in the literature [9, 10]. For example, one can
+find some studies on theoretical measurements of mass spectra of heavy mesons [11], harmonic oscillators [12, 13, 14],
+potential wells [16, 15], and solid-state physics problems (e. g., position-dependent mass problems) [17, 18, 19].
+Generally, depending on the type of interaction used to describe the quantum-mechanical system, there may
+be some difficulties in the analytical description. Thus, to bypass these difficulties, some techniques are applied.
+1Email address: carlos@fisica.ufc.br
+2Email address: collinsokonedet@gmail.com
+3Email address: cleiton.estevao@fisica.ufc.br
+4Email address: norshamsuri@unimap.edu.my
+5Email address: muhammad.asjad@ku.ac.ae
+Preprint submitted to Annals of Physics
+January 30, 2023
+arXiv:2301.11627v1  [hep-ph]  27 Jan 2023
+
+Among these techniques, there is the well-known Nikiforov-Uvarov (NU) method [3, 20, 21, 22], the Nikiforov-
+Uvarov Functional Analysis (NUFA) [23, 24, 25], the Laplace transformation method (LTM) [26], the asymptotic
+interaction method [27], the WKB approximation method [28]. However, it is relevant to highlight that among the
+techniques mentioned, the NU method stands out [29, 30, 31]. Indeed, the NU method stands out because it allows
+parameterizing Schr¨odinger’s equation [32], making it possible to find the solution for different physical potentials
+of interest [32]. Thus, we will conveniently use the NU method in this work.
+A relevant class of particles is mesons.
+The historical framework of these particles began in 1974 [33].
+It
+was in 1974 that the Brookhaven National Laboratory announced the discovery of a new particle called J [33].
+Simultaneously, the Stanford Linear Accelerator announced the existence of another particle called ψ [34].
+A
+relevant feature of these particles found is that they have similar properties. So has become possible to interpret these
+particles as a new quark (namely, a charm quark). With the discovery of these particles, it was possible to explain the
+suppression of weak kaon decays that change the flavor [35, 36]. Posteriorly, the fourth quark existence, the so called
+bottom quark, was confirmed [35, 36]. Only in 1978 Cornell’s model was developed. In its original proposal, Cornell’s
+potential took into account the quark-antiquark interaction in the heavy sector [37, 38, 39, 40]. Essentially the initial
+assumptions were that interactions arise SU(3) color gauge symmetry with flavor broken only by quark masses.
+Besides, this type of interaction admitted contributions from the Coulomb interaction (induced by a gluon exchange
+interaction).
+Furthermore, Cornell’s potential considered contributions from a phenomenological interaction of
+confinement, considered linear. The interactions were independent of flavor and spin, thus implementing the well-
+known Heavy Flavor Symmetry and Heavy Quark Spin Symmetry [37, 38, 39, 40].
+Seeking to investigate the quarkonia eigenstates, we will consider Cornell’s potential in our theory. This poten-
+tial proves to be effective in explaining quark confinement. Furthermore, this potential explains the mass of the
+quarkonium states and its relation with the angular momentum of the hadron (known as the Regge trajectory)
+[37, 38, 39, 40]. Briefly, the Cornell potential has the following form:
+V (r) = −4
+3
+αs
+r + σr + v0,
+(1)
+where r is the effective radius of the quarkium state, αs is the running coupling, σ is the QCD string tension, and
+v0 ≃ −0.3GeV. Note that this potential has two contributions. The first contribution (i.e., − 4
+3
+αs
+r ) dominates at
+short distances, in general for r < 0.1fm [41]. This short-range contribution is known as Yukawa’s potential. In this
+case, this contribution is due to the exchange of a gluon between the quark and its antiquark. Meanwhile, the second
+potential contribution (i.e., σr) is the linear confinement term responsible for describing the non-perturbative QCD
+effects that result in color confinement. In this scenario, σ is the tension of the QCD string that forms when the
+gluonic field lines collapse in a flow tube. One estimates that the tension value of the QCD string is σ ≃ 0.18GeV2
+[42]. Furthermore, it is important to mention that Cornell’s potential (1) has been gaining supporters in recent
+years. For example, the Cornell potential has been used in spectroscopy studies of mesons [43] and S-wave heavy
+quarkonia [44], among other studies.
+Not far from this discussion, we found, in literature, some works discussing the influence of defects in quantum
+eigenstates [45, 46, 47]. Indeed, the study of topological defects in quantum mechanics problems has aroused the
+2
+
+interest of several researchers [48, 49, 50], and this interest is due to its wide application. Generally speaking, there
+are three classes of defects, i. e., the cosmic string defect, the global monopole, and domain walls [51, 52]. Usually,
+one-dimensional cosmic string defects [53] and global monopoles [54, 55, 56] apply widely to quantum mechanical
+systems. Indeed, in this scenario, these defects were implemented, for example, in studies of harmonic oscillators
+[57, 58], charged particle scattering [59], and models with exotic interaction [60]. Considering these applications
+arises the question: How does the monopole-like defect influence the eigenstate of heavy mesons? Throughout this
+manuscript, we will seek to answer this question.
+Quantum information entropy (QIE) is a useful tool in studies on quantum information and uncertainty mea-
+surements in quantum-mechanical systems. Indeed, the QIE arises from a reinterpretation of Shannon’s information
+entropy [61]. Essentially, this reinterpretation, i. e., the QIE, and the Bialynicki-Birula and Mycielski [62] relation,
+gives us a generalized uncertainty measure of Heisenberg’s principle.
+Due to this, the QIE has gained search-
+light in studies on quantum-mechanical systems. Thus, one can find several works discussing the information of
+quantum-mechanical systems through the QIE formalism. For example, some applications of the QIE in the quan-
+tum system appear in investigations of hyperbolic potential wells [63], the Aharonov-Bohm effect [64], and effective
+mass problems [16].
+Considering the applications presented, two questions naturally arise. First, how do point-like global monopole
+(PGM) defects change the mesons’ eigenstates? Second, how does the quantum information modifies due to this
+defect? As far as we know, our work is the first to address these questions. Thus, the main purpose of this paper
+is to study the QIE of heavy mesons in the presence of a point-like global monopole defect.
+To reach our purpose, we organize the work as follows: In Sec. II, the quantum description of heavy mesons is
+exposed, assuming Cornell’s potential. In Sec. III, we present a discussion and numerical results of the QIE of the
+Charmonium and Bottomonium states of heavy mesons. Finally, in Sec. IV, the findings are announced.
+2. Quantum description of heavy mesons
+As discussed in Ref. [54], allow us to start by considering background spacetime with PGM defect. Particularly,
+let us assume a spherically symmetric spacetime described by the line element
+ds2 = −dt2 + ηijdxidxj,
+with
+i, j = 1, 2, 3,
+(2)
+where the metric tensor ηij is
+ηij =
+�
+�
+�
+�
+�
+1
+α2
+0
+0
+0
+r2
+0
+0
+0
+r2 sin2 θ
+�
+�
+�
+�
+�
+and
+ηij =
+�
+�
+�
+�
+�
+α2
+0
+0
+0
+1
+r2
+0
+0
+0
+1
+r2 sin2 θ
+�
+�
+�
+�
+� .
+(3)
+Here the α parameter is related to the PGM defect, which depends on the energy scale. Mathematically, α2 =
+1 − 8πGλ2 where λ is the energy scale and G the gravitational constant.
+Indeed, several researchers have adopted the PGM defect in their studies. For example, in Ref. [65], to investigate
+spin-0 particles in the presence of dyons and Aharonov-Bohm effect with scalar interaction, the authors adopted the
+3
+
+existence of a PGM defect. Furthermore, Ahmed [47] discussed the topological effects produced by this background
+on spin-0 particles subject to the Kratzed potential. Particularly the interest in this defect is because it induces
+topological effects that influence the particle’s quantum dynamics. Motivated by this, we consider that spinless
+mesons are in the spacetime (2).
+Considering the non-relativistic scenario, one writes Schr¨odinger’s equation as
+− ℏ2
+2µ∇2
+LBψ(r, t) + V (r, t)ψ(r, t) = iℏ∂ψ(r, t)
+∂t
+.
+(4)
+Here, µ is the particle’s mass and ∇2
+LB is the Laplace-Beltrami operator defined as
+∇2
+LB ≡
+1
+√g ∂i(√ggij∂j),
+(5)
+where g ≡ det(gij). Furthermore, we will assume, in principle, that the interaction V (r, t) = V (r), i. e., an arbitrary
+central potential.
+Assuming the Laplace-Beltrami operator (5) and the metric signature (2), we write Schr¨odinger’s equation as
+follows:
+− ℏ2
+2µr2
+�
+α2 ∂
+∂r
+�
+r2 ∂
+∂r
+�
++
+1
+sin θ
+∂
+∂θ
+�
+sin θ ∂
+∂θ
+�
++
+1
+sin2 θ
+� ∂2
+∂ϕ2
+��
+ψ(r, θ, ϕ, t) + V (r)ψ(r, θ, ϕ, t) = iℏ∂ψ(r, θ, ϕ, t)
+∂t
+.
+Allow us to particularize our study for the case of spinless heavy mesons. In this case, the interaction of the
+theory is
+V (r) = W1r − W2
+r .
+(6)
+This interaction is called Cornell’s potential. Essentially, Cornell’s potential is employed to model the quarkonium
+interaction. Besides, one finds, in literature, several investigations on quantum-mechanics systems adopting Cornell’s
+potential, e. g., see Refs. [66, 67].
+To study the wave eigenfunctions that the theory describes, let us assume that the particular solutions of Eq.
+(6) in terms of the eigenvalues of the angular momentum operator ˆL2 are
+ψ(r, t) = e−
+iEnlt
+ℏ
+Rnl(r)
+r
+Ylm(θ, ϕ)
+(7)
+where Ylm(θ, ϕ) are spherical harmonics and Rnl(r) is the radial wave eigenfunction.
+Substituting Eq. (7) into (6), we have that the radial part of Schr¨odinger’s equation for Cornell’s potential in
+the presence of the PGM defect is
+d2Rnl(r)
+dr2
++
+�2µEnl
+α2ℏ2 − 2µW1r
+α2ℏ2
++ 2µW2
+α2ℏ2r − l(l + 1)
+α2r2
+�
+Rnl(r) = 0.
+(8)
+Eq. (8) is not solvable in its current form. To solve this, allow us to adopt r → x−1. This change of coordinates
+leads us to
+d2Rnl(x)
+dx2
++ 2
+x
+dRnl(x)
+dx
++ 1
+x4
+�2µEnl
+α2ℏ2 − 2µW1
+α2ℏ2x + 2µW2x
+α2ℏ2
+− l(l + 1)x2
+α2
+�
+Rnl(x) = 0.
+(9)
+4
+
+Here let us implement an approximation scheme (AS) on the term W1
+x by assuming that there is a characteristic
+radius r0 of the meson. In this case, one obtains the expansion of W1
+x
+in a power series around r0, i. e., around
+δ ≡ 1/r0, up to the second order. By setting y = x − δ around y = 0, we obtain that
+d2Rnl(x)
+dx2
++ 2x
+x2
+dRnl(x)
+dx
++ 1
+x4
+�
+−˜ϵ + ˜β1x − ˜β2x2�
+Rnl(x) = 0,
+(10)
+where
+−˜ϵ = 2µEnl
+ℏ2α2 − 6µW1
+α2ℏ2δ ,
+˜β1 = 2µW2
+α2ℏ2 + 6µW1
+α2ℏ2δ2 ,
+and
+˜β2 = 2µW1
+α2ℏ2δ3 + l(l + 1)
+α2
+.
+(11)
+Eq. (10) is the solvable form of the NU method. The major equation closely related to this method is
+P ′′(x) + �τ(x)
+σ(x)P ′(x) +
+�σ(x)
+(σ(x))2 P(x) = 0.
+(12)
+Comparing to Eq. (12) with Eq. (10), it follows that
+˜τ(x) = 2x,
+σ(x) = x2
+and
+˜σ(x) = −˜ϵ + ˜β1x − ˜β2x2.
+(13)
+That explicitly shows that Eq. (10) satisfies the requirement of the NU approach. Furthermore, it is also worth
+noting that ˜σ(x) and σ(x) are polynomials of at most second degree, and ˜τ(x) is at most polynomials of the first
+degree. The NU method is popular among mathematical scientists and related disciplines. Several authors have
+used this method to solve problems of similar interest [29, 68, 69]. Although the method is quite popular, it will
+be useful to highlight some details to make our article self-contained. For this reason, we will detail this approach
+in the appendix (please, verify the appendix). Following the steps described in the appendix [Eqs. (A.1-A.8)], one
+obtains that the self-energies and eigenfunctions of our system are, respectively,
+Enl = 3W1
+δ
+− α2ℏ2
+8µ
+�
+�
+6µW1
+α2δ2ℏ2 + 2W2
+α2ℏ2
+n + 1
+2 +
+�
+1
+4 +
+2µW1
+δ3α2ℏ2 + l(l+1)
+α2
+�
+�
+2
+(14)
+and
+Rnl(r) = Nnlr
+˜
+β1
+2
+√
+˜ϵ e−rL
+˜
+β1
+√
+˜ϵ
+n (2
+√
+˜ϵr).
+(15)
+Figs. 1 and 2 expose the behavior of the radial wave eigenfunctions for the Charmonium and Bottomonium states
+of heavy mesons.
+3. The quantum information entropy
+Since the seminal paper by Claude E. Shannon on the mathematical theory of communication [61], there has
+been a growing interest in the studies of information entropies, e. g., see Refs. [16, 45, 70, 71, 72]. In quantum
+mechanics, part of this interest is because the quantum information entropies are related to the uncertainty measures
+of the quantum system [73, 74, 75]. Furthermore, this entropy tells us how good the description of the quantum
+states of the theory is [16, 45, 71, 72].
+5
+
+0
+5
+10
+15
+20
+25
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+r
+R00(r)
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+0
+5
+10
+15
+20
+25
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+r
+R10(r)
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+0
+5
+10
+15
+20
+25
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+r
+R11(r)
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+(a)
+(b)
+(c)
+Figure 1: Radial wave eigenfunctions of heavy mesons (Charmonium) when mc = 1.209 Gev, µ = 0.6045 Gev, W1 = 0.20 Gev, W2 =
+1.244 Gev, δ = 0.231 Gev, and ℏ = 1. Fig. (a) corresponds to the state n = 0 and l = 0. On the other hand, Fig (b) is the state n = 1
+and l = 0. The state n = 1 and l = 1 is in Fig. (c).
+0
+5
+10
+15
+20
+25
+0.0
+0.1
+0.2
+0.3
+0.4
+r
+R00(r)
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+0
+5
+10
+15
+20
+25
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+r
+R10(r)
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+0
+5
+10
+15
+20
+25
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+r
+R11(r)
+α=0.5
+α=0.6
+α=0.7
+α=0.8
+(a)
+(b)
+(c)
+Figure 2: Radial wave eigenfunctions of heavy mesons (Bottomonium) when mb = 4.823 Gev, µ = 2.4115 Gev, W1 = 0.2 Gev, W2 =
+1.569 Gev, δ = 0.378 Gev, and ℏ = 1. Fig. (a) corresponds to the state n = 0 and l = 0. On the other hand, Fig (b) is the state n = 1
+and l = 0. The state n = 1 and l = 1 is in Fig. (c).
+Quantum information entropies are related to uncertainty measures through the well-known Bialynicki-Birula
+and Mycielski (BBM) relation [62], i. e.,
+Sr + Sp ≥ D (1 + lnπ).
+(16)
+Here, Sr is the information entropy at position space; Sp is the information entropy at momentum space (or
+reciprocal space). Besides, D describes the spatial dimension of the system, e. g., D = 1 for the one-dimensional
+case, D = 2 for two-dimensional one, and D = 3 for three-dimensional one. In particular, the BBM relation to the
+quantum theory of the heavy mesons in the presence of point-like defects is Sr + Sp ≥ 6.43419. It is interesting to
+mention that the Bialynicki-Birula and Mycielski relation (16) is a sophisticated version of Heisenberg’s uncertainty
+principle. Thus, the results of quantum information entropy must respect this condition.
+For the quantum system discussed earlier (see Sec. 2), the information entropy is
+Sr = −
+ˆ 2π
+0
+ˆ π
+0
+ˆ ∞
+0
+|ψ(r, t)|2 ln[|ψ(r, t)|2]r2 sin θ dr dθ dφ.
+(17)
+6
+
+Meanwhile, the information entropy at the reciprocal space is
+Sk = −
+ˆ 2π
+0
+ˆ π
+0
+ˆ ∞
+0
+|ψ(k, t)|2 ln[|ψ(k, t)|]2 k2 sin θk dkr dkθ dkφ.
+(18)
+The spherical wave eigenfunction at reciprocal space is
+ψ(k, t) = (2π)3/2
+k−1/2
+∞
+�
+l=0
+(−i)l
+l
+�
+m=−l
+Ylm(kθ, kφ)
+ˆ ∞
+0
+J 1
+2 +l(krr)r3/2 dr
+ˆ 2π
+0
+ˆ π
+0
+ψ(r, t)Ylm(θ, φ)r2 sin θ dθ dφ.
+(19)
+Eq. (19) is Hankel’s transform of the wave eigenfunction ψ(r, t). Hankel’s transform is a transform whose kernels are
+Bessel functions. Thus, Hankel’s transform also is called Bessel’s transform. In general, Hankel’s transform is the
+two-dimensional Fourier transform of a circularly symmetric function or also spherical three-dimensional functions
+[76].
+Considering the definitions presented in Eqs. (17-18), we can study the quantum information entropy of heavy
+mesons. Thus, we begin our study by discussing the quantum information entropy of the Charmonium state.
+3.1. Quantum information entropy for the Charmonium
+Quarkonia, in particle physics, are hadronic states made of a heavy quark-antiquark pair. For the vector case,
+they are the charmonium meson J/ψ (made up of a cc pair) and bottomonium meson Υ (made up of a bb pair)
+and their corresponding radial excitations.
+In fact, in the literature, the word quarkonium refers to the state
+charmonium and bottomonium [77]. Briefly, Charmonium is a bound state of a quark and charmed antiquark. We
+found, in the literature, several investigations discussing the Charmonium states. For example, some investigations
+study about properties of hot gluonic plasma [78], the classification of states of mesons and their electromagnetic
+decays [79], and the collision of heavy ions [80].
+Motivated by these applications, let us now study the quantum information of this state. Thus, we will now
+particularize our study to the quantum information entropies of the Charmonium state. To inspect Charmonium’s
+information it is necessary to assume mc = 1.209 Gev, µ = 0.6045 Gev, W1 = 0.20 Gev, W2 = 1.244 Gev, δ =
+0.231 Gev, and ℏ = 1. Using these values, we describe eigenfunctions (7) of the Charmonium eigenstates. With
+the wave eigenfunctions (7), one constructs the probability densities of the theory, i. e., |ψ(r, t)|2. Substituting
+the probability density in the definition of information entropy (17) comes to the numerical results of the quantum
+information entropy at position space. In Tab 1, we expose the numerical result of the information at the position
+space.
+To calculate the information entropy associated with the momentum of the Charmonium, the Hankel
+transform (19) of the eigenfunctions (7) is calculated. Substituting the spherical wave eigenfunctions (19) in the
+definition (18), one obtains the numerical results of quantum information entropy at the momentum space shown
+in Tab. 1.
+The numerical results in Tab. 1 suggest that the quantum information entropy decreases at position space as
+the topological defect increase. In other words, when the point-like global monopole defect increase, the greater
+the quantum information of the Charmonium states. As a consequence of this behavior, the uncertainty measures
+related to the position measurements of the heavy mesons will be smaller when the α parameter increases. In
+7
+
+Table 1: Quantum entropy information for the heavy mesons (Charmonium state)
+Eigenstates (n, l, m)
+Parameter α
+Sr
+Sp
+Sr + Sp
+BBM relation
+(0,0,0)
+0.5
+9.35657
+-0.01643
+9.34014
+6.43419
+0.6
+8.67692
+-0.00799
+8.66893
+6.43419
+0.7
+8.11763
+0.01172
+8.12935
+6.43419
+0.8
+7.64474
+0.04550
+7.69024
+6.43419
+(1,0,0)
+0.5
+8.61339
+-1.65703
+6.95636
+6.43419
+0.6
+8.11752
+0.09796
+8.21548
+6.43419
+0.7
+7.72505
+0.13561
+7.86066
+6.43419
+0.8
+7.40575
+0.17689
+7.58264
+6.43419
+(1,1,0)
+0.5
+8.18144
+-0.99643
+7.18501
+6.43419
+0.6
+7.68558
+-0.90431
+6.78127
+6.43419
+0.7
+7.29310
+-0.83069
+6.46241
+6.43419
+0.8
+7.16383
+-0.72070
+6.44313
+6.43419
+(1,1,1)
+0.5
+8.20998
+-1.47165
+6.73833
+6.43419
+0.6
+7.78339
+-1.05141
+6.73198
+6.43419
+0.7
+7.43900
+-1.00400
+6.43500
+6.43419
+0.8
+7.15430
+-0.69458
+6.45972
+6.43419
+contrast, as the point-like global monopole defect becomes large, the information entropy associated with the
+momentum of the mesons decreases. Thus, one can conclude that when the PGM defect increase, the momentum
+uncertainty measures of the particles in the Charmonium eigenstate decrease. Furthermore, we note that the BBM
+relation to the Charmonium state of the quarks remains valid for all eigenstates. Finally, it is possible to note that
+for higher energy levels, when the parameter α → 1, BBM’s relation tends to the minimum uncertainty relation, i.
+e., Sr + Sp ≈ 6.43419.
+3.2. Comments on Bottomonium Quantum Information Entropy
+For the Charmonium state mc = 1.209 Gev, µ = 0.6045 Gev, W1 = 0.20 Gev, W2 = 1.244 Gev, δ =
+0.231 Gev, and ℏ = 1. For the Bottomonium status, these parameters assume the following values: mb = 4.823 Gev, µ =
+2.4115 Gev, W1 = 0.2 Gev, W2 = 1.569 Gev, δ = 0.378 Gev, and ℏ = 1. Notice that from the Charmonium state to
+the Bottomonium state, there is an increment of these values and the wave function profile is invariant. Thus, by
+inspection, one perceives that the quantum information entropy of the Bottomonium state has a similar behavior
+to the Charmonium state. Indeed, the quantum information entropy Sr decreases at the position space as the
+parameter α (the PGM defect) increases. Meanwhile, the information entropy Sp increases at the momentum space
+when the parameter α decreases. The variation of the Sr and Sp information must vary so that the BBM relation
+remains valid. These results of the quantum information entropy of the Bottomonium state (and Charmonium pre-
+8
+
+sented previously) of the quarks agree with the predictions presented in Ref. [77] using the configurational entropy
+formalism.
+4. Conclusion
+In this work, we investigated the Charmonium and Bottomonium states that describe heavy mesons using
+Schr¨odinger’s theory in the presence of a PGM defect. To carry out this study, we considered a curved background
+coupled with the PGM defect. Using the Nikiforov-Uvarov (NU) formalism, the quantum eigenstates of the heavy
+mesons were studied. Then, considering the wave functions that describe the quarkonium eigenstates, we investigate
+the QIE of heavy mesons using Shannon’s formalism.
+Analyzing quantum self-states that describe the heavy mesons, one notes that the radial eigenfunctions are the
+associated Laguerre polynomials. Furthermore, to investigate the self-states, the NU method is considered. This
+method leads us to the energy spectrum shown in Eq. (14). This spectrum tells us that in the classical limit
+En,l→∞ ≈ 3W1/δ, i. e., the energy will depend purely on W1, the contribution of short-range of the potential.
+Meanwhile, the fundamental states will depend on short- and long-range contributions of Cornell’s potential. Be-
+sides, one perceives that the PGM defect alters the quantum eigenstates of the particles as it varies. Indeed, when
+α increases, the wave function amplitude increases. Thus, the larger the α parameter, the greater the probability
+of finding quarkonium in regions close to the PGM defect.
+Analyzing the QIE, we perceive that the quantum information decreases (at the position space) as the PGM
+defect increases. On the other hand, it increases if the contribution of the PGM defect decreases. In contrast,
+at the reciprocal space, an opposite behavior is observed. Furthermore, it is necessary to mention that behaviors
+of the quantum information are consequences of the wave function profiles that describe the Charmonium and
+Bottomonium states. As a direct consequence, also, of this result, it is noted that the momentum’s uncertainty
+measures of the heavy mesons decrease as the position uncertainties increase. Thus, it is possible to notice that the
+BBM uncertainty relation remains valid for all energy levels. Furthermore, the BBM relation tends to a minimum
+value for higher energy states when α → 1. This occurs because, in this scenario, the PGM defect traps the heavy
+meson.
+This work has some prospects. Among them, a direct perspective of this work is the study of the quantum
+dynamics of heavy mesons in the presence of other classes of defects, e. g., string-like or domain wall defects.
+Another perspective is the study of the thermodynamic and statistical properties of these particles when subjected
+to thermal baths. We hope to carry out these studies soon.
+Authors Declaration
+Funding
+This research has been carried out under LRGS Grant LRGS/1/2020/UM/01/5/2 (9012-00009) Fault-tolerant
+Photonic Quantum States for Quantum Key Distribution provided by Ministry of Higher Education of Malaysia
+(MOHE).
+9
+
+Conflicts of interest/Competing interest
+All the authors declared that there is no conflict of interest in this manuscript.
+Acknowledgements
+C. O. Edet acknowledges eJDS (ICTP). C. A. S. Almeida thanks to Conselho Nacional de Desenvolvimento
+Cient´ıfico e Tecnol´ogico (CNPq), no 309553/2021-0. F. C. E. Lima is grateful to Coordena¸c˜ao de Aperfei¸coamento
+de Pessoal de N´ıvel Superior (CAPES), no 88887.372425/2019-00.
+Appendix A. Review of the Nikiforov-Uvarov (NU) Method
+Initially proposed by Nikiforov and Uvarov, the NU method seeks to solve Hypergeometric-type differential
+equations of the form of Eq. (12) [81, 82]. One can obtain the solutions of Eq. (12) employing the trial wave
+function:
+P(x) = Φ(x)yn(x),
+(A.1)
+which reduces the differential equation (12) to a Hypergeometric-type differential equation of the form:
+σ(x)y′′
+n(x) + τ(x)y′
+n(x) + λyn(x) = 0.
+(A.2)
+Allow us to highlight that the function Φ(x) is defined as
+Φ′(x)
+Φ(x) = π(x)
+σ(x),
+(A.3)
+where π(x) is a polynomial of first-degree. Meanwhile, the second term in Eq. (A.1) is the hypergeometric function
+with its polynomial solution given by Rodriques relation, i. e.,
+yn(x) = Bn
+ρ(x)
+dn
+dsn [σnρ(x)]
+(A.4)
+The term Bn is the normalization constant, and ρ(x) is known as the weight function, which in principle must
+satisfy the condition given;
+d
+ds [σρ(x)] = τ(x)ρ(x)
+(A.5)
+where τ(x) = ˜τ(x) + 2π(x).
+Naturally, we note here that the derivative of τ(x) should be τ(x) < 0. Thus, one can obtain the eigenfunctions
+and eigenvalues using the expression defined by π(x) and λ, i. e.,
+π(x) =σ′(x) − ˜τ(x)
+2
+±
+��σ′(x) − ˜τ(x)
+2
+�2
+− ˜σ(x) + kσ(x)
+(A.6)
+and
+λ = k + π′(x).
+(A.7)
+Considering the discriminant of the square root [in Eq. (A.6)] equal to zero, we obtain the value of k. In this
+case, the new eigenvalue equation is
+λ + nτ ′(x) + n(n − 1)
+2
+σ′′(x) = 0,
+with
+n = 0, 1, 2, . . .
+(A.8)
+10
+
+Appendix B. Solutions in Detail
+Substituting ˜σ(x) = −˜ϵ + ˜β1x − ˜β2x2 into Eq. (A.6), we arrived at
+π(x) = ±
+�
+˜ϵ − ˜β1x + (˜β2 + k)x2.
+(B.1)
+The discriminant of the quadratic expression under the square root above is
+k =
+˜β2
+1 − 4˜β2˜ϵ
+4˜ϵ
+.
+(B.2)
+Substituting Eq. (B.2) in (B.1), one obtains
+π(x) = ±
+� ˜β1x
+2
+√
+˜ϵ
+− ˜ϵ
+√
+˜ϵ
+�
+,
+(B.3)
+where
+π′(x) = −
+˜β1
+2
+√
+˜ϵ
+.
+(B.4)
+Thus, we can conclude that
+τ(x) = 2x −
+˜β1x
+√
+˜ϵ
++ ˜ϵ
+√
+˜ϵ
+and
+τ ′(x) = 2 −
+˜β1
+√
+˜ϵ
+,
+(B.5)
+which leads us to
+˜β2
+1 − 4˜β2˜ϵ
+4˜ϵ
+−
+˜β1
+2
+√
+˜ϵ
+= n˜β1
+2
+√
+˜ϵ
+− n2 − n
+(B.6)
+Eq. (B.6) yields the energy equation of the Cornell potential presented in Eq. (14)
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+Material vs. structure: Topological origins of band-gap truncation resonances in
+periodic structures
+Matheus I. N. Rosa1, Bruce L. Davis2, Liao Liu2, Massimo Ruzzene1,2, Mahmoud I. Hussein2,3
+University of Colorado Boulder, Boulder, CO, 80303
+Abstract
+While resonant modes do not exist within band gaps in infinite periodic materials, they may appear as in-gap localized
+edge modes once the material is truncated to form a finite periodic structure. Here, we provide an analysis framework that
+reveals the topological origins of truncation resonances, elucidating formally the conditions that influence their existence
+and properties. Elastic beams with sinusoidal and step-wise property modulations are considered as classical examples of
+periodic structures. Their non-trivial topological characteristics stem from the consideration of a phason parameter that
+produces spatial shifts of the property modulation while continuously varying how the boundaries are truncated. In this
+context, non-trivial band gaps are characterized by an integer topological invariant, the Chern number, which is equal to
+the number of truncation resonances that traverse a band gap as the phason is varied. We highlight the existence of multi-
+ple chiral edge states that may be localized at opposite boundaries, and illustrate how these can be independently tuned by
+modified boundary-specific phason parameters. Boundary phasons modify the truncation of only one boundary at a time.
+Furthermore, we show that the frequency location of a truncation resonance is influenced by the modulation wavelength,
+modulation volume fraction, boundary conditions, and number of cells comprising the finite structure, thus quantify-
+ing its robustness to these factors. Non-topological in-gap resonances induced by a defect are also demonstrated, with
+their frequency dependence on the phason investigated to elucidate their contrast to truncation resonances. A coupling
+between topological and non-topological modes is shown to be possible when the defect is located at an edge. Finally,
+experimental investigations on bi-material phononic-crystal beams are conducted to support these findings. Our results
+provide a fundamental perspective on the topological character of truncation resonances in periodic structures and how
+this character relates to the underlying periodic material properties. The tunability of these unique structural resonances
+through material-property modulation may be exploited both in applications where in-gap resonances are not desired,
+such as vibration attenuation and thermal conductivity reduction, or where in-gap resonances provide a functional role,
+such as filtering, waveguiding, energy harvesting, and flow control by phononic subsurfaces.
+Keywords: Phononic materials, band-gap resonances, topological protection, phasons, experimental phononics
+1. Introduction
+The study of elastic wave propagation in a continuous periodic medium is a classical problem in mechanics that can be
+traced back to Rayleigh in 1887 [1]. With the advent of composite materials, the interest in this problem surged with early
+contributions in the 1950s [2] and 1960s [3] formulating dispersion relations for wave propagation in laminated compos-
+ites, and other forms of periodic media [4, 5], followed by extension to multi-dimensional composites in the 1970s [6]. The
+field re-emerged in the early 1990s with the study of phononic crystals [7, 8] and the establishment of formal connections
+with lattice dynamics in crystals [9], and gathered further pace with the rise of acoustic and elastic metamaterials [10].
+In all these studies, periodicity is utilized enabling dynamic characterization by considering a representative unit cell, as
+commonly done in condensed matter physics [11]. Calculating the dispersion relation, or the band structure, using the
+Floquet/Bloch theorem [12, 13] formally enforces the assumption of an extended medium with an infinite number of unit
+cells. This is not only computationally rewarding, but physically provides a fundamental description of the modal wave
+propagation properties of the medium under investigation−removing any influence of overall size and external bound-
+ary conditions. In this framework, the medium under consideration is rendered a material with characteristic intrinsic
+properties, such as band gaps (whose locations may be predicted analytically [14–17]) and other key features revealed
+Email addresses: massimo.ruzzene@colorado.edu (Massimo Ruzzene ), mih@colorado.edu (Mahmoud I. Hussein )
+1Paul M. Rady Department of Mechanical Engineering
+2Ann and H.J. Smead Department of Aerospace Engineering Sciences
+3Department of Physics
+Preprint submitted to Elsevier
+arXiv:2301.00101v1  [cond-mat.mtrl-sci]  31 Dec 2022
+
+by the nature of the band structure. The thermal conductivity, for example, is an intrinsic material property that is di-
+rectly influenced by the band structure−determined by analysis of only a single atomic-scale unit cell [18, 19]. Effective
+dynamic properties, such as effective density and Young’s modulus [20], provide another example of intrinsic material
+properties. On the other hand, unless a medium practically comprises thousands or millions of unit cells (as in a bulk
+crystal for example), realistic realizations are formed from a relatively small finite number of unit cells, yielding a periodic
+structure, rather than a material, with extrinsic properties. This is particularly the case in engineering problems such as
+sound [21] and vibration [22] isolation, and other similar applications [23, 24], and also the case in nanoscale thermal
+transport [25] where unique dynamical properties emerge primarily from the presence of finite size along the direction of
+transport.
+1.1. Truncation resonances
+A periodic structure in practice may still consist, in some cases, of a relatively large but tractable number of unit cells,
+and in other cases, of only a few unit cells along the direction of vibration transmission. The number of cells impacts the
+degree of attenuation within a band gap [26]. However, the contrast between the material and structure behavior may not
+be limited to only quantitative differences but also to fundamental qualitative distinctions. One noticeable anomaly be-
+tween the material and structure responses is the possibility of existence of resonances inside band gaps, i.e., resonance
+peaks in the frequency response function (FRF) of a finite periodic structure that appear within band-gap frequency ranges
+of the corresponding infinite periodic material. These resonances are often referred to as truncation resonances [27, 28] be-
+cause they emerge from the truncation of a medium that is otherwise formed from an infinite number of unit cells. These
+resonances are associated with mode shapes that localize at the truncation junction, and are thus also commonly referred
+to as edge or surface modes [29–37]. The presence of these modes has been uncovered theoretically by Wallis [29] in his
+study of a finite discrete diatomic chain of atoms with free ends. This followed the work of Born on finite atomic chains [38]
+which was motivated by the study of the influence of lattice vibrations on X-ray scattering. Recent studies extended Wallis’
+theory of finite discrete chains to more general conditions [28, 39, 40] and experiments on chains of discrete-like coupled
+spheres validated the theory [35].
+The problem of truncation resonances in continuous periodic media−the focus of this paper−has also been inves-
+tigated extensively. Early studies examined one-dimensional wave propagation in periodically layered/lamenated com-
+posites, also referred to as superlattices. Existence conditions for truncation resonances were derived for semi-infinite
+superlattices for out-of-plane [30, 32, 34] and in-plane [31, 33, 37] waves. It was shown that surfaces modes in some in-
+stances may appear below the lowest bulk band, i.e., the band that hosts conventional resonances. Investigations of the
+truncation phenomenon were also done on finite layered phononic crystals examining transverse waves [36, 41], on finite
+beam-based phononic crystals [42, 43] and locally resonant elastic metamaterials [44, 45], and on rod-based phononic
+crystals [46, 47]. Among the factors that influence the frequency location of the truncation resonances are the unit-cell
+symmetry and the boundary conditions [41, 43, 45–47]. When there is more than one layer in the unit cell, the number
+of surface states increases [34, 37]. Techniques proposed for control of the truncation resonances also include tuning of
+unit-cell spatial material distribution or volume fraction [42], and the anomalous addition of a “cap layer" [32, 34] or a
+“tuning layer" [42, 48] at the edge of the structure. A cap layer is simply a homogeneous layer, whereas a tuning layer is
+a purposefully truncated single unit cell. The concept of truncation resonances is also relevant to other areas in applied
+physics such as photonic crystals [49] and quantum lattices [41].
+1.2. Connection to topological physics
+The principle of a truncation resonance is fundamentally connected to the periodic structure’s topological proper-
+ties; this connection forms the core focus of the present study. Inspired by the emergence of topological insulators in
+condensed matter physics [50], classical analogues have been developed in photonics [51] and phononics [52], demon-
+strating the features of robust topological waves. In passive elastic materials, topological interface modes are created
+by contrasting two materials with band gaps existing at the same frequencies, but characterized by different topological
+invariants. Examples include interface modes in one-dimensional (1D) structures [53–56] in analogy to the Su-Schrieffer-
+Heeger model [57], and waveguiding along interfaces in two-dimensional (2D) materials in analogy to the Quantum Spin
+Hall Effect [58, 59] or to the Quantum Valley Hall Effect [53, 60, 61]. These effects rely on symmetry breaking by interfacing
+two domains whose unit cells have opposite symmetries, which results in contrasting topological properties in the recip-
+rocal space. Hence, an actual interface between two materials is required, which presents a contrast to the truncation
+resonances we explore in this paper. We will show an intriguing connection that stems from a stronger type of topolog-
+ical effect associated with the Quantum Hall Effect (QHE) [62, 63]. The QHE manifests in 2D lattices of electrons under
+the presence of a strong magnetic field, which leads to robust edge waves that propagate along the boundaries of a finite
+sample (structure), without backscattering at corners or defects. It is therefore sufficient to exploit the interface between
+a single material medium and vacuum. However, such a strong effect requires breaking time reversal symmetry, which in
+2
+
+the quantum case is achieved through the magnetic field. Emulating similar features on 2D elastic materials is possible
+through active components that break time reversal symmetry, such as rotating frames [64] or gyroscope spinners [65, 66].
+An alternative that has emerged later, and which we adopt here, is to map the QHE to 1D passive structures that have
+extended dimensionality emanating from their parameter spaces [67, 68]. This has been achieved by using patterned me-
+chanical spinners [69], spring-mass lattices [70], acoustic waveguides [71, 72], and continuous phononic crystals or elastic
+metamaterials with modulations of inclusions such as ground springs [73], stiffeners [74], and resonators [75, 76]. In these
+examples, edge states localized at the boundaries of 1D periodic and quasi-periodic finite domains are observed to ap-
+pear in correspondence to non-zero topological inavariants called Chern numbers. The boundary at which the localization
+occurs can be determined by a phason parameter that is associated with spatial shifts in the medium’s modulated prop-
+erties. This feature leads to possibilities for topological pumping by varying the phason parameter continuously along
+time [77–79] or along a second spatial dimension [70, 80], inducing a transition of the edge states from being localized at
+one boundary to the other. Thus, energy can be "pumped" between two boundaries of a system through a transition of a
+topological edge state. The application of the field of topology to elastic and acoustic material systems has been attracting
+much interest in recent years [52, 81, 82].
+In this paper, we provide a formal framework for the identification of the topological character of truncation reso-
+nances in periodic structures, drawing on concepts from the QHE. We consider a family of periodic elastic beams with
+either sinusoidal or step-wise property modulations. The modulations offer key parameters that expand the structure’s
+property space and allow us to readily apply the concepts of topological band theory. In particular, the variation of a peri-
+odic beam’s spectral properties with respect to the modulation wavelength allows us to extract the Chern numbers of the
+band-gaps and identify the locations of truncation resonances. Then, the phason parameters associated with spatial shifts
+of the modulations further characterize the truncation resonances as topological edge states spanning the band gaps. The
+frequency dependence of the location of a truncation resonance on the phason has recently been predicted, for periodic
+rods, by means of a closed-form transfer-matrix-based mathematical formulation [47]. Here, we investigate, for periodic
+flexural beams, the topological origins of this class of relations. We show that the number of truncation resonances within
+a gap is equal to the predicted Chern number, for any set of boundary conditions, although the particular features of their
+branches as they traverse the gaps may vary. We elucidate how additional boundary phason parameters can be defined,
+formalizing the notion of the tuning layer [42, 48], to manipulate the edge states localized at different boundaries indepen-
+dently. Furthermore, we examine the convergence of the truncation resonant frequencies as a function of the number of
+unit cells−a matter of significant practical importance especially when this number is relatively small. The fundamental
+differences, and the possibility of coupling, between truncation resonances and corresponding non-topological defect-
+mode resonances are then investigated. Next, we provide laboratory results using a bi-material phononic-crystal beam
+as experimental validation of some of the key features of truncation resonances and their association with topological
+theory. Finally, we use our experiments to explore yet another important factor in the design space, namely the role of the
+materials’ volume fraction within the unit cell in influencing the frequency locations of the truncation resonances.
+The paper is organized as follows: following this introduction, Section 2 provides a description of the considered pe-
+riodic flexural beams and their boundary truncation through phasons. Next, Section 3 develops the theory and compu-
+tational analysis to characterize the topological properties of truncation resonances and those of non-topological defect
+resonances, and the coupling of the two types of resonances, followed by Section 4 which provides experimental results
+and further analysis. Finally, Section ?? has a general discussion on the key findings and their broader implications to
+related areas of research, and Section 6 provides a closing summary and outlines possible future research directions.
+2. Modulated phononic-crystal beams: Truncation characterization by phasons
+We consider elastic beams undergoing flexural motion described by transverse displacement w = w(x) and angle of
+rotation ϕ = ϕ(x), where x is the axial position, as classical examples of 1D periodic materials or structures. The properties
+of the beam are the Young’s modulus E = E(x), shear modulus G = G(x), density ρ = ρ(x), cross-sectional area A = A(x),
+and second moment of area I = I(x). These properties are modulated in space as illustrated in Fig. 1. Two scenarios are
+considered; in the first the Young’s modulus is modulated according to a cosine function, i.e. E(x) = E0[1+αcos(2πθx −
+φ)], while other parameters remain constant (Fig. 1(a)). This cosine-modulated phononic crystal (CM-PnC) serves as an
+idealized continuous periodic waveguide used to illustrate the behavior of interest in a simple setting. It is characterized
+by a unit cell of length a = 1/θ, where α is the amplitude of the modulation with respect to the mean value E0 and θ may
+be viewed as the modulation wavenumber. The second case corresponds to a beam modulated in a step-wise fashion,
+which we refer to as step-wise modulated phononic crystal (SM-PnC). It generically represents a periodic material of
+two alternating layers of lengths a1 and a2, with different constituent material or geometrical (e.g. cross-sectional area)
+3
+
+properties. In this case, the material or geometrical properties are modulated through a step-wise function of period
+a = 1/θ = a1 + a2 that takes two different values in the intervals of length a1 and a2.
+The appearance of in-gap resonances stems from the truncation of the boundaries. The truncation details are here
+characterized by phason parameters that are connected to non-trivial topological properties. The most natural choice of
+the phason is simply the phase φ of the property modulations, which rigidly shifts the modulation in space. Thus it results
+in a simultaneous change of the local properties of the beam at both boundaries. This is illustrated in the schematics of
+Fig. 1 for both the sinusoidal and step-wise modulations. The blue boxes highlight the region of the modulations selected
+to form the properties of the finite beams. From a given initial configuration, a change in phason over the range 0 < φ < 2π
+(higher values of φ do not need to be considered due to the periodicity) can be interpreted as simultaneously adding a
+segment of length φa/2π to the left boundary, while removing the same length from the right boundary. This will naturally
+influence any vibration mode localized at either boundary. It’s effect can be further understood as the superpostion of two
+independent parameters which we call boundary phasons. A change in the right boundary phason φr corresponds to
+removing a length φr a/2π from the right boundary while keeping the left boundary unchanged, while a change in the
+left boundary phason φl corresponds to adding a length φl a/2π to the left boundary while keeping the right boundary
+unchanged. Hence, changing the phason φ corresponds to changing both the left and right boundary phasons by the
+same amount (as illustrated in the figure). As we will show, the boundary phasons independently tune the topological
+truncation resonances at their respective boundary, and their superimposed effect leads to the variation of the resonances
+with respect to the conventional phason φ.
+Herein, the flexural motion of the beam is modeled through Timoshenko theory as governed by the following two
+coupled equations:
+ρA ∂2w
+∂t2 − q(x,t) = ∂
+∂x
+�
+κsAG
+�∂w
+∂x −ϕ
+��
+,
+(1a)
+ρI ∂2ϕ
+∂t2 = ∂
+∂x
+�
+EI ∂ϕ
+∂x
+�
++κsAG
+�∂w
+∂x −ϕ
+�
+,
+(1b)
+where κs denotes the shear coefficient, and t and q = q(x,t) represent time and the external forcing, respectively. Equa-
+tions 1a and 1b are combined to yield a single fourth-order partial differential equation with only w as the dependent
+variable [83]. In our investigation, we consider three types of problems: a Bloch dispersion analysis problem for a unit-cell
+representing an infinite material, an eigenvalue analysis problem for a finite structure with arbitrary boundary conditions
+(BCs), and a harmonic forced-response problem for a finite structure with arbitrary BCs. In the first two problems, we set
+Figure 1: Elastic periodic beams with (a) sinusoidal and (b) step-wise property modulation whose spatial distribution is defined by a phason φ or
+boundary phasons φr and φl . A modulation characterized by φ is a superposition of modulations characterized by φr and φl .
+4
+
+(a)
+1
+E(x),p
+w(x,t)
+X
+D
+Φ(b)
+1
+E(x),p(x)
+X
+-q = 0 and
+w(x,t) = ˆwei(µx−ωt),
+(2)
+where ω denotes the frequency. In Eq.2, we set 0 ≤ µ ≤ π/a for the Bloch dispersion problem, where µ = 0 is used for
+the finite periodic-structure eigenvalue with arbitrary BCs. The results are obtained by a finite-element discretization of
+the equations of motion. The implementation details of these methods are omitted here for brevity since they are widely
+available in the literature (for example, see Ref. [84]).
+Motivated by the experimental portion of this work (see Section 4), we select the following parameters. The SM-PnC
+consists of a bi-material beam composed of alternating layers of Aluminum (Al) and the polymer acrylonitrile butadiene
+styrene (ABS). These materials are selected due to the contrast of mechanical properties leading to wide band gaps. Their
+properties are as follows: Young’s moduli EAl = 68.9 GPa and EABS = 2.4 GPa, shear moduli GAl = 25.9 GPa and GABS = 0.872
+GPa, and densities ρAl = 2700 kg/m3 and ρABS = 1040 kg/m3, respectively. While we will allow the unit-cell length to vary
+through the θ parameter, the ABS polymer length filling fraction is fixed as aABS/a = 0.2; this ratio will be changed only in
+Section 4.3. For purposes of comparison, the properties of the CM-PnC are then chosen to make it statically equivalent [26]
+to the SM-PnC by selecting a fixed density ρ0 = (0.2ρABS + 0.8ρAl) and elastic modulus modulation with a mean value of
+E0 = (0.2/EABS +0.8/EAl)−1. We consider a Poisson’s ratio of ν = 0.33, which consequently determines the shear modulus
+through the relation G = E/(2(1+ν)). Throughout this paper, the CM-PnC modulation amplitude is fixed at α = 0.9, and
+the beams have a square cross-section geometry with side length h = 2.54cm. The finite-element analysis follows by
+discretizing the beams with linear Timoshenko beam elements with a shear coefficient of 5/6. The beam element length
+varies according to the case studied but does not exceed a maximum length of ¯a/100, where ¯a = 203 mm is the unit-cell
+size of the experimental beams and is used as a reference unit-cell length throughout the paper.
+Figure 2 presents a comparison between the properties of the CM-PnC and SM-PnC for the reference unit-cell size
+¯a = 203 mm, highlighting the contrast between material and structure. Panels (a) and (b) display their dispersion diagrams
+in a frequency range of interest from 0-9 kHz, which is a material feature. Both CM-PnC and SM-PnC exhibit the same
+long-wave static limit that approaches the dispersion of the homogenized beam with material property constants ρ0,E0
+(dashed lines), but display different band-gaps (shaded gray regions). In particular, the SM-PnC has wider gaps due to
+its discrete nature and the contrast of both densities and elastic moduli, while the CM-PnC has smaller gaps due to a
+fixed density and a continuous variation of the elastic modulus only. On the right side of the dispersion diagrams, the
+eigenfrequencies of representative finite beams with 15 unit cells and free-free BCs are plotted as black dots, with φ = 0.2π
+and φ = 0.4π selected for the CM-PnC and SM-PnC beams, respectively. Truncation resonances are observed to appear
+in band gaps, a feature which is unique to the structure, non-existing at the material level. An arbitrary phason value
+is chosen here to produce a large number of truncation resonances as an example, but the behavior with the full range
+of φ will later be explored and explained. The truncation resonances are localized at one of the two boundaries of the
+finite beams, with selected mode shapes displayed in Figs. 2(c-f). By looking at such isolated cases (as has been largely
+done in previous studies), there is no apparent reason or pattern pertaining to the appearance of in-gap resonances, why
+are they localized at one boundary instead of the other, and why these features can change by selecting different BCs or
+different numbers of unit cells, etc. In the following sections, we will shed light on all of these questions by illustrating the
+topological character of in-gap truncation resonances associated with non-zero Chern numbers, and consequently how
+they can be manipulated through the phason and other parameters or design features.
+3. Topological properties of modulated phononic-crystal beams
+In this section we develop the theoretical tools for the topological characterization truncation resonances by examin-
+ing their behavior inside band gaps. We begin by investigating the effect of the modulation wavenumber θ, which allows
+us to extract the topological invariants (Chern numbers). We then show how the Chern numbers are related to in-gap
+truncation resonances through the variation of the phason parameters. We also study the effect of the number of unit
+cells comprising the finite structure on the convergence of the truncation resonance frequencies. Finally, we provide a
+comparison between topological truncation resonances and non-topological defect resonances, highlighting their key
+differences and demonstrating the possibility of their coupling as a defect is moved towards a boundary.
+3.1. Topological characterization by the Chern number
+In principle, the Chern number characterizes the topology of a vector field defined over a two-dimensional torus. For
+2D periodic materials the torus is composed of two orthogonal wavenumber coordinates κx and κy and describes the
+reciprocal space Brillouin zone [53, 58, 59, 61, 85]. For 1D modulated materials such as the considered beams, the phason
+φ serves as an additional dimension and replaces the missing wavenumber component to form a torus based on κ and
+φ. [70]. The eigenvector field is the Bloch mode displacement ˆwn(κ,φ) corresponding to the nth band defined over the
+torus (κ,φ) ∈ T2 = [0,2π]×[0,2π], recalling that the dispersion is 2π-periodic in both φ and κ, with κ = µa defined as the
+5
+
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+0
+��ā
+Wavenumber, ��(m-1)
+I
+II
+0
+��ā
+Wavenumber, ��(m-1)
+I
+II
+0
+1
+-10
+5
+10
+15
+Displacement, w
+Position, x/ā
+0
+1
+-10
+5
+10
+15
+Position, x/ā
+Mode I
+Mode II
+0
+5
+10
+15
+Position, x/ā
+Mode I
+0
+5
+10
+15
+Position, x/ā
+Mode II
+(a)
+(b)
+(e)
+(f)
+(c)
+(d)
+Displacement, w
+15 unit cells
+15 unit cells
+15 unit cells
+15 unit cells
+Structure
+Material
+Structure
+Material
+Figure 2: Material versus structure properties. Dispersion diagrams (material) for the CM-PnC and the SM-PnC models are displayed in (a-b) as solid
+lines, while dashed lines correspond to the homogenized beam dispersion. Band-gap frequency ranges are shaded grey. A finite structure with 15 unit
+cells exhibits in-gap truncation resonances as illustrated alongside the dispersion diagrams, with selected mode shapes displayed in (c-f). For both
+models, the unit-cell length is ¯a = 203 mm.
+non-dimensional wavenumber. Due to the continuous nature of the beams, the dispersion frequency bands are invariant
+with φ, which only produces a shift in the choice of the unit cell. However, the variation of φ produces changes in Bloch
+eigenvectors, which may reflect in non-trivial topological properties. The Chern number Cn for the nth band is defined as
+Cn =
+1
+2πi
+�
+D
+βn dD,
+(3)
+where D = T2, βn = ∇ × An is called the Berry curvature, and An = ˆw∗
+n · ∇ ˆwn is the Berry connection, with ()∗ denoting
+a complex conjugate. The Chern number is an integer that quantifies the topological properties of the bands; these are
+robust to small perturbations in the system’s unit cell as long as these perturbations do not close the gaps separating the
+bands. Among other features, the Chern number is related to discontinuities (or vorticities) in the eigenvector field [85],
+localization of the Berry curvature [53], and to phase accumulation of the Bloch modes along cyclic paths in the torus
+Brillouin zone [62, 70].
+Of particular relevance to the present work is the bulk-boundary correspondence principle that relates the existence
+of in-gap edge states in finite systems to the Chern numbers [86]. This is done through the computation of a gap label Cg
+given by the summation of the Chern numbers of the bands below the gap, i.e. C (r)
+g
+= �r
+n=1Cn, which is equal to the num-
+ber of truncation resonances found inside such gap when the phase φ varies in an interval of 2π (see Section 3.2 for more
+details). However, the computation of the Chern number as given by Eq. (3) is often challenging due to phase or gauge
+ambiguities [87]. Furthermore, it has to be done for each θ value that defines a different unit-cell size (see, for example,
+Refs. [70, 80]). Here, we take an alternative, and more generic, approach that produces the gap labels Cg without direct
+computation of the band Chern numbers Cn, and for all θ values at once. Such approach relies on density of states com-
+putations based on the spectral variation with θ, which has been developed using mathematical principles of K-theory in
+the context of periodic and aperiodic topological insulators [88, 89], and later extended to quasi-periodic acoustic/elastic
+metamaterials [71–76]. This approach has not yet been extended to continuous elastic periodic waveguides such as the
+6
+
+0
+5
+10
+15
+20
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+Modulation wavenumber,���(m-1)
+0
+5
+10
+15
+20
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+25
+IDS
+0
+5
+10
+15
+20
+0
+5
+10
+15
+IDS
+0
+5
+10
+15
+IDS
+0
+8
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+f 
+��
+IDS
+f 
+��
+IDS
+f 
+1+�
+1+2�
+1+3�
+1+�
+1+2�
+1+3�
+0
+2.5
+5
+0
+2.5
+5
+1+4�
+1+5�
+1+6�
+1+8�
+Modulation wavenumber,���(m-1)
+Modulation wavenumber,���(m-1)
+Modulation wavenumber,���(m-1)
+Modulation wavenumber,���(m-1)
+Modulation wavenumber,���(m-1)
+Periodic BCs
+Periodic BCs
+Zoom
+Zoom
+Cg=1
+Cg=2
+Cg=3
+Cg=1
+Cg=2
+Cg=3
+Cg=1
+Cg=2
+Cg=3
+Cg=8
+Cg=4
+Cg=5
+Cg=6
+Figure 3: Eigenfrequencies of finite beam with L = 100 ¯a and PBCs for (a) sinusoidal and (b) step-wise modulation, with zoomed view in (c). Black dots
+represent eigenfrequencies while white areas denote band gaps. The corresponding IDS plots are displayed in the bottom panels (d-f), where selected
+fitted lines have colors corresponding to the gaps marked and labeled in (a-c).
+beams studied here.
+3.1.1. Extraction of the Chern number by varying the modulation wavenumber
+To begin, we investigate the variation of the beams’ spectral properties as a function of the modulation wavelength θ.
+The procedure relies on a large finite structure of fixed size L = 100 ¯a, and the computation of its eigenfrequencies under
+periodic boundary conditions (PBCs). The results are illustrated in Fig. 3(a,b) for the CM-PnC and SM-PnC configurations,
+where the eigenfrequencies are plotted as a function of θ as black dots. In the computation, the considered range of θ is
+discretized in intervals of ∆θ = 1/L, i.e., θn = n/L, such that each considered structure has an integer number n of unit
+cells. By doing so, the resulting eigenfrequencies sample the Bloch dispersion bands defined for the considered θ value,
+and no frequencies are found inside the gaps due to the PBCs and the "perfect" periodicity emanating from an integer
+number of unit cells [73]. The resulting spectrum provides a map for the location of the bands (black regions) and band
+gaps (white regions) as a function of θ, and consequently of unit-cell length a = 1/θ. We note that SM-PnC produces a
+more complex spectrum (Fig. 3(b)) with a larger number of gaps when compared to CM-PnC (Fig. 3(a)), in particular for
+lower values of θ as illustrated in the zoomed view of Fig. 3(c).
+The band-gap Chern numbers can be extracted by computing the Integrated Density of States (IDS) of the spectrum.
+It is defined as
+IDS(θ, f ) = lim
+L→∞
+�
+n[fn ≤ f ]
+L
+,
+(4)
+where [·] denotes the Iverson Brackets, which provides a value of 1 whenever the argument is true. In simple terms, for
+a given θ and frequency f , the IDS is the summation of all the eigenfrequencies below f , normalized by the structure
+size L. It theoretically converges as the structure size tends to infinity, but it is practically sufficient to consider large
+structures such as the one with L = 100 ¯a considered in our investigation. The IDS is displayed for the CM-PnC medium in
+Fig. 3(d), and for the SM-PnC medium in Fig. 3(e) with a zoomed view for the lower θ range in (f). In this representation,
+the z-axis and the associated colormap represent frequency f as a function of IDS and θ. The insets in (d,e) illustrate
+the 3D views highlighting sharp discontinuities in the surface plot, which are visualized as straight lines in the top view
+colormaps. Each straight line is associated with a band gap and occurs since the IDS does not change inside the gap.
+7
+
+1Hence, a jump in frequency (color) occurs as the IDS changes from the last mode before the gap to the first mode right
+after the gap. According to the theory [71], and confirmed by our findings, the variation of the IDS with θ inside the gaps
+identify straight lines expressed as
+IDS(f ) = n0 +Cg θ,
+(5)
+with the gap Chern number Cg corresponding to the slope. The lines of the most prominent gaps in Fig. 3 are fitted and
+overlaid to the IDS plots, allowing the extraction of the Chern gap labels from the slopes as marked in the top panels, with
+different colors used to represent different gaps. These gap labels are defined generically for any θ value that defines the
+band gap, and are related to the truncation resonances as described in the following section.
+3.2. Topological edge states and their control by phasons
+The non-zero Chern gap labels indicate the presence of in-gap edge states existing for structures with truncated
+boundaries, i.e., the truncation resonances. Their properties are illustrated in Figs. 4 and 5 for the CM-PnC and SM-PnC
+configurations, respectively. The figures display the frequencies of a finite structure of fixed length L = 15 ¯a as a function
+of modulation wavenumber θ and phase φ, for different BCs such as free-free and pinned-pinned. The frequencies are
+color-coded according to a localization factor p to identify modes localized at the boundaries, which is defined as
+p =
+�
+Lr |w|dx −
+�
+Ll |w|dx
+�
+L |w|dx
+,
+(6)
+where L denotes the domain of the beam, and Lr and Ll correspond to a smaller portion of length 0.15L at the right and
+left boundaries, respectively. With this definition, positive (red) and negative (blue) p values indicate modes localized at
+the right and left boundary, respectively, while values that are close to zero (black) indicate non-localized bulk modes.
+The left panels in Figs. 4 and 5 display the eigenfrequencies of the finite beam as a function of θ, for different BCs
+as illustrated by the schematics. The spectra are overall similar to the bulk spectra exhibited in Fig. 3, with black regions
+also defining the bulk bands, but with additional modes appearing inside the band gaps. These modes are the topological
+edge states, corresponding to the truncation resonances which are localized at one of the boundaries of the beam. The
+modes localized at the right boundary (red) traverse the band gaps multiple times as they migrate from the band above
+to the band below their respective gaps. Although not the focus of the present investigation, this behavior stems from the
+positive gap labels Cg > 0 and can be explained by density of states arguments [73]. Furthermore, the modes localized at
+the left boundary (blue) do not migrate between bands and instead remain inside the gap for the considered range of θ.
+The different behavior between left- and right-localized modes occur due to the way the finite structure is constructed,
+where the change in θ produces a qualitative change at the right boundary (the modulation is truncated at different places
+for different θ), but not of the left boundary (the modulation is always truncated at the same place).
+The gap label Cg dictates the number of left- and right-localized edge modes that span the band gap as the phason
+φ varies within an interval of 2π, for a fixed θ value. This is illustrated for selected θ values (marked as vertical dashed
+green lines) in the middle and right panels of Figs. 4 and 5, which display the variation of the eigenfrequencies with the
+phason φ. As previously mentioned, variations of φ do not affect the frequencies of the dispersion bands, and therefore
+the boundaries of the band gaps (material property) remain unchanged with φ. However, the phason influences how
+both boundaries of a finite structure are truncated (Fig. 1), and its variation causes the eigenfrequency branches of the
+truncation resonances to traverse the gaps. The first selected value θ1 = 1/ ¯a corresponds to the modulation wavelength
+for the reference unit-cell size ¯a. In the CS-PnC case (Figs. 4(b,e)), this unit-cell size produces two small gaps with Chern
+labels Cg = 1 and Cg = 2, which were extracted from the procedure in Fig. 3. For both types of BCs (free-free in (b) and
+pinned-pinned in (e)), one left- and one right-localized edge state traverse the first gap, and two edge states traverse the
+second gap, as the phason φ varies from 0 to 2π. In the SM-PnC case (Figs. 5(b,e)), the choice θ1 = 1/ ¯a corresponds to
+the case investigated in the experimental section of this paper (see Section 4), which produces three band-gaps with Cg
+values ranging from 1 to 3. Regardless of the type of boundary condition, the number of left- and right-localized edge
+modes spanning the band gaps is equal to the corresponding Chern gap label. In addition, the gap label sign is related
+to the direction the edge modes cross the gap [89]. A positive Cg > 0 indicates that |Cg | left-localized branches will cross
+the gap from the lower band to the upper band, and an equal number of right-localized states will cross from the upper
+band to the lower band. Although no examples are found in this paper, a negative sign |Cg | < 0 produces transitions in
+opposite directions [70]. Also note that the eigenfrequencies have a periodic behavior with φ, and are actually continuous
+at φ = 0 = 2π. Therefore a few branches of the truncation resonances traverse the gap through that point; for example, see
+the second right-localized mode in the second gap of Fig. 5(e). Indeed, the phason variable φ defines a continuous ring,
+with no start or ending point, with the beginning and end at φ = 0 and φ = 2π, respectively, being arbitrary choices for the
+plots.
+8
+
+Other examples are shown to demonstrate the generality of the approach and give more insights into the behavior
+of the edge states. The case of θ2 = 2.5/ ¯a (panels (c,f) in Figs. 4 and 5) corresponds to a unit-cell size 2.5 times smaller
+than the reference ¯a, and therefore the finite length L = 15 ¯a now comprises 37.5 unit cells. Even without an integer
+number of unit cells, the number of edge sates inside each gap matches the corresponding gap labels, for both CS-PnC
+and SM-PnC, and both types of BCs considered. In fact, this behavior is general and holds for any arbitrary θ value. The
+last row in Fig. 5 focuses on the lower θ range, where the SM-PnC features additional gaps with higher Chern gap labels.
+The examples θ3 = 2 m−1 and θ4 = 3 m−1 correspond to unit cell sizes of 0.5m and 0.33m respectively, and form finite
+structures with 6.09 and 9.135 unit cells for the fixed length L = 15 ¯a. They feature gap labels as high as Cg = 8, and the
+behavior of the edge states spanning the gaps with φ is in agreement with the extracted gap labels, again even without
+an integer number of unit cells. Among many edge states, two transitions experienced by the modes as a function of φ
+are highlighted by thicker lines and dots in Fig. 4(f) and in Fig. 5(h), and have their mode shape variation displayed in
+Figs. 6(a,b) respectively. These examples illustrate a transition between a right- and left-localized mode that occurs as
+a function of φ, with an intermediate state as a non-localized bulk mode when the eigenfrequency branch tangentially
+approaches the boundary of the gap. This type of transitions have been exploited for topological pumping applications,
+where the phason φ is varied along an additional spatial [70, 80] or temporal [77–79] dimension to induce a migration of
+localized modes between two boundaries.
+These results reveal that the truncation resonances are in fact topological edge states that traverse the band gaps for
+variations of the phason φ. The number of truncation resonances that traverse a gap is equal to the corresponding gap
+label Cg . This holds true for any set of BCs, although the particular shape of the branches of the edge states as they traverse
+the gap may be different. In addition, while the number of in-gap resonances can be predicted, one cannot guarantee the
+existence of truncation resonances for a particular phason value φ, but only that |Cg | branches will traverse the gap when
+φ varies in an interval of 2π. For example, the finite structure considered in Fig. 2(a) correspond to a phason value φ = 0.2π,
+which intersects both the right- and left-localized edge state branches of Fig. 4(b), and therefore one resonance localized
+at each boundary is found in this case. In contrast, for a phason value φ = π, the same gap in Fig. 4(b) does not exhibit
+any edge states, and therefore no truncation resonances would be found. Similarly, the modes I and II in Fig. 2(b) are
+intersections of the left- and right-localized edge state branches in the first and third gap of Fig. 5(b), respectively, for
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+(a)
+��
+��
+0
+5
+10
+15
+20
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+Modulation wavenumber, ��(m-1)
+(d)
+��
+��= 1/ā
+(b)
+-0.4
+0.4
+0
+p
+(c)
+����2.5/ā
+��
+0
+0.5
+1
+1.5
+2
+Phason,����
+(e)
+0
+0.5
+1
+1.5
+2
+-0.4
+0.4
+0
+p
+(f)
+Free-Free
+Pinned-Pinned
+Cg=1
+Cg=2
+Cg=1
+Cg=1
+Cg=2
+Cg=2
+Cg=1
+Cg=2
+Phason,����
+��= 1/ā
+����2.5/ā
+(right)
+(left)
+(right)
+(left)
+Figure 4: Eigenfrequencies of finite CM-PnC structure with length L = 15 ¯a and free-free (top) or pinned-pinned (bottom) BCs. The left panels (a,d)
+display the variation of the eigenfrequencies with θ, while the middle (b,e) and right (c,f) panels display the variation with φ for the selected θ values
+highlighted as vertical dashed green lines in (a,d). The frequencies are color-coded according to the polarization p, and the gap labels Cg are added for
+reference.
+9
+
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+(a)
+��
+��
+0
+5
+10
+15
+20
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+Modulation wavenumber, ��(m-1)
+(d)
+��
+(b)
+-0.4
+0.4
+0
+p
+(c)
+��
+(e)
+-0.4
+0.4
+0
+p
+(f)
+(g)
+��
+��
+0
+0.5
+1
+1.5
+2
+Phason, ���
+��= 2m-1
+(h)
+0
+0.5
+1
+1.5
+2
+-0.4
+0.4
+0
+p
+��= 3m-1
+(i)
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+0
+2.5
+5
+Free-Free
+Pinned-Pinned
+Zoom
+Cg=1
+Cg=1
+Cg=1
+Cg=1
+Cg=2
+Cg=2
+Cg=8
+Cg=3
+Cg=3
+Cg=3
+Cg=3
+Cg=4
+Cg=4
+Cg=5
+Cg=5
+Cg=6
+Phason, ���
+Modulation wavenumber, ��(m-1)
+��= 1/ā
+����2.5/ā
+��= 1/ā
+����2.5/ā
+(right)
+(left)
+(right)
+(left)
+(right)
+(left)
+Pinned-Pinned
+Figure 5: Eigenfrequencies of the finite SM-PnC structure with length L = 15 ¯a and free-free (top row) or pinned-pinned (middle row) BCs. The left panels
+(a,d) display the variation of the eigenfrequencies with θ, while (g) displays a zoom of (d) in the low θ range. The middle (b,e,h) and right (c,f,i) panels
+display the variation with φ for the selected θ values highlighted as vertical dashed green lines in (a,d,g). The frequencies are color-coded according to
+the polarization p, and the gap labels Cg are added for reference.
+φ = 0.4π, while other phason choices would define different truncation resonances or the their absence. Therefore, to
+better understand the behavior of the truncation resonances one needs to consider the entire family of structures defined
+for variations of φ, instead of separately considering particular cases.
+3.2.1. Boundary phasons
+As described, the phason φ simultaneously modifies the properties of both boundaries of a finite structure (Fig. 1), and
+therefore influence the truncation resonances localized at both boundaries. A higher degree of control over the truncation
+resonances is achieved by using the right- and left-boundary phasons introduced in Fig. 1, which modify only one bound-
+ary at a time. This is equivalent to adding a tuning layer at one end of the structure as done in Refs. [42, 48]. The effect of
+boundary phasons is demonstrated in Fig. 7, which repeats the eigenfrequency variation with φ of Fig. 3(f) and Fig. 4(d)
+in the left panels, and compares them to the the variation as a function of right-boundary phason φr and left-boundary
+phason φl displayed in the middle and right panels, respectively. The plots clearly show evidence of how the boundary
+phason only causes the edge states localized at the corresponding boundary to traverse the gap, while the superimposed
+effect of both boundary phasons lead to the effect caused by the phason φ. Indeed, as φr varies (Figs. 7(b,e)), only the
+10
+
+0.4
+0.6
+0.8
+1
+���
+0
+0
+-1
+1
+1
+2
+3
+x (m)
+w
+f
+0
+1
+2
+3
+x (m)
+0
+-1
+1
+w
+0.4
+0.6
+0.8
+���
+0.2
+f
+(a)
+(b)
+Pinned-Pinned
+���2.5/ā
+�= 2m-1
+Pinned-Pinned
+Figure 6: Examples of mode shape transitions as a function of phason φ for the (a) CM-PnC and (b) SM-PnC structures, corresponding to the branches
+highlighted in Fig. 4(f) and Fig. 5(h), respectively.
+right-localized modes traverse the gaps, producing the same branches as the ones in Figs. 7(a,d). Any left-localized modes
+that were defined for φ = 0 (the starting point) appear as roughly flat bands inside the gap, since the left boundary is not
+changing with φr . A similar effect is observed for the variation with φl in Figs. 7(c,f). For a structure that has a sufficient
+number of unit cells (i.e., has reached convergence as described Section 3.3 to follow), the right- and left-localized edge
+states form a set of decoupled chiral bands [89], the number of which corresponds to the gap label magnitude |Cg | and
+whose slopes are associated with the gap label sign.
+3.3. Effect of number of unit cells on frequency convergence of topological truncation resonances
+Next, we investigate the effect of the number of unit cells on the behavior of the truncation resonances. As shown
+earlier, truncation modes exhibit an exponential decay away from the boundary since their frequency lies inside a band
+gap, and therefore correspond to a complex wave number. For structures with a large number of unit cells, the in-gap
+truncation modes are only mildly affected by further addition of unit cells since their displacement tend to zero away
+from the boundary. In that scenario, a further increase in number of unit cells will produce a larger number of bulk
+modes, while the branches of the edge states spanning the band gaps with φ will remain the same. However, for structures
+with a small number of unit cells, the truncation resonances are more likely to be influenced by the opposing edge and by
+other effects such as mode coupling and veering with bulk modes or another edge state.
+This behavior and the convergence with the number of unit cells is elucidated by the results of Fig. 8. The SM-PnC
+structure with θ1 = 1/ ¯a is chosen to exemplify these features, with the first and second row corresponding to free-free and
+pinned-pinned BCs, respectively. The panels (a,d) display the variation of the eigenfrequencies with φ for a structure with
+5 unit cells, while the right panels (c,f) correspond to a larger structure comprising 15 cells. In the middle panels (b,e), the
+variation of the frequencies with the number of unit cells is displayed for the fixed phason value highlighted by the vertical
+dashed-line intersections in the other panels. Overall, the number of bulk modes increase with the number of unit cells as
+expected, and the edge state branches traversing the gaps are similar but exhibit small differences. These differences are
+amplified for phason values that are close to mode couplings as illustrated in the top row. At the selected phason value,
+there is a strongly coupled avoided crossing between the right- and left-localized edge states for the case with 5 unit cells
+shown in (a), and therefore the eigenfrequencies defined for that phason value are more separated when compared to the
+structure shown in (c) with 15 cells and without the avoided crossing. Therefore, the frequencies of the edge states for
+this phason value vary as a function of the number of unit cells and converge to a fixed value at approximately 10 unit
+cells as illustrated in Fig. 8(b). In contrast, in the case of the bottom row with pinned-pinned BCs, the chosen phason
+value intersects the edge state mode and an adjacent mode that is well isolated, and therefore the truncation frequency
+converges quicker at around four unit cells. These results illustrate that while convergence is always achieved, the required
+number of unit cells may vary between different structures depending on the BCs and the presence of coupling effects at
+the phason value of interest.
+11
+
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+(a)
+(b)
+-0.4
+0.4
+0
+p
+(c)
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+0
+0.5
+1
+1.5
+2
+Phason, ���
+(d)
+0
+0.5
+1
+1.5
+2
+(e)
+0
+0.5
+1
+1.5
+2
+-0.4
+0.4
+0
+p
+(f)
+Right boundary phason, �r��
+Left boundary phason, �l��
+Pinned-Pinned
+Pinned-Pinned
+Cg=1
+Cg=2
+Cg=1
+Cg=2
+Cg=1
+Cg=2
+Cg=8
+Cg=3
+Cg=4
+Cg=5
+Cg=6
+Cg=8
+Cg=3
+Cg=4
+Cg=5
+Cg=6
+Cg=8
+Cg=3
+Cg=4
+Cg=5
+Cg=6
+(right)
+(left)
+(right)
+(left)
+����2.5/ā
+��= 2m-1
+Figure 7: Eigenfrequency variation as a function of phason φ (a,d), right-boundary phason φr (b,e) and left-boundary phason φl (c,f) for finite beam
+with L = 15 ¯a and pinned-pinned BCs. The top row consists of a CM-PnC structure with θ2 = 2.5/ ¯a while the bottom row consists of a SM-PnC structure
+with θ3 = 2 m−1.
+3.4. Topological truncation resonance versus non-topological defect resonance
+Truncation resonances, with their topological character, are not the only type of resonances that appear due to trun-
+cation or breakage of symmetry in a periodic medium. Another type of resonance, that is also of localized nature, is that
+associated with defect modes [90–92]. Under the developed framework, the band gaps characterized by non-zero Chern
+labels are guaranteed to support |Cg | truncation resonances spanning the gaps as a function of phason or boundary pha-
+son parameters. Although we do not present an example in this paper, in some cases a band gap may be characterized
+by Cg = 0, which is referred to as a topologically trivial band gap. In this case, the presence of in-gap resonances is not
+guaranteed, although they may appear. Since there is no topological explanation or origin to their appearance, these trun-
+cation resonances are usually categorized as defect modes. One example can be found in reference [75], where a central
+trivial gap with Cg = 0 does not exhibit in-gap resonances under pinned-pinned BCs (Fig. 2a), but exhibits truncation
+resonances under clamped-free BCs (Fig. 3a). Note that the truncation resonances in this second case do not traverse the
+band gap, which is a key feature expected from topological modes as we highlight in this work.
+We here illustrate another important scenario where a physical defect is introduced to a finite structure in order to
+create an in-gap resonance, although in this case a non-topological resonance as we will show. As an example, we con-
+sider a finite SM-PnC structure comprising 15 unit cells with θ1 = 1/ ¯a, and introduce a defect initially located at the 8th
+unit cell by "skipping" the ABS portion within this unit cell, making it entirely out of aluminum. The results displayed in
+Fig. 9 show the variation of the eigenfrequencies with φ, with the defect unit cell highlighted in the schematics at the top
+and identified by the larger white segment, which represents aluminum. As the phason varies, material is added to the left
+boundary and removed from the right boundary (Fig. 1), which causes the defect to continuously drift towards the right
+boundary. The defect moves by one unit cell with every change in 2π; these increments are marked by the vertical dashed
+lines in the figure. After a change in phason of 14π, the defect is at the last unit cell, and finally for 16π it exists the structure
+and a perfect periodic domain is restored. In a defect-free structure, the variation of the eigenfrequencies with φ is trivially
+periodic in intervals of 2π. With the inclusion of the defect, additional modes are found inside the gaps and co-exist with
+the truncation resonances. The interplay between the in-gap defect mode and the truncation resonances is highlighted
+by the selected mode shapes displayed in the bottom panels. In the initial configuration, the in-gap defect resonance
+is localized at the center (8th unit cell) of the structure and is completely decoupled from the truncation resonances, as
+evidenced by the plots in stage I. As the phason varies, the trajectory of the defect modes remain almost flat inside the
+12
+
+gaps, in sharp contrast to the behavior of the topological states which transverse the gaps. Indeed, the truncation reso-
+nances exhibit the expected periodic behavior as their branches traverse the gaps in a pattern that repeats periodically in
+intervals of 2π. However, as the defect physical position approaches the right boundary, the in-gap defect modes progres-
+sively couple with the truncation resonances localized at the right boundary, this is seen in all three gaps viewed in the
+figure. Focusing on the third band gap as an example, the frequency curves in stage II exhibit a weak coupling, while in
+stage III a larger coupling is observed causing an avoided crossing with relatively strong repulsion between the defect and
+truncation resonances. As the defect moves within the last unit cells (13th-15th), it slowly transforms to capture, itself, the
+characteristics of a truncation resonance localized at the right boundary, with a mode shape example displayed for stage
+IV. At this last stage, the branches of the right-localized truncation resonances are very different from the periodic pattern
+of the perfect periodic structure, since they are created by a truncation near a defect.
+These results highlight key differences between the truncation resonances and defect modes. The defect mode defines
+a flat branch inside the gap as a function of φ, until it starts to couple with the topological truncation resonances−which
+happens as the position of the defects nears the boundary. It is interesting to note that when the coupling takes place, the
+shape of the coupled truncation resonance branch changes as it traverses the gap. However, the counting principle given
+by the gap Chern labels is still valid. This can be verified as in every interval of φ = 2π, there is a net number of 1, 2 and
+3 right-localized modes transversing the first, second, and third gap, respectively. Therefore, the truncation resonances
+retain this key topological property even with the interference of a defect at the boundary. We should also stress that the
+topological classification of an in-gap mode is always relative to a given set of parameters. The defect mode introduced
+here is non-topological in the context of the phason degree of freedom, which causes it to remain confinded inside the
+gap as a flat band. However in some cases this type of defect mode might find a topological classification under a different
+set of parameters and analysis framework [93].
+4. Experimental investigation of modulated phononic crystal beams
+4.1. Experimental set-up and measurements
+For the experimental investigation, we focus on the SM-PnC beam structure, again composed of alternating layers
+of Al and ABS with a ratio of layer lengths of 4:1 (Al:ABS) for the baseline unit-cell configuration. The unit-cell length
+0
+2
+3
+4
+5
+Frequency, f  (kHz)
+(d)
+1
+0
+0.5
+1
+1.5
+2
+Phason, ���
+(e)1
+5
+10
+15
+20
+Number of unit cells
+(f)0
+0.5
+1
+1.5
+2
+Phason, ���
+0
+2
+3
+4
+5
+Frequency, f  (kHz)
+(a)
+1
+-0.4
+0.4
+0
+p
+-0.4
+0.4
+0
+p
+(b)
+(c)
+15 unit cells
+15 unit cells
+5 unit cells
+5 unit cells
+Free-Free
+Pinned-Pinned
+(right)
+(left)
+(right)
+(left)
+Figure 8: Eigenfrequency variation with φ for structure with θ1 = 1/ ¯a comprising 5 cells (a,d) and 15 cells (c,f). The middle panels (b,e) show the variation
+with the number of unit cells for the fixed phason values highlighted as vertical dashed lines in the other panels. Top and bottom rows correspond to
+free-free and pinned-pinned BCs, respectively. Band-gap frequency ranges are shaded grey.
+13
+
+.
+.
+.
+.
+.
+.
+.
+.
+.
+.
+.
+.
+.
+.
+...
+.
+.
+..
+0000
+.
+.
+.
+.
+.
+..
+..
+....
+.
+.
+.
+.
+.
+.
+.
+0......
+.
+.
+.
+000000
+......
+.
+...00
+..
+.
+.
+000000
+000000
+000000
+000000....
+....
+....
+.--
+....--
+---.
+.
+....
+..
+.
+.
+..
+..
+....
+.
+.
+.
+.
+.
+...
+...
+0000
+.
+.
+..00
+00000
+00000
+00000
+90000
+00000
+2
+4
+6
+8
+Frequency, f  (kHz)
+0
+16
+14
+12
+10
+8
+6
+4
+2
+Phason,����
+-0.4
+0.4
+0
+(right)
+(left)
+p
+8th cell
+10th cell
+13th cell
+15th cell
+0
+5
+10
+15
+Position, x/ā
+0
+1
+-1
+Displacement, w
+0
+5
+10
+15
+Position, x/ā
+0
+5
+10
+15
+Position, x/ā
+0
+5
+10
+15
+Position, x/ā
+I
+I
+II
+III
+IV
+IV
+III
+II
+Figure 9: Eigenfrequencies as a function of phason φ for a finite SM-PnC structure with θ1 = 1/ ¯a, L = 15 ¯a and a defected unit cell. The location of the
+defect changes by one unit-cell increments with every change of 2π in φ, as marked by the vertical dashed lines and illustrated in the top schematics.
+Band-gap frequency ranges are shaded grey. Selected mode shapes are displayed in the bottom panels, whose colors correspond to the polarization of
+the mode, with dashed and solid lines representing the mode with open and closed circle markers, respectively.
+and cross-sectional area are selected as ¯a = 203 mm and A = 645 mm2, except in Section 4.3 where the unit-cell length
+is varied. The values of these geometric parameters are chosen to allow for the generation of several band gaps below
+9 kHz for practical reasons; however, all conclusions are scale invariant and hence applicable to periodic structures that
+are orders of magnitude smaller in size (with the limit that they are appropriately represented by continuous models). In
+this section, we show additional FE results for direct comparison with the experiments, where we use the same FE model
+details as in Section 3 with specifically 100 finite elements being used per unit-cell. For our experimental set-up, a set of
+Al and ABS solid blocks were fabricated and connected to each other by an adhesive to form the periodic structure. The
+test articles were suspended using thin nylon wires to simulate free-free BCs as depicted in Fig. 10(a).
+First we show the complex band structure of the unit cell, which is shown in Fig. 10(b)−the real part of which is identical
+to Fig. 2(b). This calculation shows that three relatively large band gaps exist between 0 and 9 kHz. Figure. 10(c) shows
+a corresponding FRF obtained theoretically (solid line) and experimentally (dashed line) for a 5-unit-cell version of the
+structure, in which the “input” force excitation and the “output” displacement evaluation are at the extreme ends. For
+the experimental results, the test article was excited at the tip of the structure using a force hammer. The impulse forcing
+data F from the force hammer was used in conjunction with the response data U obtained by a sensing accelerometer
+connected at the other end of the structure, to generate the receptance U/F over the frequency range 0-9 KHz. The
+amplitude of the experimental response was calibrated to match the average of all theoretical data points over the 0-9
+KHz frequency range. An excellent correlation is observed between the theoretical and experimental FRF curves. It can be
+seen, however, that the correlation generally degrades at higher frequencies along with an increasing level of noise. This
+is due to the difficulty of stimulating high frequencies with a force hammer as well as the reduced resolution when using a
+constant sampling rate over all frequencies.
+4.2. Effects of modulation wavenumber, boundary phasons, and number of unit cells by experiment
+In Fig. 5(a), we have shown the effect of the modulation wavenumber (i.e., unit-cell length) on the locations of the
+truncation resonances. Here we repeat our computational investigation focusing on the range 0.18 ≤ a ≤ 0.22 m and over-
+lay the data of the experimental case of a = 0.2 m (θ = 5). The results, which are shown in the inset of Fig. 10(c), indicate
+very good agreement between theory and experiments. Another approach that keeps the unit-cell geometric configura-
+tion intact is the addition of a single tuning layer (or a partial unit-cell) at the end of the finite periodic structure [42, 48],
+as demonstrated in Section3.2.1. As illustrated in Fig. 1, the addition of a tuning layer corresponds to the application of a
+14
+
+boundary phason φl. The material and geometrical configuration of the tuning layer should be chosen such that it would
+generally form a physically cropped unit-cell, i.e., it would form a partial unit-cell when its length is less than a and a full
+unit-cell when its length is a. Figure 10(d) displays a plot of the resonant frequencies as a function of the length of the
+tuning layer, denoted by lTL and ranging from lTL = 0 (φl = 0, 5 unit-cells) to lTL = a (φl = 2π, 6 unit-cells) for the same
+baseline design of Fig.10(a)−this corresponds partially to the results shown in Fig. 5(b) but now with the addition of exper-
+imental data points. With the addition of a tuning layer, band-gap resonances rapidly traverse the band gaps. However,
+once they reach the band-gap boundaries they behave like regular structural resonances (bulk modes) with slower levels
+of variation as a function of lTL.
+Given the localization nature of truncation resonances, the measured amplitude at the far end of the SM-PnC structure
+is expected to be less than at the edge where the mode is localized and where the excitation is applied. In Fig. 11, we show
+using both theory and experiment an FRF comparison between 5- and 6-unit-cell structures in (a) and 5- and 15-unit-
+cell structures in (b). A truncation resonance peak clearly exists inside the second band gap. We also observe a stronger
+(b)
+-80
+-60
+-40
+-20
+Response, w/F  (dB)
+Wavenumber, � (m-1)
+0
+-π/a
+π/a
+0
+0.5
+1
+1.5
+2
+Left boundary phason, �l��
+-0.4
+0.4
+0
+p
+�l=0
+Cg=1
+Cg=2
+Cg=3
+Theory
+Experiment
+0
+2
+4
+6
+8
+Frequency, f  (kHz)
+(right)
+(left)
+0.18
+0.20
+0.22
+1
+2
+3
+4
+5
+a (m)
+Frequency, f (kHz)
+aABS/a = 0.2
+Experiment
+(c)
+(d)
+(a)
+Figure 10: Experimental validation: (a) Photograph of the experimental setup showing a 5-unit-cell SM-PnC beam structure consisting of layers of
+Aluminum and ABS polymer with a ABS volume fraction of 20% and ¯a = 203 mm. The structure was excited on the far left side (on the first ABS polymer
+layer) with a force hammer and measured with an accelerometer on the other far end. (b) Frequency band diagram of the infinite (material) constituent
+of the SM-PnC beam and (c) corresponding FRF response of the finite structure. Inset: Resonance frequency (thin solid lines, theory; dots, experiment)
+versus unit-cell length a for the 5-unit-cell periodic beam structure. (d) Corresponding resonance frequency (solid lines, theory; dots, experiment)
+versus left boundary phase (i.e., length of a tuning layer attached at the far left end). At φl = 0.4π, the tuning layer transitions from ABS to Al. At φl = 2π,
+the tuning layer is a full regular unit cell and the total structure is rendered a 6-unit-cell structure. In (a), the solid lines represent propagation modes,
+and the dashed lines represent attenuation modes. Band-gap frequency ranges are shaded grey.
+15
+
+Not addedT-100
+-80
+-60
+-40
+-20
+0
+Response, w/F  (dB)
+5 Unit cells
+6 Unit cells
+2
+3
+4
+5
+-80
+-60
+-40
+-20
+Experiment
+Frequency, f (kHz)
+Response, w/F (dB)
+Theory
+0
+2
+4
+6
+8
+Frequency, f (kHz)
+5 Unit cells
+15 Unit cells
+2
+3
+4
+5
+-80
+-60
+-40
+-20
+Experiment
+Frequency, f (kHz)
+Response, w/F (dB)
+Theory
+0
+2
+4
+6
+8
+Frequency, f (kHz)
+(a)
+(b)
+Figure 11: Frequency response function comparison for the finite SM-PnC beam structure with different number of unit cells. The results show a
+truncation resonance in the second band gap. Compared to the baseline case of 5 unit cells, the tructioan resonance is observed to experience negligible
+shift in frequency for (a) a 6 unit-cell structure and (b) a 15 unit-cell structure. Strong spatial attenuation in displacement amplitude across the structure
+is observed as the number of unit cells is increased. These results are for the same unit-cell configuration considered in Fig. 10. Band-gap frequency
+ranges are shaded grey.
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+2
+4
+6
+8
+ABS length-fraction, aABS/a
+Frequency, f  (kHz)
+Theory
+Experiment
+Figure 12: Experimental validation: Resonance frequency (thin solid lines, theory; dots, experiment) versus ABS length-fraction for the 5-unit-cell SM-
+PnC beam structure with ¯a = 203 mm. The experimental data points correspond to an ABS length-fraction of 0.1, 0.15, 0.2, 0.25 and 0.3, respectively. The
+thick solid lines represent the band-gap boundaries for the corresponding infinite periodic materials.
+attenuation from edge-to-edge as the number of unit cells (and total structure length) increases. As for the effect of the
+number of unit cells on the frequency of the truncation resonance, we note that there is a negligible shift from 5 to 15 unit
+cells. These results are to be compared with the eigenfrequency versus phason plot shown in Fig. 8(b) for free-free BCs. It is
+shown in that figure that beyond 5 unit cells, the change in the frequency of the truncation resonances become negligible.
+In contrast, the frequencies of the conventional resonances demonstrate substantial shifts, as shown in both Fig. 8(b) and
+Fig. 11. We also observe in Fig. 11(b) that while the amplitude of the truncation resonance peak drops significantly as the
+number of unit cells is increased from 5 to 15, the amplitudes of all the conventional resonances do not experience any
+noticeable drops.
+4.3. Effect of unit-cell material volume fraction by experiment
+In addition to property modulation wavenumber and phasons, an alternative approach for controlling the frequency
+locations of truncation resonances is alternation of the unit-cell design, e.g., by changing its material composition and/or
+16
+
+spatial distribution or its geometry. This can result in achieving a total exit of a truncation resoance from a band-gap
+frequency range, as illustrated in Fig. 12 for a 5-unit-cell SM-PnC structure, which shows that when aABS/a is set to 0.25
+or higher, no in-gap resonances appear in any of the three gaps covered by both computation and experiment. In this
+figure, we consider the full range aABS/a, which at one extreme (aABS/a = 0) represents a homogenous Al beam, and at
+the other extreme (aABS/a = 1) represents a beam composed of only ABS polymer. This figure also allows us to examine
+the sensitivity of the truncation resonances’ frequencies to smooth variations in the material volume fraction. It can
+be seen that the truncation resonances are noticeably more sensitive to varying the unit-cell layer dimensions than the
+conventional resonances. Once they exit the band gaps however, these unique resonances become less sensitive to varying
+aABS/a, and their sensitivity becomes similar to that of the conventional resonances.
+5. Further reflection on the material vs. structure theme
+The distinction and interconnection between a material and a structure may be examined and classified at various
+levels. A basic distinction is that of intrinsic versus extrinsic properties or characteristics, e.g., the Young’s modulus and
+density being intrinsic material properties in contrast to the stiffness and total mass as extrinsic structural characteristics.
+The distinction may also be made based on physical response. In this context, an elementary classification may be based
+on the behavior of static deformation, such as the length scale of deformation or spatial span of tangible force interactions.
+For example, consider a lattice configuration of beams forming a truss that lies at the core of a larger structural frame. If the
+length scale of deformation at, say, the center of the core is much larger than the individual beam elements and negligible
+force interaction occurs with the boundaries formed by the frame, then this deformation may be viewed as a form of
+material behavior. On the other hand, if the length scale of the deformation is on the order of the beam elements, and
+non-negligible interaction occurs with the boundaries, then the “periodic network of beams behaves as a structure, such
+as a frame in a building or a truss in a bridge [94]."
+In this work, we have addressed the material-versus-structure correlation problem at a more fundamental level; that is,
+by examining the characteristics pertaining to finite size in comparison to the properties associated with idealized infinite
+size, and doing so from a topological elastodynamics perspective. Here, the dispersion curves represent material proper-
+ties and the natural frequencies represent structural characteristics. In this context, finite size along the direction where
+the physical phenomenon of interest takes effect (in this case, wave propagation) is what distinguishes the material versus
+structure character. Finite dimensions in other lateral dimensions (such as the thickness of a beam, for exmaple) may play
+a significant role in altering the material properties or structural characteristics, but not in altering the classification of ma-
+terial versus structure. As a periodic material is truncated, and rendered a structure, both bulk and truncation resonances
+emerge; the latter being intimately connected to the nature of the truncation. This investigation focuses specifically on
+this aspect.
+6. Conclusions
+In this paper, we have investigated using theory and experiments the fundamental question of the relation and inter-
+play between material and structure. We provided a formal connection between topological physics and truncation reso-
+nances in finite periodic structures. Periodic structures can be understood and topologically characterized using property
+modulation parameters such as the modulation wavelength θ and phason φ. These parameters expand the physical space
+and allow for a rigorous study of the nature of truncation resonances.
+The Chern number is a material property obtained from unit-cell analysis, considering a large number of unit cells
+with periodic boundary conditions applied. It allows us to predict the behavior of a periodic medium through the bulk-
+boundary correspondence principle, which in fact is itself a manifestation of the interconnection between the notion of
+a material and a structure, originated in the quantum realm−which we bring here to elastic media. In the QHE theory,
+for example, the Chern number is a material invariant that predicts the existence of edge currents propagating along the
+edges of truncated finite samples. Similarly, for our elastic structures, the gap labels predict the number of truncation
+resonances that span a band gap as φ is varied for a finite structure with any prescribed BCs.
+We have shown that the existence of in-gap truncation resonances cannot be guaranteed for any φ and that the topo-
+logical character is understood only when sweeping through φ. This brings a more comprehensive perspective rather
+than analysing particular truncation cases, and provides a methodology for designing for truncation resonances or their
+absence. The boundary phasons, which is a concept we introduce in this work, provide an additional tool to control trun-
+cation resonances, albeit at different boundaries independently. We have also investigated the effect of the number of
+unit cells in a finite structure, elucidating that the left- and right-boundary phasons become independent only when a
+sufficient number of unit cells is present. We similarly demonstrated that the frequency location of truncation resonances
+converge only when the structure is comprised of a sufficiently large number of unit cells, at least five cells in most cases.
+17
+
+Mode couplings—whose locations are influenced by the boundary conditions among other factors—impact the rate of
+convergence of the truncation resonances. The impact of the unit-cell constituent material composition was also stud-
+ied, showing that a truncation resonance may be forced to exit a band gap with an appropriate choice of material volume
+fraction.
+We have also examined another important type of localized mode in finite structures, the defect mode. We have shown
+it to be non-topological, since it remains flat with change of φ inside the band gap unless it couples with a truncation
+resonance. In a perfect “undefected" periodic structure, there can only be one mode localized at each boundary for any
+given phason value. By coupling with a defect, it is possible to have two modes localized at the same boundary for a given
+structure, living inside a band gap, with different frequencies.
+This study, we expect, will inspire future work on multiple fronts. For example, similar principles may be extended
+to 2D and 3D periodic structures and their truncation resonances, which may manifest as localized modes at points,
+edges, and surfaces, having connections to topological physics and possibly to higher-order Chern numbers and higher-
+order topological modes (such as corner modes). Another domain of potential applicability is coiled phononic crystals
+for space saving [95]. A further angle to be explored in the question of material versus structure is the static regime,
+where similar connections may be established for topological floppy modes [96]. Other areas to be investigated are the
+interplay with nonlinearities [97], the applicability to damage mechanics such as the effect of number of unit cells on the
+fracture toughness [98], and the role of size effects in nanoscience where small finite dimensions have profound impact
+on thermal transport [25] and other physical properties. Implications to quasiperiodic media [73–76, 99] or nonperiodic
+media described statistically by representative volume elements may also be explored. Finally, the framework presented
+for connecting between topology and truncation may potentially be applied to finite systems in other branches of physics,
+such as photonics [49] and quantum mechanics [41].
+Acknowledgement
+The authors acknowledge the students Andrew S. Tomchek and Edgar A. Flores for their assistance in conducting the
+experiments.
+References
+[1] J. W. S. Rayleigh, Xvii. on the maintenance of vibrations by forces of double frequency, and on the propagation of
+waves through a medium endowed with a periodic structure, Philos. Mag. 24 (1887) 145–159.
+[2] S. M. Rytov, Acoustical properties of a thinly laminated medium, Sov. Phys.-Acoust. 2 (1956) 68–80.
+[3] C. T. Sun, J. D. Achenbach, G. Herrmann, Continuum theory for a laminated medium, ASME J. Appl. Mech. 35 (1968)
+467–475.
+[4] M. Heckl, Investigations on the vibrations of grillages and other simple beam structures, J. Acoust. Soc. Am. 36 (1964)
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+arXiv:2301.00572v1  [gr-qc]  2 Jan 2023
+Spacetime singularities and curvature
+blow-ups
+István Rácz ∗
+Wigner RCP
+H-1121 Budapest, Konkoly Thege Miklós út 29-33, Hungary
+January 3, 2023
+Abstract
+The singularity theorems of Penrose, Hawking, and Geroch predict the exis-
+tence of incomplete inextendible causal geodesics in a wide range of physically
+adequate spacetimes modeling the gravitational collapse of stars and the ex-
+panding universe. Here, using results on spacetime extensions, it is shown that
+if a suitable low regular form of the strong cosmic censor hypothesis holds, then
+parallelly propagated blow-up of either the tidal force or frame-drag part of
+the curvature must occur in “generic” timelike geodesically incomplete maximal
+Cauchy developments.
+1
+Introduction
+During the first five decades of Einstein’s theory of gravity, singular behavior popped
+up in many of the physically relevant exact solutions. Notably, in these spacetimes,
+“singularities” were always signified by unbounded curvature. Nevertheless, the gen-
+eral covariance of the theory made even the determination of spacetime-singularity to
+be one of the most intriguing and challenging issues in general relativity [23]. In the
+1960s, this yielded a fierce debate concerning the relevance of spacetime singularities
+found in models admitting symmetries [30, 14]. In 1965, by applying methods of
+global differential geometry, Roger Penrose proved that singularities must occur irre-
+spective of symmetries in spacetimes modeling gravitational collapse [33, 34]. More
+precisely, Penrose, in his seminal work, demonstrated that a spacetime cannot be null
+geodesically complete if the following three conditions are satisfied:
+(1) the null convergence condition holds, i.e., Rabkakb ≥ 0 for all null vectors ka ,
+∗E-mail address: racz.istvan@wigner.hu
+1
+
+(2) the spacetime is globally hyperbolic with a non-compact Cauchy surface Σ ,
+(3) there exists a closed trapped surface T .
+The proof of this theorem is derived by contradiction (see, e.g., [47, 48]). It starts
+by assuming that the spacetime is null geodesically complete. Then, condition (3)
+shows that these geodesics begin to focus, whereas condition (1) guarantees that they
+keep focusing, and focal points must develop. As opposed to this, if the spacetime
+is null geodesically complete, condition (2) can be used to exclude the appearance of
+such focal points. The contradiction is avoided by dropping the indirect assumption,
+verifying that the spacetime cannot be null geodesically complete.
+It formulates the expectation that gravitational interaction is attractive because
+null geodesic congruences inevitably get focused by the curvature. The first half of
+condition (2) is also moderate, as Einstein’s theory of gravity is known to possess a
+well-posed initial value problem.
+Note also that condition (1) is very mild.
+It formulates the expectation that
+gravitational interaction is attractive in the sense that, if it holds, null geodesic con-
+gruences inevitably get focused by the curvature. The first half of condition (2) is also
+moderate as Einstein’s theory of gravity is known to possess a well-posed initial value
+problem [3, 4, 53]. The second half assumes only that the spacetime represents the
+history of an isolated object. Assumption (3) is more demanding. It presumes that
+the gravitational field is so strong in a domain bounded by a two-dimensional spatial
+surface T that the outward-directed null rays, starting at T perpendicularly, have
+already negative expansion. This latter condition was verified to hold if a sufficient
+amount of energy/matter is concentrated in the spatial region bounded by T [46, 11].
+The strength of this theorem is rooted in that all the three conditions above are of
+pure geometrical character. The same assumptions could be imposed in a wide range
+of metric theories of gravity. Notably, by adopting the new technical elements applied
+in Penrose’s theorem, Hawking immediately proved the existence of incomplete causal
+geodesic curves in cosmological models [20]. Soon after, a series of novel singularity
+theorems by Penrose, Hawking, and Geroch concluded that gravitational singularities
+occur in many physically realistic situations [20, 21, 18]. Note, however, that all these
+theorems share a shortcoming.
+Namely, the existence of incomplete, inextendible
+causal geodesics is used as a synonym of spacetime-singularity [23].
+In the following decades considerable efforts had been made to over-bridge the gap
+between physical intuition and the conclusion of the singularity theorems. All these
+aimed to verify that some physically relevant quantities do indeed become infinite
+along the incomplete inextendible causal geodesics predicted by the singularity theo-
+rems. Nevertheless, still, no such satisfactory reasoning exists yet. Our main goal in
+this paper is to bring the intuitive picture of singularities and the predictions of the
+singularity theorems closer to each other.
+2
+
+While making the scope of the discussions a bit wider, recall first that in met-
+ric theories of gravity, such as Einstein’s theory, a spacetime, (M, gab) is supposed
+to be a smooth Hausdorff, paracompact, connected, orientable manifold M. It is
+also assumed that a smooth metric gab of Lorentzian signature is also given on M
+[18, 53]. The base manifold M is also assumed to be chosen sufficiently large to rep-
+resent all the events compatible with the history of the investigated physical system.
+In contrast, the presence of incomplete inextendible causal geodesics —the singular-
+ity theorems predicted these— might be considered as a warning signal indicating
+that some parts of those spacetimes which describe the expanding universe and the
+gravitational collapse of stars are missing [23].
+After getting aware that spacetimes may not be complete, it is natural to ask if
+they can be extended. In answering this question, recall first that the above determi-
+nation of a spacetime refers merely to its essential mathematical structures. In many
+cases, mainly for mathematical conveniences, it is assumed that a spacetime possesses
+a smooth differentiable structure and the other fields are also smooth. Recall that
+in the smooth setting, a seminal result by Choquet-Bruhat and Geroch [2] guaran-
+tees the existence of a unique (up to diffeomorphisms) maximal Cauchy development
+(see also [44]).
+Note also that recently the existence and uniqueness of maximal
+global hyperbolic developments of vacuum general relativistic initial data sets (h, K)
+in Sobolev spaces Hs ⊕ Hs−1 was proven in [6], for s ∈ N with s > n/2 + 1, where
+n (≥ 3) stands for the dimension of the initial data surface. Given these results, at
+least in the case of globally hyperbolic spacetimes, we would expect that it is entirely
+satisfactory to work with the maximal Cauchy development.
+Note, however, that things are much more intricate. Namely, causal geodesically
+incomplete spacetimes exist such that they contain as a part “the maximal” Cauchy
+development of suitable smooth initial data given on an otherwise also “maximal” ini-
+tial data surface. These Cauchy developments can be continued beyond the Cauchy
+horizon such that no curvature blow-up occurs while crossing this horizon. As imme-
+diate examples, think of the maximal analytic extensions of the Kerr and Taub-NUT
+spacetimes (see, e.g., [18]). In both cases, the predictive power of general relativity
+is lost while crossing the Cauchy horizon of the maximal Cauchy development. It
+is also fair noting that this type of behavior has been found to occur exclusively in
+spacetimes with symmetries.
+The strong version of the cosmic censorship conjecture of Penrose emerged from
+these troublesome circumstances [37, 38]. Penrose’s strong cosmic censorship conjec-
+ture claims that the maximal Cauchy development of a “generic” compact or asymp-
+totically flat initial data is never part of a larger spacetime [38]. If true, the corre-
+sponding Cauchy development cannot have a Cauchy horizon, especially no extension
+beyond it could make sense. While investigating issues related to the strong cosmic
+censor hypothesis and the main dilemmas on spacetime singularities, the following
+3
+
+questions arose: How could it be shown that something violent happens “there”.
+Where and what sort? (For detailed discussions on many fundamental issues, see,
+e.g., [18, 35, 23, 53, 47, 48].) In attempting to answer some of these questions, it is
+rewarding to have a glance at the main cornerstones we already have in our hands.
+These are the singularity theorems, the existence of maximal Cauchy development,
+and we may also adopt the strong cosmic censorship conjecture.
+Inspecting these fundamental concepts for some time, the following argument,
+based on contradiction, develops: Consider a causal geodesically incomplete space-
+time. Assume that it is the maximal Cauchy development of some “generic” compact
+or asymptotically flat initial data and that the strong cosmic censor conjecture holds
+for this class of spacetimes. These assumptions guarantee that such spacetime cannot
+be extended within the considered differentiability class. Assume, in addition, that
+nothing violent happens along either of the incomplete inextendible causal geodesics.
+In particular, assume that all the tidal-force and frame-drag parts of the curvature
+tensor remain bounded while approaching the “ideal endpoints” of the incomplete
+inextendible causal geodesics. This regularity of the curvature permits a global ex-
+tension of the otherwise maximal Cauchy development. If the strong cosmic censor
+hypothesis holds, this contradiction allows us to conclude that the causal geodesic
+incompleteness in a maximal Cauchy development must always be accompanied by
+the singular behavior of some of the tidal-force and frame-drag parts of the curvature
+tensor.
+In this paper, attention will be restricted to the timelike case. In other words,
+spacetimes admitting incomplete inextendible timelike geodesics will be considered.
+One of the main results of this paper can be summarized as follows: Consider an
+n-dimensional smooth “generic” globally hyperbolic spacetime (M, gab) and assume
+that γ is an incomplete timelike geodesic that is inextendible in (M, gab). Assume
+also that there exists an (n−1)-parameter congruence of causal geodesics, G, spanning
+an open neighborhood of a final segment of γ, such that the tidal force and frame-
+drag (or the electric and magnetic) 1 parts of the curvature tensor, along with the line
+integral of the first-order transversal covariant derivatives of the tidal (or electric)
+part, —measured with respect to a parallelly propagated synchronized basis field—
+are guaranteed to be uniformly bounded along the members of G. Then (M, gab) is
+extendible within the class of C0 Geroch-Traschen regularity class.
+The paper is organized as follow: In Section 2 basic results concerning the ex-
+tendibility of real function, defined on bounded subsets of Rn, is considered.
+In
+Section 3 the basic notion of spacetime extensions and some of the relevant results
+are recalled. A specific choice for the differentiability class of metric is made and
+1For the definition of the tidal force and frame-drag (or the electric and magnetic) parts of
+curvature see equation (5.5) in subsection 5.1 below.
+4
+
+the low differentiable version of the strong cosmic censor hypothesis is discussed in
+Section 4. The main results of the present paper are presented in Section 5, while our
+conclusions and the final remarks are given in Section 6.
+2
+Extensions of real functions
+In demonstrating that a spacetime is part of a larger one, we need results on the
+extendibility of spacetimes. Nevertheless, as a preparation, it is rewarding to glance
+at some of the essential ingredients of this notion.
+Recall first that spacetimes are represented by n-dimensional smooth differentiable
+manifolds on which suitably regular metrics are also given. Note that as a manifold
+locally is Rn, and the components of the metric in the corresponding local coordinates
+are real functions, it is advantageous to know if real functions, given on a subset of
+Rn, can be extended. Exactly this problem was studied by Whitney in the early
+30’s [55, 56] (see also [22]). He considered a real-valued function F, say of class Cm,
+defined on a subset A of Rn, and asked under which conditions exists a function �
+F,
+of class Cℓ, with ℓ ≤ m, on the entire of Rn, such that �F = F on A ?
+In answering the above addressed issues Whitney in [55] introduced the term
+“property P”, to characterize subsets in Rn, defined as follows.2
+Definition 1. A point set A ⊂ Rn is said to possess the property P if there is a
+positive real number ω such that for any two points x and y of A can be joined by a
+curve in A of length L ≤ ω · ρ(x, y), where ρ(x, y) denotes the Euclidean distance of
+the points x, y ∈ Rn.
+The main result by Whitney can be summarized by the following:
+Theorem 1. Assume that A ⊂ Rn has property P, and that F(x1, ..., xn) is of class
+Cm, for some positive integer m ∈ N, in A . Suppose that ℓ ∈ N is so that ℓ ≤ m,
+and also that each of the ℓth order derivatives ∂ℓ1
+x1 · · · ∂ℓn
+xnF, with ℓ1 + · · · + ℓn = ℓ, can
+be defined on the boundary ∂A of A so that they are continuous in A = A ∪ ∂A .
+Then there exists an extension �
+F of F so that �
+F is of class Cℓ throughout Rn.
+Note that it was shown in [55, 56] that �F can be chosen smooth (or, if needed, it
+can be analytic) in Rn \ A .
+To demonstrate that property P plays an essential role in Theorem 1, it is il-
+luminating to recall Example 4.1. of [40]. A smooth real function F on a bounded
+subset A ⊂ R2 is constructed there such that F, along with its partial derivatives up
+2In the literature, property P is also often termed quasi-convex.
+5
+
+to any fixed order, are uniformly bounded in A , nevertheless, F cannot even have a
+continuous extension to the closure of A .
+Note, finally, that the property P as a concept has nothing to do with the causal
+structure of an underlying spacetime, even if it is applied to characterize coordinate
+patches therein. Nevertheless, it is worth emphasizing that in the case of globally
+hyperbolic spacetimes —these are at the center of the investigations in this paper—
+the coordinate domains applied in our constructions, in Section 5, are guaranteed to
+possess the property P (see Proposition 4.1. in [40]).
+3
+Spacetime extensions
+Note first that while giving the notion of spacetime extensions, various differentiability
+assumptions on the metric have been applied depending on the context. In addition,
+as we have not yet fixed a preferred differentiability class, for the moment, it is
+advantageous to keep the applied differentiability class as flexible as it is possible.
+Accordingly, in the definition below CX will signify either the class of analytic, C∞,
+Ck, Ck−, Ck,α functions. It may also stand for more involved classes such as the
+differentiability class C0−,α applied in [9]. With this notation, spacetime extensions
+can be defined as follows:
+Definition 2. Let M and �
+M be n-dimensional connected, paracompact, Hausdorff,
+smooth differentiable manifolds, and (M, gab) and (�
+M, �gab) be time oriented spacetimes
+with metrics at least of class CX. Then (�
+M, �gab) is called to be a CX-extension of
+(M, gab) if there exists an embedding Φ : M → �
+M such that Φ[M] is a proper subset
+of �
+M and Φ is a diffeomorphism between M and Φ[M] ⊂ �
+M, and such that Φ carries
+the metric gab into �gab|Φ[M], i.e., Φ∗gab = �gab|Φ[M]. If (M, gab) admits a CX-extension
+it is said to be CX-extendible. If such an extension does not exist (M, gab) is called
+to be CX-inextendible.
+Note that the involved manifolds cannot be rougher than C1 to permit (at least)
+continuous tangent spaces and also to be able to host a continuous metric. Neverthe-
+less, as it is argued in Theorem 2.9 in [19] if a Cr-differentiability structure, r ≥ 1 is
+given on M, then for every s, r < s ≤ ∞ there exists a compatible Cs-differentiability
+structure on M such that it is unique up to Cs-diffeomorphisms, and such that it is
+Cr-diffeomorphic to the original one. Therefore, without loss of generality, we as-
+sumed above that both of the manifolds, M and �
+M, admit smooth differentiable
+structure. Note also that the differentiability class of gab need not to be exactly CX,
+i.e., gab may belong to some higher differentiability class.
+We close this section by briefly recalling some of the most important results on
+spacetime extensions.
+In doing so, note that the first systematic investigation of
+6
+
+spacetime extensions was carried out by Clarke [7, 8, 9, 10]. His main result is that
+for a “generic” globally hyperbolic C0− causal geodesically incomplete spacetime, there
+is a C0−,α extension provided that the Riemann tensor is also Hölder-continuous. (A
+spacetime, in [7][9], was considered generic if its b-completion was not D-specialized at
+any of the b-boundary points attached to it to represent singularities.) Besides the in-
+disputable importance of these pioneering investigations, there are some drawbacks to
+the above-recalled result. Firstly, Clarke’s results are based on an extensive use of the
+b-boundary construction, which is known to have severe defects even for the simplest
+Friedman-Robertson-Walker cosmological model (for more details, see section 5.2 of
+[9]). Secondly, it may happen that a given spacetime cannot be extended within the
+C0−,α class; in contrast that the curvature remains finite everywhere, simply because
+it fails to be Hölder-continuous at the points of the b-boundary.
+Given the indicated drawbacks, it became of obvious interest to construct space-
+time extensions using the regular geometrical structures of the spacetime to be ex-
+tended exclusively. In particular, it is preferable to avoid using boundary construc-
+tions.
+Keeping these ideas in the forefront, the present author also conducted a
+systematic study of local and global extensions of causal geodesically incomplete
+spacetimes [39, 40]. The main result in [40] can be summarized as follows: Consider
+an n-dimensional smooth “generic” (i.e., locally algebraically non-special) globally hy-
+perbolic spacetime (M, gab) and assume that γ is an incomplete causal geodesic that
+is inextendible in (M, gab). Assume also that there is an (n−1)-parameter congruence
+of causal geodesics, G, spanning an open neighborhood of a final segment of γ, such
+that the components of the curvature tensor, along with its covariant derivatives up to
+order (k −1), and also the line integrals of the components of the kth-order covariant
+derivatives are finite along the members of G —stipulated with respect to a parallelly
+propagated synchronized basis field— are guaranteed to be uniformly bounded along
+the members of G. Then (M, gab) is Ck−-extendible. Comparing these results with
+those covered by the present paper, it is transparent that much lower differentiability
+requirements suffice to show the extendibility of smooth global hyperbolic spacetimes
+within the class of continuous Geroch-Traschen metrics. Note also that the restric-
+tions on curvature are more optimized here as, on the way of proving our main results
+(see, i.e., Theorems 2 and 3 below), we refer merely to the tidal force and frame-drag
+parts of the curvature.
+4
+The choice of differentiability class
+In proceeding, we now fix the differentiability class of the metric that suits most of
+the following discussions. In doing so, recall first that general relativity is a physical
+theory; thereby, the field equations and their solvability or the possible breakdown of
+the field equations are of fundamental interest to us. On these grounds, it is desirable
+7
+
+to admit spacetime models with metrics and other fields that are less well behaved
+than smooth. It is also apparent that the wider the differentiability class of involved
+metrics and matter fields is, the wider the class of gravity-matter systems that can
+be studied within the selected framework.
+These observations immediately suggest involving the widest class of spacetime
+models, allowing us to make sense of the Einstein equations at least as distributions.
+Geroch and Traschen investigated this fundamental issue in [24]. They showed that
+the widest possible class of metrics for which the Riemann, Einstein, and Weyl tensors
+make sense as distributions is the space of “regular metrics” or, as we also refer to
+it, the “Geroch-Traschen regular metrics”. In particular, gab is said to be a Geroch-
+Traschen regular metric if
+(1) gab locally bounded with a locally bounded inverse gab, 3
+(2) the “weak derivatives” of gab are locally square-integrable.
+The metric gab and its inverse gab are locally bounded if the scalar densities gabtab
+and gabuab are bounded for all test fields tab and uab. A test field is supposed to be a
+smooth tensor density of compact support. The tensor distributions are continuous
+linear maps from the vector space of test fields to the real numbers. The derivative
+of the locally bounded metric gab, as distribution, is the distribution ∇egab such that
+�
+M
+(∇egab)teab = −
+�
+M
+gab(∇eteab)
+(4.1)
+for any test field tabc, where ∇a is a smooth torsion-free covariant derivative operator
+on M. Note that the integrals in (4.1) make sense as test fields are assumed to be
+tensor densities of weight −1 [24]. Then a locally integrable 4 tensor field µabc is called
+the weak-derivative of gab if
+�
+M
+gab(∇eteab) = −
+�
+M
+µeab teab
+(4.2)
+holds for all test fields tabc. The weak-derivative µabc of gab is called locally square
+integrable if the scalar density µabc µdef tabcdef is locally integrable for all test fields
+tabcdef.
+3Here, as in [24], to have a well-defined inverse, it is tacitly assumed that gab is non-degenerate.
+In the present context, this can be guaranteed if, on compact sets, a lower uniform bound exists on
+the determinant [29].
+4A tensor field νa...bc...d, defined almost everywhere on M, is called locally integrable if for
+every test field ta...bc...d the scalar density νa...bc...d ta...bc...d is Lebesgue measurable and its Lebesgue
+integral converges.
+8
+
+It was proved in [24] that if the metric gab is Geroch-Traschen regular, then the Rie-
+mann, Einstein, and Weyl tensors make sense as distributions. Geroch and Traschen
+also showed that if a regular metric, gab, is not only locally bounded with a locally
+bounded inverse but is continuous, then it can be approximated by sequences of
+smooth metrics {(i)gab} so that the associated curvature tensors {(i)Rabcd} converge
+in L2 to the curvature distribution assigned to the continuous regular metric gab.
+Returning to our main issue, given the results in [24] it is tempting to choose the
+C0 Geroch-Traschen regularity class. This choice is also supported by the fact that
+it is considered the broadest class of metrics such that weak solutions of the Einstein
+equations exist (see, e.g., [18, 24, 4, 13]). It is also frequently claimed (see, e.g., [4, 13])
+that local existence and uniqueness of weak solutions to the initial value problem
+of the vacuum Einstein’s equations are expected to exist in a setting with metrics
+that belong to the C0 Geroch-Traschen regularity class. Note also that regardless
+of the differentiability class, it is an additional requirement that maximal Cauchy
+developments should also exist. If one can extend a maximal Cauchy development of
+the data given on a maximal initial data slice, the extension is inherently ambiguous
+beyond the Cauchy horizon since, even if the vacuum Einstein equations are satisfied
+there, the data on the initial data surface no longer determine the solution beyond
+the Cauchy horizon. To over-bridge the gap between available techniques and firm
+results, we shall adopt the following C0 form of the strong cosmic censor conjecture
+proposed by Chrishtoduolus, Dafermos, Sbierski [4, 13, 45]:
+Condition 1. The maximal globally hyperbolic development of generic compact or
+asymptotically flat initial data is inextendible as a spacetime within the class of con-
+tinuous Geroch-Traschen regular metrics.
+While it appears conceivable that Condition 1 holds, this remains to be seen. As
+there are a lot of missing technical elements yet it is rewarding to have a glance of
+the regularity class of C0,1
+loc metrics. In doing so note first that the Geroch-Traschen
+regular metrics belong to the intersection H1
+loc ∩ L∞
+loc, where the relation H1 = W 1,2,
+valid for the L2-based Sobolev spaces, was used [1]. Taking then into account that
+on any domain C0,1
+loc = W 1,∞
+loc , and that the local Sobolev spaces are nested, i.e.,
+W 1,m
+loc ⊆ W 1,n
+loc for any m ≥ n hold, it follows that C0,1
+loc metrics belong to the Geroch-
+Traschen regularity class and as such they possess a distributional curvature and
+Einstein tensor. Having this on mind, it is also worth mentioning that several recent
+results holds for the regularity class of C0,1
+loc metrics. For instance, it was shown in
+[5] that C0,1
+loc differentiability of the metric suffices for many vital results of the C∞
+causality theory. Analogously, within the C0,1
+loc regularity class global hyperbolicity
+makes sense, allowing to represent Cauchy hypersurfaces as level sets of C∞ time
+functions [6, 41]. It is also worth noting that the global existence and uniqueness of
+solutions to linear wave equations can also be guaranteed if the background metric
+9
+
+belongs to the C0,1
+loc regularity class [6, 43]. In this context, it is also worth mentioning
+that the existence of maximal Cauchy development for the “3 + 1” vacuum Einstein
+equations requires a metric with critical Sobolev exponent s = 2 [26].
+Finally, it is also worth emphasizing that many physically adequate solutions
+belong to the class of spacetimes with C0 Geroch-Traschen regular metric. They in-
+clude, for instance, gravitational shock waves (where the curvature has a δ-function
+behavior on a null three-surface) [36], thin mass shells (where the curvature has a
+δ-function behavior on a timelike three-surface) [25]. Analogously, this class also in-
+volves solutions containing pressure-free matter where the geodesic flow lines have
+two- or three-dimensional caustics and shell-crossing singularities [51].
+These ex-
+amples demonstrate that whenever Condition 1. holds the existence of incomplete
+timelike geodesics, within the class of spacetimes with C0 Geroch-Traschen regular
+metrics, cannot simply be explained by referring to the presence of certain non-smooth
+wavefronts or caustics of flow lines. Instead, they indicate more serious breakdowns
+of physics.
+Indeed, our principal aim in the present paper is to show that given
+a “generic” (i.e., locally algebraically non-special) timelike geodesically incomplete
+globally hyperbolic spacetime with a smooth metric, either the spacetime can be ex-
+tended within the class of C0 Geroch-Traschen regular metrics or some of the tidal
+force or frame-drag parts of the curvature tensor, or the first-order transversal co-
+variant derivatives of the tidal forces —measured by a (n − 1)-parameter family of
+synchronized observers 5 — become unbounded.
+5
+The main results
+This section is to present our main results. The first one read as:
+Theorem 2. Consider an n-dimensional smooth locally algebraically non-special max-
+imal globally hyperbolic timelike geodesically incomplete spacetime. Assume that Con-
+dition 1 is satisfied, i.e., the C0 form of the strong cosmic censor hypothesis holds.
+Denote by γ one of the incomplete timelike geodesics, and consider a synchronized
+(n − 1)-parameter family of timelike geodesics, G, spanning a neighborhood of a final
+segment of γ. Then, in any arbitrarily small neighborhood of γ, there exists a member
+γ of G such that either one of the tidal force or frame-drag components of the curva-
+ture or one of the first-order transversal covariant derivatives of one of the tidal force
+components blows up along γ.
+Proof: The proof of this theorem is based on contradiction. It starts by assuming
+that the tidal force and frame-drag parts of the curvature, along with the first-order
+5In Subsection 5.1 below, these type of observers are modeled by applying (n − 1)-parameter
+families of synchronized timelike geodesics and frame fields parallelly propagated along them.
+10
+
+transversal covariant derivatives of the tidal forces, —measured with respect to a
+synchronized basis field defined along the (n − 1)-parameter family of synchronized
+timelike geodesics, G,— remain uniformly bounded along the members of G. This
+allows proving Theorem 3 below claiming that the spacetime can be extended within
+the C0 Geroch-Traschen regularity class. This, however, is incompatible with the as-
+sumptions we made. Namely, a “generic” smooth maximal globally hyperbolic time-
+like geodesically incomplete spacetime cannot be extended when the C0 version of
+the strong cosmic censorship hypothesis is also assumed to hold.
+Accordingly, proving Theorem 3 below, given the above reasoning, completes the
+proof of Theorem 2.
+✷
+Theorem 3. Consider an n-dimensional locally algebraically non-special smooth glob-
+ally hyperbolic timelike geodesically incomplete spacetime (M, gab). Denote by γ one
+of the incomplete timelike geodesics, and consider a synchronized (n − 1)-parameter
+family of timelike geodesics, G, spanning a neighborhood of a final segment of γ. As-
+sume that the tidal force and frame-drag parts of the curvature, along with the line
+integrals of the first-order transversal covariant derivatives of the tidal forces, are
+uniformly bounded along the members of G. Then (M, gab) can globally be extended
+within the C0 Geroch-Traschen regularity class.
+Proof: The proof of this theorem is given by performing the following sequence of
+steps.
+1. Assume that γ : (t1, t2) → M is a future directed and future incomplete timelike
+geodesic that is inextendible in (M, gab). For definiteness we assume that γ is
+future incomplete, the other case then follows by a time reversal.
+2. Choose an (n − 1)-parameter family of synchronized future directed timelike
+geodesics, G, such that γ belongs to G.
+3. Choose U ⊂ M such that U is the union of the images of the members of G,
+and such that a final segment γ|[t0,t2), for some t0 ∈ (t1, t2) is contained in U.
+4. Show, under the assumption in our theorem, that there exist �U ⊂ Rn and a
+smooth embedding φ : U → �U such that φ ◦ γ is continuously extendible in �U.
+5. Extend the metric gab from U to �U within the C0 Geroch-Traschen regular-
+ity class, and denote by φ : (U, gab|U) → ( �U, �gab) the intermediate extension
+obtained.
+6. The desired Φ : (M, gab) → (�
+M, �gab) global extension is defined then by gluing
+(M, gab) and ( �U, �gab) at their “common parts”, i.e., applying a quotient space
+induced by φ, whereas the metric �gab gets to be determined by gab and �gab.
+11
+
+5.1
+Synchronized Gaussian coordinates and reference frames
+This section is to introduce the synchronized Gaussian coordinates and synchronized
+orthonormal basis fields that play a central role in constructing the desired interme-
+diate and global extensions.
+Consider a future directed and future incomplete timelike geodesic γ : (t1, t2) →
+M. Assume that t is the proper time parameter along γ, and that va is the correspond-
+ing unit tangent field va = (∂/∂t)a along γ. Then synchronized Gaussian coordinates
+can be defined, in a sufficiently small neighborhood of any point p = γ(t0) of γ, with
+t0 ∈ (t1, t2), as follows [39, 40]. Choose first a sufficiently small open neighborhood,
+Q, of the origin in the linear subspace T ⊥
+p (M), spanned by the (n − 1)-dimensional
+subspace of spacelike vectors orthogonal to va, such that the exponential map6 is
+guaranteed to be a local diffeomorphism between Q and exp[Q] ⊂ M. Denote by
+Σ the image of Q under the action of the exponential map, i.e., Σ = exp[Q]. Ac-
+cordingly, Σ is generated by spacelike geodesics starting at p = γ(t0) with tangent
+orthogonal to va. Extend then va from p = γ(t0) to a smooth future directed nor-
+malized, gabvavb = −1, timelike vector field on Σ which is also everywhere normal
+to Σ. Chose (x1, . . . , xn−1) to be arbitrary local coordinates on Σ and consider the
+(n − 1)-parameter congruence of timelike geodesics, G, starting at the points of Σ
+with tangent va. Since Σ and va are smooth these geodesics do not intersect in a
+sufficiently small neighborhood V of Σ. Extend the functions x1, . . . , xn−1 to V , by
+keeping their values constant along the geodesics in G, and chose the proper time
+t = xn on V as the nth coordinate that is synchronized such that xn = t0 on Σ. The
+functions (x1, . . . , xn) give rise to local coordinates on V .
+In these synchronized Gaussian coordinates, the spacetime metric can be seen to
+take the form [35, 53]
+ds2 = −dt2 + gαβ dxαdxβ ,
+(5.3)
+where gαβ is a (n − 1) × (n − 1) positive definite matrix the components of which are
+smooth functions of all the coordinates (x1, . . . , xn), and the Greek indices take the
+values 1, 2, . . . , n − 1.
+It is also worth mentioning here that, by construction of the Gaussian coordinates,
+the coordinate basis fields Ea
+α = (∂/∂xα)a, with α = 1, 2, . . . , n − 1 are Jacobi fields
+along the (n − 1)-parameter congruence of timelike geodesics in G. Accordingly, the
+Ea
+α coordinate basis fields are subject to the Jacobi equation
+ve∇e(vf∇fEa
+(α)) = Refg
+aveEf
+(α)vg ,
+(5.4)
+where ∇a denotes the metric compatible covariant derivative operator.
+6The exponential map exp : Tp(M) → M assigns a point q ∈ M to a vector Xa ∈ Tp(M) such
+that q is in unit affine parameter distance from p along the geodesic starting at p with tangent Xa.
+12
+
+A synchronized basis field, {ea
+(a)}, along the members of an n−1-parameter family
+of timelike geodesics G, can also be chosen as follows. Start with an orthonormal
+basis {ea
+(a)} ⊂ Tp(M), with name index a, taking the values 1, 2, . . . , n, such that
+ea
+(n) = va at p. Extend then {ea
+(a)} from p by parallelly propagating it first along
+the spacelike geodesics generating Σ. If ea
+(n) happens to be different from the already
+defined va vector field on Σ apply the simplest boost transformation point-wise such
+that ea
+(n) = va holds for the yielded smooth 7 basis field {ea
+(a)} on Σ. Finally, extend
+this smooth basis field by parallelly propagating {ea
+(a)} from Σ, along the members of
+the (n − 1)-parameter family of timelike geodesic congruence G. Note also that, by
+construction, the relation ea
+(n) = va holds along the members of G.
+Utilizing the above defined synchronized basis field, {ea
+(a)}, for instance, the tidal
+force and frame-drag (or the electric and magnetic) parts of the Riemann tensor,
+with respect to the observers moving along the members of G with velocity va, can
+be given, see, e.g., [31, 32], as
+Ranbn = Rabcd ea
+(a)vbec
+(b)vd
+and
+Rabcn = Rabcd ea
+(a)eb
+(b)ec
+(c)vd ,
+(5.5)
+respectively, where the indices a, b, c take the values 1, 2, . . . , n − 1. Note that the
+tidal force and frame-drag parts of curvature, as given in (5.5), are related to the
+electric and magnetic parts of the curvature, defined with respect to the unite timelike
+vector field va [31][32]. In the 4-dimensional case, the electric and magnetic parts are
+represented by the symmetric tensors Eab and Bab defined as
+Eab = Ranbn = Rabcd ea
+(a)vbec
+(b)vd
+and
+Bab = 1
+2 ǫefaRef
+bn = 1
+2 ǫefaRef
+bc ea
+(a)eb
+(b)vc ,
+(5.6)
+respectively, where ǫabc stands for the contraction ǫabc = ǫabceve of the 4-volume ele-
+ment ǫabce and the unite timelike vector field va.
+5.2
+The selection of U
+Since our principal aim is to perform spacetime extensions based on a suitable choice
+of synchronized Gaussian coordinates, it is of obvious interest to know how far from
+Σ these coordinates and the synchronized basis field can be applied. Our aim in this
+subsection is to answer this question.
+Start by choosing U ⊂ Rn consisting of those n-tuples, (x1, . . . , xn), to which
+there exists a timelike geodesic γ : (t1, t2) → M in G such that xn ∈ (t1, t2) and
+(x1, . . . , xn−1, t0) are the coordinates of the intersection γ ∩ Σ. Define the map ψ :
+7The smoothness of the basis field {ea
+(a)} on Σ follows from that of va, the process of parallel
+propagation, along the generators of Σ, and also that of the boost transformations applied point-wise
+on Σ.
+13
+
+U → M such that ψ(x1, . . . , xn) = γ(xn) with assuming that (x1, . . . , xn−1, t0) are
+the coordinates of γ ∩ Σ. Although, ψ, by construction, is smooth, in general, it is
+not necessarily one-to-one. If it is not one-to-one V is a proper subset of ψ[U ].
+Recall also that the choice we have made for Σ guarantees that the second fun-
+damental form χab of Σ vanishes at p = γ(t0). Therefore, by choosing a sufficiently
+small open neighborhood σt0 of p with compact closure in Σ it can be guaranteed
+that there exists a small positive number κ, depending on χab and σt0, such that for
+a suitable norm —for its definition see Section 3. in [40]— ∥χab∥ < κ holds on σt0.
+Our aim is to show that, in suitable circumstances, there exist t0, σt0 ⊂ Σ and
+ε > 0 such that on the subset U[σt0, ε] of U ⊂ Rn, defined as
+U[σt0, ε] := {(x1, . . . , xn) ∈ U | ψ(x1, . . . , xn−1, t0) ∈ σt0 and xn ∈ [t0, t2 + ε)} , (5.7)
+the map ψ : U[σt0, ε] → M is one-to-one. In verifying this claim first we refer to the
+proof of Proposition 3.2.5 of [40] to the individual members γ ∈ G, the boundedness
+of the tidal force (or electric) part of the curvature tensor implies that there exist
+ǫ > 0 such that no conjugate point along γ, in the parameter interval [t0, t2 + ε), can
+occur to σt0. This, with reference to the claims in the proof of Proposition 3.1 of
+[40], implies that to any point q in U[σt0, ε] there must exist an open neighborhood Oq
+such that ψ is a local diffeomorphism between Oq and its image Oq = ψ[Oq]. In other
+words, this guarantees that the Gaussian coordinates are locally well-defined on Oq.
+The goal is now to show that under suitable conditions the Gaussian coordinates
+are only locally well-defined but also globally well-defined throughout ψ[U[σt0, ε]]. To
+understand the difficulties at this point recall that spacetimes with “quasi-regular”
+singularities do also exist (see, e.g., Refs. [7, 15, 16]) which can get in the way of
+getting well-defined Gaussian coordinates on ψ[U[σt0, ε]]. These spacetimes are known
+to have topological defects which prevent the existence of U[σt0, ε] ⊂ Rn on which
+ψ could be one-to-one.
+To separate these cases in [40] the notion of topological
+singularity was introduced which, in the timelike case, reads as:
+Definition 3. A future directed incomplete future inextendible timelike geodesic γ :
+(t1, t2) → M is said to terminate on a topological singularity if there is no choice for
+t0, σt0 ⊂ Σ and ε such that the set ψ[U[σt0, ε]] would be simply connected.
+It is proved then in Section 5 of [40] that the existence of a topological singularity
+in globally hyperbolic spacetimes implies that they are locally algebraically special,
+i.e., these spacetimes cannot be “generic”. It is also proved (see Theorem 3.1. in [40])
+that whenever γ : (t1, t2) → M does not terminate on a topological singularity for a
+suitable choice of U[σt0, ε] the members of G do not intersect in ψ[U[σt0, ε]], i.e., there
+exist t0, σt0 ⊂ Σ and ε > 0 such that ψ is one-to-one on the entire of U[σt0, ε].
+14
+
+Choose U to be the interior, (ψ[U[σt0, ε]])◦, of the image ψ[U[σt0, ε]] of U[σt0, ε], and
+also �U to be the Cartesian product ςt0 × [t0, t2 + ε) ⊂ Rn, where ςt0 = ψ−1[σt0]. Note
+that by construction (U[σt0, ε])◦ is a proper subset of �U, also that ψ is one-to-one on
+U = (ψ[U[σt0, ε]])◦. Denote by φ the restriction of the inverse of ψ to U = (ψ[U[σt0, ε]])◦,
+i.e., φ = ψ−1|(ψ[U[σt0 , ε]])◦. The map φ : U → �U is an embedding that will be used in
+constructing the desired intermediate extension φ : (U, gab|U) → ( �U, �gab).
+Note that, by the above choices made for U and �U, the members of G in U are
+represented by straight coordinate lines in φ[U], and also that for any member γ of
+G, that starts at σt0, and that is future directed incomplete and future inextendible
+in (M, gab), the curve φ◦γ can be continued as a straight line in the region �U \φ[U]. 8
+5.3
+Extending the metric from U to �U
+We shall need the following proposition in extending the metric from U to �U ⊂ Rn.
+Proposition 1. Assume that φ[U] is defined as above and that it has the property
+P. Consider a smooth function F on φ[U] and assume that its first-order partial
+derivatives, ∂tF and ∂xαF, where α = 1, 2, . . . (n − 1), are uniformly bounded on
+φ[U]. Then, the unique continuous extension �F of F to the closure φ[U] is Lipschitz
+function that can be further extended onto �U \ φ[U] such that �F is also Lipschitz
+throughout �U.
+Proof: First, the uniform boundedness of the first-order partial derivatives, ∂tF and
+∂xαF, α = 1, 2, . . . (n − 1), can be used to show that F is Lipschitz function on φ[U]
+as it was done in proving Proposition 3.3.1 in [39]. Note that then F is also uniformly
+continuous there.
+The proof of Proposition 4.2 in [40] can be used to verify then that the unique
+continuous extension �F of F onto the closure φ[U] is also Lipschitz function, with the
+same Lipschitz constant.
+Finally, in virtue of Kirszbraun’s theorem [27] �F also extends to �U \ φ[U] as a
+Lipschitz function with the same Lipschitz constant.
+✷
+Note that as �F is Lipschitz everywhere on �U the weak derivatives of �F exist, and
+they are bounded, whence they are also locally square integrable there.
+8Note that in our ultimate argument, in proving Theorem 3, (M, gab) is assumed to be a Cauchy
+development. Whence, if there exists a non-empty boundary to M, it has to be part of a Cauchy
+horizon H of (M, gab). In particular, the union of the future endpoints of the images of the members
+of G in �U, denote it by H+, is a subset H+ = H+|U of the (non-empty) future Cauchy horizon H+
+of (M, gab). Recall then that, in virtue of Proposition 6.3.1, along with the arguments in Section
+6.5, of [18], H+ ⊂ ∂(φ[U]) is a closed, embedded, achronal three-dimensional C1− submanifold in �U.
+15
+
+In proceeding recall first that the (n − 1) × (n − 1) matrix elements gαβ in (5.3)
+are given by the contractions
+gαβ = gabEa
+(α)Eb
+(β) ,
+(5.8)
+where Ea
+(α) stand for the coordinate basis elements (∂/∂xα)a, with α = 1, . . . , n − 1.
+In virtue of Proposition 1, the components of gαβ, that are smooth functions on φ[U],
+extend as Lipschitz functions onto φ[U] if the time- and spatial-derivatives,
+∂tgαβ = gab
+��
+ve∇eEa
+(α)
+�
+Eb
+(β) + Ea
+(α)
+�
+ve∇eEb
+(β)
+��
+(5.9)
+∂xνgαβ = gab
+��
+Ee
+(ν)∇eEa
+(α)
+�
+Eb
+(β) + Ea
+(α)
+�
+Ee
+(ν)∇eEb
+(β)
+��
+,
+(5.10)
+can be guaranteed to be uniformly bounded along the members of G.
+Inspecting the individual terms on the right hand sides in (5.9) and (5.10) it
+appears that it is completely satisfactory to show that the norms ∥Ea
+(α)∥, ∥ve∇eEa
+(α)∥
+and ∥Ee
+(ν)∇eEa
+(α)∥ of the vector fields Ea
+(α), ve∇eEa
+(α) and Ee
+(ν)∇eEa
+(α) are uniformly
+bounded with respect to the synchronized orthonormal basis fields {ea
+(a)} defined along
+the members of G in U. Here the norm ∥Xa∥ of a vector field Xa, with respect to the
+synchronized basis field {ea
+(a)} and the Lorentzian metric gab on U, is defined as
+∥Xa∥ :=
+�
+�
+�
+�
+4
+�
+b=1
+�
+gabXaeb
+(b)
+�2
+.
+(5.11)
+Note that, by applying a straightforward adaptation of Corollary 3.3.5. of [39], the
+norm ∥Ea
+(α)∥ can be guaranteed to be bounded on U provided that the tidal force
+components of the curvature tensor, Rabcd ea
+(a)vbec
+(b)vd, —measured with respect to
+a parallelly propagated synchronized orthonormal frame field along the members of
+G— are uniformly bounded along the members of G. Recall also that in a Gaussian
+coordinate system the coordinate basis fields Ea
+(α) = (∂/∂xα)a, by construction, are
+subject to the the Jacobi equation (5.4). Applying then Lemma 3.3.6. of [39], to the
+individual members of G, we get
+∥ve∇eEa
+(α)∥γ(t) ≤ ∥ve∇eEa
+(α)∥γ(t0) +
+� t
+t0
+∥Rbcd
+avbEc
+(α)vd∥γ(t′) dt′ .
+(5.12)
+Combining this with the linearity of the integrand in Ea
+(α), we get that both of the
+terms ∥Ea
+(α)∥ and ∥ve∇eEa
+(α)∥ are uniformly bounded along the members of G when-
+ever the tidal force components of the Riemann tensor —defined with respect to
+a synchronized orthonormal basis field— are also guaranteed to remain uniformly
+bounded along the members of G.
+16
+
+The characterization of the norm ∥Ee
+(ν)∇eEa
+(α)∥ of the vector field Ee
+(ν)∇eEa
+(α)
+requires a bit more care.
+Using the linearity of the curvature terms, along with the Leibniz rule, and the
+vanishing of the commutator of the coordinate basis fields Ea
+(α) and va, we get 9
+ve∇e(vf∇f[Eh
+(ν)∇hEa
+(α)]) = − Ek
+(ν)∇k[Refh
+aveEf
+(α)vh] + Refh
+aEe
+(ν)vf[vk∇kEh
+(α)]
++ vk∇k[Refh
+aEe
+(ν)vfEh
+(α)] .
+(5.13)
+This, along with
+Ea
+(α) =
+n−1
+�
+i=1
+�
+gklEk
+(α) el
+(i)
+�
+ea
+(i) ,
+(5.14)
+(which follows from the orthogonality of Ea
+(α) and va), yields
+d2
+dt2
+�
+gklEk
+(α) el
+(i)
+�
+= −
+n−1
+�
+j=1
+�
+Rabcd va eb
+(j)vc ed
+(i)
+� �
+gklEk
+(α) el
+(j)
+�
++
+n−1
+�
+j=1
+Aj
+�
+Rabcd va eb
+(j)vc ed
+(i)
+�
++
+n−1
+�
+j,h=1
+�
+Bjh
+�
+Rabcd ea
+(i) eb
+(h) ec
+(j)vd�
++ Cjh
+d
+dt
+�
+Rabcd ea
+(i) eb
+(h) ec
+(j)vd�
++ Djh ek
+(h)∇k
+�
+Rabcd va eb
+(j)vc ed
+(i)
+��
+,
+(5.15)
+where the coefficients Ai, Bij, Cij and Dij depend exclusively on the fields Ea
+(α),
+vf∇fEa
+(α) and Ef
+(α)∇f ea
+(i). Note that the uniform boundedness of ∥Ea
+(α)∥ and ∥vf∇fEa
+(α)∥
+has already been guaranteed by uniform boundedness of the tidal forces. The uniform
+boundedness of ∥Ef
+(α)∇f ea
+(i)∥, however, requires the uniform boundedness of the line
+integral of the frame-drag part of the curvature. To see this, note that because of the
+commutation of the coordinate basis fields Ea
+(α) and va, also as the basis fields ea
+(i),
+i = 1, 2, . . . (n − 1), are parallelly propagated with respect to va,
+ve∇e(Ef
+(α)∇f ea
+(i)) = −Refh
+aveEf
+(α)eh
+(i)
+(5.16)
+holds, which implies
+d
+dt
+�
+gkl(Ef
+(α)∇f ek
+(i)) el
+(j)
+�
+=
+n−1
+�
+h=1
+[ gklEk
+(α) el
+(h) ] [ Rabcdea
+(i)eb
+(j)ec
+(h)vd ] .
+(5.17)
+9Equation (5.13) is a compact form of the generalized Jacobi equation that was introduced in
+[39] (see also [40]) to characterize the propagation of various order of covariant derivatives of the
+coordinate basis fields along members of causal geodesic congruences.
+17
+
+Accordingly, the norm of the ∥Ee
+(ν)∇eEa
+(α)∥ of the vector field Ee
+(ν)∇eEa
+(α) will be
+uniformly bounded along the members of G in U whenever the the tidal force and
+frame-drag parts of the curvature 10 and the line integrals of the first-order transversal
+covariant derivatives of the tidal forces
+� t
+t0
+�
+ek
+(c)∇k[Rabcd ea
+(a)vbec
+(b)vd]
+�
+|γ(t′) dt′ ,
+(5.18)
+are all guaranteed to be uniformly bounded along the members of G in U, where the
+indices a, b, c take the values 1, 2, . . . , n − 1.
+The critical point here is that it suffices to restrict merely the line integrals of the
+first-order transversal covariant derivatives of the tidal forces which follow from the
+discussion in [40], starting below (4.15), including (4.16) and the paragraph below
+there. 11
+Note also that the finiteness of these line integrals does not require the
+integrands’ boundedness. They can be uniformly bounded if the integrands do not
+blow up faster than (tp − t)−1+ǫ, for some ǫ > 0, where tp stands for the supremum
+of the affine parameter value along γ ∈ G.
+Utilizing all the above observations, we get that both the time- and spatial-partial
+derivatives, ∂tgαβ and ∂xνgαβ of the metric will be uniformly bounded along the mem-
+bers of G whenever the tidal force and frame-drag parts of the curvature, along with
+the line integrals of the first-order transversal covariant derivatives of the tidal forces,
+are guaranteed to be uniformly bounded along the members of G.
+If this happens, combining all the above observations, it follows then that the
+gαβ components of the metric gab, in Gaussian coordinates, are Lipschitz functions
+throughout φ[U]. Proposition 1 also implies that the metric tensor components gαβ, in
+(5.3), extend from φ[U] onto the entire of �U = ςt0 ×[t0, t2+ε) such that the extensions
+�gαβ are guaranteed to be Lipschitz functions throughout �U.
+Note also that the extended metric �gαβ is C0 Geroch-Traschen regular metric on
+�U. Firstly, since the components �gαβ are guaranteed to be Lipschitz functions in �U the
+boundedness (not merely the local boundedness) of �gαβ and that of �gαβ immediately
+follows.
+For the same reason, as the components �gαβ of the metric are Lipschitz
+functions on �U each of the weak derivatives of �gαβ are well-defined and bounded,
+thereby, they are also square integrable. This completes the proof of the following:
+10The frame-drag part of the curvature gets involved via the line integral of various terms on the
+right-hand side in (5.15). However, in the fourth term, the first order t-derivative of the frame-drag
+part is involved, which, in turn, leads to an algebraic restriction on this part of the curvature.
+11One has to consider here the line integrals of the last four terms on the right-hand side in (5.15).
+Nevertheless, as discussed in footnote 5.3, the line integral of the second, third and fourth terms
+leads to an algebraic restriction on the frame-drag part of the curvature. Indeed, it is the fifth term
+that yields restrictions on the line integrals of the first-order transversal covariant derivatives of the
+tidal forces.
+18
+
+Proposition 2. Consider the embedding φ : U → �U defined as above. Assume that
+the tidal force and frame-drag parts of the curvature tensor —measured with respect
+to a parallelly propagated synchronized orthonormal frame field along the members of
+G— , along with the line integrals of the first-order transversal covariant derivatives
+of the tidal forces, are uniformly bounded along the members of G. Then there exist
+an extension φ : (U, gab|U) → ( �U, �gab) such that �gab belongs to the C0 Geroch-Traschen
+regularity class.
+Applying the auxiliary extension φ : (U, gab|U) → ( �U, �gab), the desired global
+extension can then be given as follows: Chose U′ to be a compact subset of U in
+M. The differentiable structure of U induces a manifold with boundary on U′. Let,
+furthermore, �
+O ⊂ �U be comprised by the union of φ[U′] and a sufficiently small
+neighborhood of the endpoint of φ ◦ γ in �U ⊂ Rn. Define then �
+M to be the factor
+space
+�
+M = (M ∪ �O)/φ ,
+(5.19)
+i.e., �
+M is yielded by identifying x ∈ U′ and y ∈ �O if φ(x) = y. Then �
+M has the
+structure of a (Hausdorff) manifold without boundary while the spacetime (�
+M, �gab)
+is a C0 extension of (M, gab), where the metric �gab is determined on �
+M by gab and
+�gab, and, thereby, it belongs to the C0 Geroch-Traschen regularity class.
+Combining all the partial results outlined above verifies then that in the case
+of a smooth geodesically incomplete generic globally hyperbolic spacetime can be
+extended within the class of C0 Geroch-Traschen regular metrics if the tidal force
+and frame-drag parts of the curvature tensor, along with the line integrals of the first-
+order transversal covariant derivatives of the tidal forces —defined for a synchronized
+basis field— are guaranteed to be uniformly bounded along the members of G, which
+completes the proof of Theorem 3.
+✷
+6
+Final remarks
+The two main results of this paper are intimately related. First, “generic” (locally
+algebraically non-special) smooth globally hyperbolic timelike geodesically incomplete
+spacetime (M, gab) were considered. We showed then that such a spacetime could
+(globally) be extended within the C0 Geroch-Traschen regularity class if the tidal
+force and frame-drag parts of the curvature tensor, along with the line integrals of the
+first-order transversal covariant derivatives of the tidal forces, are uniformly bounded
+along the world-lines of an (n − 1)-parameter family of synchronized observers.
+The second result essentially aims to over-bridging the gap between the intuitive
+picture of spacetime singularities and the predictions of the singularity theorems. This
+is made by considering the “generic” smooth globally hyperbolic timelike geodesically
+19
+
+incomplete spacetime that is a maximal Cauchy development, and assuming that the
+C0 form of the strong cosmic censor hypothesis holds. Combining this with the first
+result’s conclusion immediately drives to a contradiction, which allows to conclude
+that there must exist worldlines of observers in arbitrarily small neighborhoods of the
+one represented by γ, such that either some of the tidal force or frame-drag compo-
+nents of the curvature, or some of the first-order transversal covariant derivatives of
+the tidal force components blow up, at least at the order (tp − t)−ǫ, for some ǫ > 0,
+or (tp − t)−1, respectively, along the corresponding worldlines of observers.
+It is worth emphasizing that tidal force and frame-drag parts of the curvature
+tensor are among the physically most adequate quantities in characterizing spacetime
+singularities. To see this, note first that the applied (n−1)-parameter family timelike
+geodesics do represent the history of a free-falling small body. Whenever tidal force
+part of the curvature become unbounded along a member of such a congruence, the
+small body may undergo an infinite pull-apart in one direction, whereas compression
+in another. Similarly, one would expect that the blowing up of the integrand in (5.18)
+involving first-order transversal covariant derivatives of the tidal forces, implies that
+significant shears will be exerted on the internal structure of the aforementioned small
+body. The frame-drag (or magnetic) part of the curvature tensor is also of physical
+interest as it is directly related to the frame-dragging angular velocity at some event
+¯p ∈ γ, with respect to the inertial directions at a nearby spatially separated event
+p ∈ γ. Consider a small body composed of densely arranged tiny gyroscopes. Then,
+the frame-drag part of the curvature measure the relative precession of the nearby
+gyroscopes [32]. Accordingly, the frame-drag part is related to the relative twisting
+exerted on nearby parts of such a small body, which must undergo infinite wringing
+if the frame-drag part blows up.
+Note that mainly for simplicity in this paper, considerations were restricted to
+the case of timelike geodesics. Nevertheless, the results summarized above general-
+ize straightforwardly (for details, see Refs. [39, 40]) to the case when geodesically
+incomplete lightlike geodesics are involved.
+It is worth mentioning that in addition to the geodesic congruence-based ap-
+proach applied in this paper, there are attempts to give alternative characterizations
+of spacetime singularities. For instance, there is an approach aiming to characterize
+spacetime singularities by studying obstructions to the evolution of test fields. The
+primary motivation behind this approach was that determination of geodesics may
+become vague in the case of metrics of low regularity [12, 42, 54, 52]. Note, however,
+that even the applied low regularity case the method proposed by Clark in [12] (see
+also [54, 42]) required the construction of congruences of timelike geodesics, whose
+tangent vectors admit bounded weak derivatives. This vector field was applied to
+define a suitable energy inequality from which the uniqueness and existence of test
+fields could be deduced.
+20
+
+The last remark immediately raises the following question: Would it be possible
+to replace the smoothness of the primary metric in Theorems 3 and 2 by allowing
+metrics belonging to the class of C0 Geroch-Traschen regular metrics? In answering
+this question, recall, first, that if a Geroch-Traschen regular metric is continuous, 12
+then it can be approximated by sequences of smooth metrics {(i)gab} such that the
+associated curvature tensors {(i)Rabcd} converge in L2 to the curvature distribution
+assigned to the C0 Geroch-Traschen regular metric gab. To recover the main conclusion
+in Theorems 3 and 2, one has to find a way to represent geodesics with respect to
+C0 Geroch-Traschen regular metrics as an accumulation of sequences of smooth (i)gab-
+geodesics, as well as, it has to be shown that weak solutions to the Jacobi equation
+make sense (almost everywhere) along congruences of timelike geodesics. 13 Notably,
+many of the needed techniques had already been developed and applied in working
+out the above-mentioned test fields based characterization of spacetime singularities
+(for details, see [9], in particular, the appendix of [42]).
+It is also an appealing issue whether the estimates we applied in proving Theorem
+3 were optimal. For instance, it is important to know whether the construction of the
+intermediate extension could be carried out requiring only the boundedness of the
+integral
+Iγ(t) =
+� t
+t0
+∥Rbcd
+avbEc
+(α)vd∥γ(t) dt .
+(6.20)
+One of the main obstacles in doing so is that when metrics belonging to the C0
+Geroch-Traschen regularity class are used, the term Rbcd
+avbEc
+(α)vd make sense only as
+distribution. It is conceivable that this can be done, but it remains to be seen.
+Acknowledgments
+This project was partly supported by the NKFIH grants K-115434 and K-142423.
+The author is also indebted to the unknown referees for helpful comments on the
+previous version of this paper.
+12It is worth pointing out that even if a Geroch-Traschen regular metric would not continuous,
+it could still be approximated by sequences of smooth metrics provided that a specific stability
+condition, c.f., Section 4 in [50], holds on it.
+13Note that approximating geodesics of merely continuous or Lipschitz metrics is a subtler matter,
+since there is no standard way to solve the geodesic equation in these cases. To overcome these
+difficulties interesting attempts were made by using Fillipov-solutions in case of C0,1 regular metrics
+in [28][49]. Strong results on approximating geodesics of globally hyperbolic C1-metrics were also
+proved in [17].
+21
+
+Data Availability
+Data sharing not applicable to this article as no datasets were generated or analyzed
+during the current study.
+Conflict of Interest Statement
+The author declares no conflicts of interest.
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+

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@@ -0,0 +1,3397 @@
+arXiv:2301.03361v1  [math.QA]  9 Jan 2023
+FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS
+OVER FINITE SIMPLE GROUPS OF LIE TYPE VII.
+SEMISIMPLE CLASSES IN PSLn(q) AND PSp2n(q)
+NICOL´AS ANDRUSKIEWITSCH, GIOVANNA CARNOVALE AND
+GAST´ON ANDR´ES GARC´IA
+Abstract. We show that the Nichols algebra of a simple Yetter-Drin-
+feld module over a projective special linear group over a finite field
+whose support is a semisimple orbit has infinite dimension, provided
+that the elements of the orbit are reducible; we obtain a similar result
+for all semisimple orbits in a finite symplectic group except in low rank.
+We prove that orbits of irreducible elements in the projective special
+linear groups could not be treated with our methods. We conclude that
+any finite-dimensional pointed Hopf algebra H with group of group-
+like elements isomorphic to PSLn(q) (n ≥ 4), PSL3(q) (q > 2), or
+PSp2n(q) (n ≥ 3), is isomorphic to a group algebra, completing work
+in arXiv:1506.06794.
+Contents
+1.
+Introduction
+1
+Acknowledgements
+5
+2.
+Racks
+5
+3.
+Algebraic groups
+8
+4.
+Split conjugacy classes
+15
+5.
+The special linear groups
+18
+6.
+Semisimple conjugacy classes represented in K
+29
+7.
+The symplectic groups
+37
+References
+41
+1. Introduction
+1.1. The problem. Let G be a finite group. The conjugacy class of x ∈ G
+is denoted by OG
+x or Ox. The subgroup of G generated by I ⊂ G is denoted
+by ⟨I⟩. Consider the following properties of a conjugacy class O of G:
+2010 Mathematics Subject Classification: 16T05, 20D06.
+Keywords: Nichols algebra; Hopf algebra; rack; finite group of Lie type; conjugacy class.
+1
+
+2
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+(C) There are H < G and r, s ∈ H ∩ O such that rs ̸= sr, H = ⟨OH
+r , OH
+s ⟩,
+OH
+r ̸= OH
+s and min{|OH
+r |, |OH
+s |} > 2 or max{|OH
+r |, |OH
+s |} > 4.
+(D) There exist r, s ∈ O such that O⟨r,s⟩
+r
+̸= O⟨r,s⟩
+s
+and (rs)2 ̸= (sr)2.
+(F) There are ra ∈ O, a ∈ I4 = {1, 2, 3, 4}, such that O⟨ra:a∈I4⟩
+ra
+̸= O⟨ra:a∈I4⟩
+rb
+and rarb ̸= rbra for a ̸= b ∈ I4.
+We say that O is of type C, D, F when the corresponding property holds.
+As explained in the Introduction to [5], the next question has profound im-
+plications for the classification of finite-dimensional pointed Hopf algebras.
+Question 1. Determine which conjugacy classes of a given finite (non-
+abelian) group G are of type C, D or F.
+Indeed, if a conjugacy class O is type C, D or F, then any Nichols algebra
+of group type with support isomorphic to O has infinite dimension; for
+brevity we say in this case that O collapses. For the purposes of this paper
+further precision on Nichols algebras is not needed.
+If O is neither of type C, D nor F then we say that it is kthulhu. It follows
+at once from the previous definitions that if O ∩ H is either abelian or a
+single conjugacy class of H for any H ≤ G, then O is kthulhu.
+Intuitively, the criteria of types C, D and F are inductive arguments that
+are more flexible in the language of racks, see Section 2. Conjugacy classes
+are the prototypical examples of racks. One may wonder whether there are
+other inductive arguments that force the collapse of a conjugacy class. In
+this respect we say that a rack is sober if every subrack is either abelian
+or indecomposable [1, §1.5]; and austere if every subrack generated by two
+elements is either abelian or indecomposable [3, 2.11]. Clearly, sober implies
+austere and austere implies kthulhu. Subsidiary to Question 1, we propose:
+Question 2. Determine which conjugacy classes of a given finite (non-
+abelian) group G are sober or austere, or kthulhu.
+1.2. Simple groups of Lie type. It is natural, and convenient, to start
+addressing Questions 1 and 2 by assuming that G is simple non-abelian, see
+[9] for the importance of this reduction. When G is alternating or sporadic,
+this was addressed in [7, 8, 14, 15]. The series of papers [1, 2, 3, 4, 5, 12] treat
+the case when G is simple of Lie type. In the first five papers, Questions 1
+and 2 were answered for non-semisimple conjugacy classes in Chevalley or
+Steinberg groups. The sixth is devoted to Suzuki and Ree groups.
+In the present paper we deal with semisimple conjugacy classes in the
+classical Chevalley groups. The main difficulty is due to the deeper influence
+of arithmetics, as opposed to the unipotent classes and the mixed classes,
+which can be reduced in most of the cases to a unipotent one in a smaller
+group. We summarize our main results and then discuss the proofs.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+3
+Theorem I. Let O ̸= {e} be a semisimple conjugacy class in a group G.
+(i) If G = PSLn(q), then any O not listed in Table 1 collapses.
+(ii) If G = PSpn(q), then any O collapses with the possible exception of
+the orbit of non-trivial involutions if n = 2 and q ∈ {3, 5, 7}.
+Part (i) is proved in Theorem 5.4; see Remark 5.3 for the cases with small
+q excluded in the statement. Part (ii) is proved in Theorem 7.1. For other
+Chevalley groups, there is substantial information in Theorems 4.1 and 6.2.
+Roughly the proofs of these results go as follows: pick a simple group G,
+a surjective morphism of groups π : G → G and conjugacy classes O and O
+in G and G respectively, such that π(O) = O. Then look at the subgroups
+H intersecting O. If H ∩ O splits as more than one conjugacy class of H
+for one H ≤ G, then work out the details to have that O is of type C or D
+and that this is preserved by the projection π : O → O. When G is of Lie
+type, the subgroup H is usually found by looking in one way or another at
+the structure of the algebraic group behind G.
+But if π(H)∩O is either abelian or one conjugacy class of π(H) for every
+H ≤ G, then O is kthulhu. When this happens, usually G is ‘small’ and
+has few subgroups.
+In the present paper, we found that it also happens to any conjugacy class
+O of an irreducible element in PSLn(q) where n is an odd prime. To show
+this we used the main result of [17] to get the list of the H ≤ G intersect-
+ing O together with some arithmetic manipulations. This outcome differs
+significantly with the results of the previous of the series and underlines the
+connection of semisimple classes with arithmetical aspects.
+Table 1. Kthulhu semisimple classes in PSLn(q).
+n
+q
+class
+Remark
+2, PSL2(2) ≃ S3
+(3)
+abelian
+3, PSL2(3) ≃ A4
+(22)
+abelian
+4, PSL2(4) ≃ A5
+(5)
+sober
+2
+5, PSL2(5) ≃ A5
+(1, 22)
+sober
+9, PSL2(9) ≃ A6
+(1, 5)
+kthulhu
+even and not a square
+irreducible, order 3
+sober
+all
+irreducible, order > 3
+sober
+odd prime
+all
+irreducible
+kthulhu
+1.3. Applications to Hopf algebras. As in previous papers, we say that
+a finite group G collapses if every finite-dimensional pointed Hopf algebra
+H with G(H) ≃ G is necessarily isomorphic to CG.
+As a corollary of
+our main Theorem and results from previous papers in the series, we obtain
+new families of groups that collapse, see Theorem III, extending [3, Theorem
+
+4
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+1.2]. For this, we first draw the complete list of kthulhu classes in the simple
+groups PSLn(q) and PSp2n(q) for n ≥ 2, any q. This information combines
+the main result of this paper with a corrected version of [1, Table 1], [3,
+Table 3] and [5, Theorem 1.1] and a careful analysis of the cases of the
+groups PSL2(q) for q = 2, 3, 4, 5, 7, 9, where some exceptions occur. We
+point out that the classes labeled (1r1, 2) in Sp2n(9), occurring in [3, Tables
+3,5] are in fact not kthulhu: they are of type C, as each of them includes a
+non-trivial unipotent class of type (2) in PSL2(9) ≃ A6 which is of type C,
+cf. Example 2.7. Also, the classes of involutions in PSL2(7) in [3, Table 1]
+are not not kthulhu: since PSL3(2) ≃ PSL2(7), they are of type C by [3,
+Lemma 2.12].
+Theorem II. Let G be either PSLn(q) or PSp2n(q), n ≥ 2 and let O be a
+non-trivial conjugacy class in G different from the class of a split involution
+in PSp4(7). Then O is kthulhu if and only if it occurs in Table 2.
+Table 2. Kthulhu classes, G = PSLn(q) or PSp2n(q), n ≥ 2.
+G
+n
+q
+type of class
+description/label
+even or else
+odd and not
+a square
+unipotent
+(2)
+5
+semisimple
+involution
+PSLn(q)
+2
+all
+semisimple
+irreducible, |x| > 3
+even
+and
+not
+a
+square
+semisimple
+irreducible |x| = 3
+3
+2
+unipotent
+(3)
+odd prime
+all
+semisimple
+irreducible
+even
+unipotent
+W(1)a ⊕ V (2)
+PSp2n(q)
+≥ 2
+odd and not
+a square
+unipotent
+(1r1, 2)
+even
+unipotent
+W(2)
+2
+3,5
+semisimple
+split involution
+It remains open to determine whether the conjugacy class of split involu-
+tions in PSp4(7) is kthulhu.
+The next result combines [1, Theorem 1.4], [3, Theorem 1.2] and [5, The-
+orem 1.2] with Theorem I.
+Theorem III. The groups PSLn(q) with n ≥ 4, PSL3(q) with q > 2, and
+PSp2n(q), n ≥ 3, collapse.
+□
+In the group PSL3(2), there is just one class that could not be treated,
+namely the regular unipotent class O, which is sober. Actually PSL3(2) ≃
+PSL2(7) and for this group, O is semisimple.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+5
+1.4. Conventions. If a ≤ b ∈ N0, then Ia,b denotes {a, a + 1, . . . , b}; also
+Ia = I1,a for simplicity.
+Let G be a group. The centraliser of X ⊂ G is denoted by CG(X). If
+x, y ∈ G, then x ⊲ y := xyx−1. We write X ≥ Y , or Y ≤ X, to express that
+Y is a subrack of X (or a subgroup, or more generally a subobject in a given
+category). The normality of a subgroup is expressed by N ⊳ G.
+Let q = pm, p a prime and m ∈ N. Let Fq be the field with q elements
+and k the algebraic closure of Fq. We denote by Gn(k) the group of n-th
+roots of unity in a field k.
+Acknowledgements
+We thank Gunter Malle for very helpful email exchanges and Andrea
+Lucchini for pointing out several references.
+N. A. was partially supported by CONICET (PIP 11220200102916CO),
+FONCyT-ANPCyT (PICT-2019-03660) and Secyt (UNC). G. A. G. was
+partially supported by CONICET (PIP 11220200100423CO), Secyt (UNLP)
+and FONCyT-ANPCyT (PICT-2018-00858). G. C. was partially supported
+by Projects BIRD179758/17, DOR2207212/22, and BIRD203834 of the Uni-
+versity of Padova. The results were obtained during visits of N. A. to the
+University of Padova, and of G. C. to the University of C´ordoba, partially
+supported by the bilateral agreement between these Universities and the
+INdAM-GNSAGA Visiting Professor program.
+2.
+Racks
+2.1. Racks. As in previous papers we use the language of racks; see [9] for
+more information. A rack is a pair (O, ⊲) where O is a non-empty set and
+⊲ : O×O → O is a self distributive operation such that ϕx := x⊲
+is bijective
+for any x ∈ O. A subset O′ ⊂ O is a subrack if O′ ⊲ O′ ⊂ O′. Let InnO
+be the group generated by the image of the map ϕ : O → SO. The main
+examples of racks considered in this paper are (unions of) conjugacy classes
+of a finite group with ⊲ being the conjugation. A rack (O, ⊲) is abelian if
+x ⊲ y = y for any x, y ∈ O. Also, a rack is indecomposable if it can not be
+presented as the disjoint union of two subracks and decomposable otherwise.
+The following observation will be useful, especially when dealing with
+orthogonal groups.
+Remark 2.1. Let G be a finite group, N ⊳ G, g ∈ G − N and ON
+g the orbit
+of g under the conjugation action of N. Then ON
+g is a subrack of OG
+g .
+This is a special case of [13, Remark 3.2] that can be verified directly.
+Notice that if N ≤ G is not normal, then ON
+g
+may fail to be a subrack
+of OG
+g . For instance, take G = S4, g = (123) and N = ⟨(12)(34)⟩; then
+ON
+g = {(123), (142)} is not closed under the rack operation.
+
+6
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+2.2. Racks of type C. The following notion was introduced in [3, Defini-
+tion 2.3] motivated by [18, Theorem 2.1].
+Definition 2.2. A rack X is of type C if there are a decomposable subrack
+Y = R � S ≤ X, r ∈ R and s ∈ S such that r ⊲ s ̸= s,
+R = OInn Y
+r
+,
+S = OInn Y
+s
+,
+min{|R|, |S|} > 2 or max{|R|, |S|} > 4.
+The group-theoretical reformulation (C) of the definition of type C is [3,
+Lemma 2.8]. We need a variation of [3, Lemma 2.8] in order to encompass
+the situation in Remark 2.1. The proof can be repeated verbatim: we recall
+it here for completeness.
+Lemma 2.3. Let G be a finite group, g ∈ G and N ⊳G. The orbit O = ON
+g
+is of type C if and only if there are H ≤ ⟨O⟩, r, s ∈ O ∩ H such that
+OH
+r ̸= OH
+s ;
+(2.1)
+rs ̸= sr;
+(2.2)
+H = ⟨OH
+r , OH
+s ⟩;
+(2.3)
+min{|OH
+r |, |OH
+s |} > 2
+or
+max{|OH
+r |, |OH
+s |} > 4.
+(2.4)
+Proof. Assume that r, s and H are as above and set R := OH
+r and S := OH
+s .
+If r′ = h ⊲ r ∈ OH
+r = R for some h ∈ H, then there exist x1, · · · , xk ∈ O
+such that h = x1 · · · xk. Hence
+r′ = x1 ⊲ (x2 ⊲ (· · · (xk ⊲ r)) ∈ O(⊲O(⊲ · · · O ⊲ O))) ⊂ O,
+so R ⊂ O∩H and similarly S ⊂ O∩H. By (2.1) the subset Y := R � S ⊂ O
+is a decomposable subrack, and r ⊲ s ̸= s is (2.2). In addition,
+OInn Y
+r
+= O⟨R,S⟩
+r
+= OH
+r = R,
+where the second equality follows from (2.3), and similarly, OInn Y
+s
+= S. The
+estimate on R and S is (2.4). Hence O is of type C.
+Conversely, let X = O and r, s, R, S, Y be as in Definition 2.2. Setting
+H := ⟨R, S⟩, we immediately have (2.3), H ≤ ⟨O⟩, R = OH
+r , S = OH
+s and
+r, s ∈ O ∩ H. Finally (2.2), (2.1) and (2.4) are straightforward.
+□
+Remark 2.4. Let G, N and O be as in Lemma 2.3.
+(a) If there exist r, s ∈ O satisfying (2.2), O⟨r,s⟩
+r
+̸= O⟨r,s⟩
+s
+and:
+min{|O⟨r,s⟩
+r
+|, |O⟨r,s⟩
+s
+|} > 2
+or
+max{|O⟨r,s⟩
+r
+|, |O⟨r,s⟩
+s
+|} > 4,
+then O is of type C by Lemma 2.3 applied to H := ⟨r, s⟩.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+7
+(b) If |g| is odd and r, s ∈ O, then for any H ≤ G containing r and s
+the estimate (2.4) follows from (2.2), since then |s| = |r| = |g| is odd,
+whence |OH
+r | ≥ |O⟨s⟩
+r | ≥ 3, and similarly for OH
+s . This generalizes [3,
+Lemma 2.7 (b)] to the situation of Remark 2.1.
+Example 2.5. Let n ≥ 5 be odd. Then the conjugacy class O of n-cycles
+in Sn is of type C. Indeed, O splits into two classes O′ and O′′ in An and
+|O′| = |O′′| = (n−1)!
+2
+> n elements. Therefore, if r ∈ O′, there exists s ∈ O′′
+such that s ̸∈ CAn(r) = ⟨r⟩ and the result follows from Remark 2.4.
+Example 2.6. The class O corresponding to the partition (12, 22) in A6 is
+of type C. Indeed, H := CA6(56) ≃ S4 and H ∩ O contains all involutions
+of the form (ab)(cd) for a, b, c, d /∈ {5, 6} and those of the form (ab)(56)
+for a, b /∈ {5, 6}. Therefore, |O′ ∩ H| = 12 and O′ contains all involutions
+in H. Now, the involutions in S4 are parted into two classes of size 6, and
+S4 contains non-commuting non-conjugate involutions. Hence, we can find
+r, s ∈ H ∩ O′ such that r ⊲ s ̸= s and OH
+r ̸= OH
+s , with |OH
+r | = |OH
+s | = 6.
+Finally, ⟨OH
+s , OH
+r ⟩ = H because S4 is generated by its involutions.
+We
+conclude by Lemma 2.3.
+Example 2.7. The class of 3-cycles in G = A3 or A4 is kthulhu because its
+intersection with any subgroup of G is either abelian or a conjugacy class.
+The class O of 3-cycles in An for n ≥ 5 and the class O′ labeled (32) in
+A6 are of type C. Indeed, O ∩ A4 splits into the classes O(123) and O(124).
+Since the representatives do not commute, O is of type C by Remark 2.4.
+Any non-inner automorphism of S6 interchanges O and O′ in A6, so O′ is of
+type C as well.
+Here is an easy but useful application of the previous Lemma.
+Lemma 2.8. Let G be a finite group, H ≤ G, x ∈ H. Assume that
+H is not abelian;
+(2.5)
+H = ⟨OH
+x ⟩;
+(2.6)
+there exists s ∈ OG
+x ∩ H : s /∈ OH
+x , |OH
+s | > 2.
+(2.7)
+Then OG
+x is of type C.
+Proof. There is r ∈ OH
+x such that rs ̸= sr; otherwise s ∈ Z(H) by (2.6),
+hence |OH
+s | = 1. Thus (2.2) holds and (2.1) and (2.3) are clear by construc-
+tion. Finally, |OH
+r | > 2 by (2.5), thus (2.4) holds.
+□
+Here is another way to detect racks of type C.
+
+8
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Lemma 2.9. Let G1 and G2 be finite groups, a1 ̸= b1 ∈ G1, a2, b2 ∈ G2.
+Set r = (a1, a2), s = (b1, b2) ∈ G := G1 × G2. Assume that
+a1b1 = b1a1,
+a2b2 ̸= b2a2;
+(2.8)
+OG1
+a1 = OG1
+b1 ,
+OG2
+a2 = OG2
+b2 ;
+(2.9)
+G2 = ⟨OG2
+a2 ⟩.
+(2.10)
+Then OG
+r = OG1
+a1 × OG2
+a2 is of type C.
+Proof. Let H = ⟨{a1} × OG2
+a2 , {b1} × OG2
+a2 ⟩ ∋ r, s; (2.2) is evident. We claim
+that OH
+r = {a1} × OG2
+a2 . Indeed, ⊆ follows from (2.8), and ⊇ from (2.10):
+y ∈ G2 =⇒ ∃x1, . . . , xt ∈ OG2
+a2 : y = x1 . . . xt
+=⇒ (a1, y ⊲ a2) = (a1, x1) ⊲ ((a1, x2) ⊲ . . . (a1, a2)) ∈ OH
+r .
+Similarly, OH
+s
+= {b1} × OG2
+b2 .
+Hence (2.1) and (2.3) follow.
+Finally, if
+|OH
+r | = |OH
+s | = |OG2
+a2 | ≤ 2, then a2 and b2 commute, contradicting (2.8).
+□
+Remark 2.10. Let G1 and G2 be finite groups, a1 ̸= b1 ∈ G1. The hypotheses
+of Lemma 2.9 on G2 hold when G2/Z(G2) is a non-abelian simple group and
+a2 ∈ G2 is not central. Namely, ⟨OG2
+a2 ⟩⊳G2, hence it is all of G2 giving (2.10).
+Furthermore there is b2 ∈ OG2
+a2 that does not commute with a2, because G2
+is not abelian, as needed in (2.8).
+2.3. Racks of type D. A rack X is of type D if it has a decomposable
+subrack Y = R � S with elements r ∈ R, s ∈ S such that r ⊲(s ⊲(r ⊲s)) ̸= s
+[7, Definition 3.5]. If O is a conjugacy class in a finite group G, then the
+rack O is of type D if and only if (D) holds, see [7].
+3. Algebraic groups
+Let G be a connected reductive algebraic group defined over k = Fq and let
+F : G → G be a Frobenius map, that is a Fq-split Steinberg endomorphism
+[22, Chapter 21]. Thus there exists an F-stable torus T such that F(t) = tq
+for t ∈ T, and GF = G(Fq) is the finite group of Fq-points. We make more
+precise assumptions on G in each Subsection below. The main objectives of
+the paper are encompassed in the following situations:
+⋄ The group G is either SLn(k) or Sp2n(k) (n ≥ 2) and
+G := GF /Z(GF) = [GF, GF ]/Z([GF , GF ])
+is a finite simple group.
+⋄ The group G is either SO2n+1(k) (n ≥ 2, p is odd) or SO2n(k) (n ≥ 4)
+and G := [GF , GF ]/Z(GF ) is a finite simple group.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+9
+In both situations we say that G is a (classical) Chevalley group. We
+allow SO5(k) whose simply connected cover is Sp6(k), and SL2(3), SL2(4),
+SL2(5), SL2(9) for flexibility in some recursive arguments.
+Let Φ be the root system of G and fix a subset ∆ of simple roots. Let Q,
+respectively Λ, be the root, respectively weight, lattice. Then Λ = ⊕i∈IZωi;
+here θ is the rank of G, I = Iθ and (ωi)i∈I are the fundamental weights. Let
+W be the Weyl group of Φ and let sα ∈ W be the reflection corresponding
+to α ∈ Φ. Also, α∨
+i : k× → T, i ∈ I, are the simple coroots. Then
+ωi(α∨
+j (ξ)) = ξδij,
+ξ ∈ k×, i, j ∈ I.
+If α ∈ Φ, then there is a monomorphism xα : k → G of abelian groups; let
+Uα = Im xα (a root subgroup) and let U, respectively U−, be the subgroup
+of G generated by the Uα’s with α ∈ ∆, respectively −α ∈ ∆.
+3.1. The classical groups. In this section we fix notation for the classical
+groups we will deal with. For m ≥ 1 we set Jm =
+�
+1
+...
+1
+�
+. We denote
+by Frq the Frobenius map GLm(k) → GLm(k) given by (aij) �→ (aq
+ij), and
+similarly the restriction to any suitable subgroup.
+We will often use the automorphism φ: GLm(k) → GLm(k) given by:
+φ(A) := Jm tA−1Jm.
+(3.1)
+The symplectic group Sp2n(k). The symplectic group Sp2n(k) is the
+subgroup of GL2n(k) leaving invariant the bilinear form
+�
+0
+Jn
+−Jn 0
+�
+. Thus
+Sp2n(k) consists of the invertible matrices
+� A B
+C D
+�
+such that
+tCJnA = tAJnC,
+tBJnD = tDJnB,
+−tCJnB + tAJnD = Jn.
+(3.2)
+In this case, F = Frq and G = Sp2n(q)/Z(Sp2n(q)) = PSp2n(q).
+The orthogonal group SO2n+1(k). Let p be odd. The orthogonal group
+SO2n+1(k) is the subgroup of SL2n+1(k) leaving invariant the bilinear form
+J2n+1. Thus SO2n+1(k) consists of the invertible matrices
+X =
+� A e B
+f k g
+C h D
+�
+,
+A, B, C, D ∈ kn×n,
+e, tf, tg, h ∈ kn×1,
+k ∈ k
+such that det X = 1 and
+tCJnA + tff + tAJnC = 0,
+tCJne + tfk + tAJnh = 0,
+tCJnB + tfg + tAJnD = Jn,
+thJne + k2 + teJnh = 1,
+thJnB + kg + teJnD = 0,
+tDJnB + tgg + tBJnD = 0.
+(3.3)
+In this case F = Frq and G = [SO2n+1(q), SO2n+1(q)] = PΩ2n+1(q).
+
+10
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+The orthogonal group SO2n(k). Let n ≥ 4.
+The orthogonal group
+SO2n(k) is the subgroup of matrices in SL2n(k) preserving the quadratic
+form �n
+i=1 xix2n−i+1. If p is odd, such elements automatically preserve the
+bilinear form with associated matrix J2n. Thus SO2n(k) consists of those
+matrices
+� A B
+C D
+�
+∈ SL2n(k) with A, B, C, D ∈ kn×n, such that
+tCJnA + tAJnC = 0, tCJnB + tAJnD = Jn,
+tDJnB + tBJnD = 0.
+(3.4)
+If p = 2, then SO2n(k) consists of those matrices
+� A B
+C D
+�
+∈ SL2n(k) satisfying
+(3.4) and such that the diagonal terms in tCJnA and tBJnD are 0. In this
+case F = Frq and
+G = [SO2n(q), SO2n(q)]/Z([SO2n(q), SO2n(q)]) = PΩ+
+2n(q).
+3.2. On normalizers. In Subsection 5.3 we shall need the finite unitary
+groups SUn(q) and GUn(q), and the following folklore fact. We consider:
+◦ G = SLn(k), q0 = pm0 with m0|m so that q is a power of q0 ;
+◦ F0 : GLn(k) → GLn(k) is defined either by F0(A) := Frq0(A) or by
+F0(A) := Frq1/2
+0 (φ(A)), for A ∈ GLn(k), φ as in (3.1), the latter occurring
+only for m0 even, in which case, we denote as usual SUn(q1/2
+0
+) = GF0 and
+GUn(q1/2
+0
+) = GLn(k)F0.
+Proposition 3.1. NGLn(q)(GF0) = Z(GLn(q))GLn(k)F0.
+Proof. We prove that NGLn(q)(GF0) ≤ Z(GLn(q))GLn(k)F0, the other in-
+clusion being immediate. Let g ∈ NGLn(q)(GF0). For any y ∈ GF0 there
+holds F0(gyg−1) = gyg−1, that is z := g−1F0(g) ∈ CGLn(q)(GF0). Now, GF0
+contains regular unipotent elements in U and U−, so it follows from [26,
+Lemma 5.3] that CGLn(q)(GF0) = Z(GLn(q)). In addition, F0 restricts to a
+Steinberg endomorphism on the connected group Z(GLn(k)) ≃ k×, hence
+Lang-Steinberg theorem [22, Theorem 21.7] is in force and there exists ζ id ∈
+Z(GLn(k)) such that ζ−1F0(ζ) id = z. Hence, ζ−1g ∈ GLn(k)F0 ≤ GLn(q)
+and so ζ id ∈ Z(GLn(k)) ∩ GLn(q) = Z(GLn(q)). The claim follows.
+□
+3.3. The subgroup K. We introduce a subgroup K of G that will be useful
+in Sections 4 and 6. In this Subsection G is one of the groups
+Sp2n(k), n ≥ 2;
+SO2n(k), n ≥ 3,
+or
+SO2n+1(k), n ≥ 3,
+where p ̸= 2 when G = SO2n+1(k). We set n′ = 2n if G = SO2n(k) or
+G = Sp2n(k) and n′ = 2n + 1 if G = SO2n+1(k), so that G ≤ GLn′(k).
+Recall φ from (3.1). Then K is the image of the injective group morphism
+j : GLn(q) → GF,
+A �→
+
+
+
+�
+A
+0
+0 φ(A)
+�
+,
+if G = SO2n(k), or Sp2n(k),
+� A 0
+0
+0 1
+0
+0 0 φ(A)
+�
+,
+if G = SO2n+1(k).
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+11
+3.4. Cuspidal classes in the Weyl group. Let S = {sα : α ∈ ∆}, so
+that (W, S) is a Coxeter group. Given J ⊂ ∆, we set
+◦ WJ = ⟨sα : α ∈ J⟩;
+◦ PJ = the standard parabolic subgroup of G determined by J;
+◦ LJ = the standard (reductive) Levi subgroup of PJ.
+Definition 3.2. [16, 3.1.1] A conjugacy class C in W is called cuspidal if
+C ∩ WJ = ∅ for all proper subsets J of S; an element is cuspidal if its
+conjugacy class is so.
+A decomposition of w ∈ W is a family Γ = (γj)j∈Il in Φ, such that
+w = sγ1 · · · sγl,
+(3.5)
+where sγj is the corresponding reflection and l is minimal (with this prop-
+erty). Then l is denoted by ℓa(w) and is called the absolute length of w. By
+a result of Kostant, see [24], Γ is then a linearly independent family and
+ℓa(w) = rk(id −w)
+(3.6)
+in the natural representation of W. By [16, Exercise 3.16], we have
+w is cuspidal
+⇐⇒ ℓa(w) = rk G.
+(3.7)
+Notice that ℓa(w) = rk G means that w has no fixed points.
+Given a decomposition Γ of w, we set
+ΨΓ = Φ ∩ (Zγ1 ⊕ · · · ⊕ Zγl) ,
+(3.8)
+GΓ = ⟨T, Uβ : β ∈ ΨΓ⟩.
+(3.9)
+Clearly, ΨΓ is a root subsystem of Φ and GΓ is a connected reductive sub-
+group of G. If Γ and Γ ′ are different decompositions of the same w, then the
+subsystems ΨΓ and ΨΓ ′, and the subgroups GΓ and GΓ ′, might be different.
+Remark 3.3. If w ∈ W is cuspidal, then GΓ is semisimple for any decompo-
+sition Γ of w, by (3.7).
+Remark 3.4. If w ∈ WJ for some J ⊂ S, then there is a decomposition Γ
+such that GΓ ≤ LJ.
+Indeed, any decomposition of w in WJ is necessarily a decomposition in
+W, by (3.6). For, w acts trivially in (RJ)⊥, hence rk(id −w) = rk(id −w)|RJ.
+3.5. F-stable tori. Here we assume that G is connected reductive and F
+is a Frobenius map. By [22, Proposition 25.1], there is a bijection from the
+set of GF-conjugacy classes of F-stable maximal tori to the set of conjugacy
+classes in W, described as follows. Let T′ be an F-stable torus in G. Then
+T′ = gTg−1 for some g ∈ G such that g−1F(g) ∈ N(T). Let
+w = class of g−1F(g) ∈ N(T)/T ≃ W.
+(3.10)
+
+12
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+The assignment T′ �→ w gives rise to the mentioned bijection. We set
+Tw := gTg−1.
+(3.11)
+Remark 3.5. Let T′ be an F-stable maximal torus in G such that T′ �→ w
+in the correspondence above and let G′ be the derived group of G. Then
+T′ ∩ G′ is an F-stable maximal torus in G′ and T′ ∩ G′ �→ w.
+Indeed, T ∩ G′ is a split torus of G′.
+The element g ∈ G such that
+T′ = gTg−1 and g−1F(g) ∈ NG(T) is a representative of w can be written
+as g = g′z where g′ ∈ G′ and z is central. Then T′ ∩ G′ = g′(T ∩ G′)(g′)−1
+and (g′)−1F(g′) ∈ NG′(T ∩ G′) is a representative of w.
+Definition 3.6. An F-stable maximal torus is cuspidal if the corresponding
+conjugacy class in W as above is cuspidal.
+Example 3.7. Let w ∈ W be a Coxeter element, i.e. a product
+w = s1 . . . sθ,
+(3.12)
+where (si)i∈Iθ is a numeration of S; this provides a decomposition Γ of w.
+Then the class of w is cuspidal and GΓ = G. If W = Sn, the conjugacy class
+of w is the only cuspidal class in W, [16, §3.1.2].
+Definition 3.8. A Coxeter torus is an F-stable maximal torus that corre-
+sponds to the conjugacy class containing a Coxeter element.
+By abuse of terminology the intersection of a Coxeter torus of G with GF
+will be called a Coxeter torus of GF.
+3.6. Semisimple classes. Here G is connected and reductive, unless oth-
+erwise stated, F is a Frobenius map and T is an F-stable torus such that
+F(t) = tq for t ∈ T.
+Let x ∈ G = GF/Z(GF ) be semisimple non-trivial and pick x ∈ GF a
+representative of x, thus x is semisimple but not central. Let O = OG
+x and
+O = OGF
+x
+; there is an epimorphism of racks O ։ O.
+Let y ∈ GF be semisimple. By [22, Proposition 26.6], there exists an
+F-stable maximal torus T′ containing y; however, not all F-stable maximal
+tori intersecting OGF
+y
+are necessarily GF-conjugated to T′. Consequently we
+assign to OGF
+y
+the set SOGF
+y
+of all conjugacy classes C in W corresponding
+to F-stable maximal tori T′ that intersect OGF
+y
+.
+Remark 3.9. Assume that G is simple and that we are not in the cases
+excluded in [22, Theorem 24.17].
+If C ∈ SOGF
+y
+, then O[GF ,GF ]
+y
+intersects
+an F-stable maximal torus T′′ in GF corresponding to an element in C.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+13
+Indeed, let g ∈ GF be such that g−1F(g) ∈ NG(T) represents an element
+in C, and let T′ = gTg−1. Then there exists l ∈ GF such that l ⊲ y ∈ T′.
+Now, GF = TF[GF , GF], [22, Corollary 24.2, Proposition 24.15, Proposition
+24.21], so l decomposes as l = t1l1 with t1 ∈ TF and l1 ∈ [GF , GF], and
+l1 ⊲ y ∈ t−1
+1 gTg−1t1 = T′′, where g−1t1F(t−1
+1 )F(g) = g−1F(g) represents an
+element in C.
+For our aim, it is convenient to introduce the following notion.
+Definition 3.10. A semisimple conjugacy class OGF
+y
+in GF is called cuspidal
+if the set SOGF
+y
+consists of cuspidal conjugacy classes in W. In other words,
+all F-stable maximal tori intersecting OGF
+y
+are cuspidal.
+Also, OGF
+y
+is called a Coxeter class if it only intersects Coxeter tori.
+Necessarily, OGF
+y
+is then cuspidal.
+Remark 3.11. Since Z(GF) = Z(G)F is contained in every torus of GF,
+the class OGF
+y
+is cuspidal, respectively Coxeter, if and only if its projection
+O′ in GF/Z(GF ) intersects only cuspidal tori, respectively Coxeter tori in
+GF/Z(GF ). We will thus call also O′ cuspidal, respectively Coxeter.
+In
+particular, if G is simply-connected and O is cuspidal, O will be called
+cuspidal.
+If G is not simply-connected, Remark 3.9 guarantees that OGF
+y
+is cuspidal,
+respectively Coxeter, if an only if O[GF ,GF ]
+y
+intersects only cuspidal tori,
+respectively Coxeter tori in GF, and we will call also O[GF ,GF ]
+y
+cuspidal,
+respectively Coxeter.
+Proposition 3.12. Assume G is simply-connected. If T′ is a maximal F-
+stable torus that intersects O, C is the conjugacy class in W corresponding to
+T′ and Γ is a decomposition of w ∈ C, then O intersects GF
+Γ . In particular,
+the following are equivalent:
+(a) O is not cuspidal.
+(b) O intersects a proper standard Levi subgroup L.
+Proof. Since F is a Frobenius automorphism, GΓ is F-stable. Pick a rep-
+resentative ˙w of w; by definition, it belongs to GΓ . By the Lang-Steinberg
+Theorem there is h ∈ GΓ such that h−1F(h) = ˙w. The tori T′ and hTh−1
+are GF-conjugate since they both map to w, cf. (3.10). That is, there exists
+y ∈ GF such that y ⊲ x ∈ hTh−1 ≤ GΓ, hence y decomposes as y = y′ for
+some y ⊲ x ∈ GF
+Γ ∩ O.
+
+14
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+If O is not cuspidal, then pick C non-cuspidal and apply Remark 3.4.
+Conversely, if y ∈ O ∩ L, then there is an F-stable maximal torus T′ of L
+that contains y. Hence T′ = uTu−1 for some u ∈ L such that σ := u−1F(u) ∈
+NL(T) ≤ NG(T); so that the class of σ belongs to the Weyl group of L.
+□
+Recall that a semisimple element y is regular if its centraliser CG(y) con-
+sists of semisimple elements, or equivalently, if the irreducible component
+CG(y)◦ of CG(y) containing the identity is a torus, [27, II.11]. This occurs
+if and only if y lives in a unique maximal torus. If our x ∈ GF is regular,
+then CG(x)◦ is the unique F-stable maximal torus containing x.
+Proposition 3.13. If x is a cuspidal element, then it is regular.
+We thank Gunter Malle for suggesting us the following proof.
+Proof. Let T0 be an F-stable maximal torus of G containing x, let g ∈ G
+be such that T0 = gTg−1 and let w be the corresponding Weyl group el-
+ement, i.e., g−1F(g) ∈ wT as in (3.10). The F-stable maximal tori in G
+containing x are also the F-stable maximal tori in the connected reductive
+group C = CG(x)◦.
+Every F-stable maximal torus in C is of the form
+cT0c−1 for some c ∈ C such that c−1F(c) ∈ NC(T0) ≤ gNG(T)g−1. Let
+WC = NC(T0)/T0 be the Weyl group of C.
+We claim that WC is triv-
+ial.
+Assume for a contradiction that WC is non-trivial.
+Let s be a re-
+flection in WC and let c ∈ C be such that c−1F(c) = ˙s, a representative
+of s in NC(T0). Then, ˙s′ := g−1 ˙sg would represent a reflection s′ in W
+and cT0c−1 = cgTg−1c−1 is an F-stable maximal torus of G, containing x
+and corresponding to g−1c−1F(c)F(g) = (g−1c−1F(c)g)(g−1F(g)) ∈ s′wT.
+Therefore, s′w is cuspidal by hypothesis on x. However, the characteristic
+polynomial of a cuspidal element is a product of cyclotomic polynomials
+different from (X − 1), therefore its value at 0 is 1, see (3.6) and (3.7). On
+the other hand, det(s′w) = − det(w). Hence, s′w and w can not be both
+cuspidal elements in W, contradicting our assumption on x. Therefore WC
+has no reflections and C = T0 is the unique maximal torus containing x.
+□
+The following well-known result is instrumental to apply Lemma 2.9.
+Lemma 3.14. Assume G is simply-connected.
+(a) xq ∈ O. (b) If xq = x, then O ∩ TF ̸= ∅.
+Proof. First, OG
+x is F-stable: if x = hyh−1 for some h ∈ G, then
+F(y) = F(h)−1xF(h) ∈ OG
+x .
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+15
+Since x is semisimple, there are t ∈ T and g ∈ G such that x = gtg−1. Thus
+tq = F(t) ∈ OG
+x and consequently xq ∈ OG
+x ∩ GF = O; here the last equality
+holds because CG(x) is connected, G being simply connected, cf. [20, §2.11,
+§8.5]. Finally, if xq = x, then tq = t ∈ TF ∩OG
+x ⊂ O by the same reason.
+□
+4. Split conjugacy classes
+We keep the notation from §3.6, namely G is simple and simply connected,
+but not of type A. Also F is a Frobenius map; T is an F-stable torus such
+that F(t) = tq for t ∈ T; e ̸= x ∈ G = GF/Z(GF ) is semisimple; x ∈ GF a
+representative of x; O = OG
+x and O = OGF
+x
+. Thus there is an epimorphism
+of racks O ։ O.
+We assume additionally that O ∩ TF ̸= ∅. Without loss of generality, we
+suppose that x ∈ TF, i.e., x is split. Adapting the proof of [3, Lemma 3.9]
+for type A, but with more work, we deal with such classes.
+We will need to consider separately the following particular situation:
+G is of type Bθ, q is odd, x satisfies sαj(x) =
+�
+x
+if j < θ,
+α∨
+θ (−1)x
+if j = θ.
+(4.1)
+Here θ ≥ 2 (as B2 = C2). When this is the case, then x has the form
+x =
+� �
+i∈Iθ−1
+α∨
+i ((−1)i)
+�
+α∨
+θ (η),
+where η2 = (−1)θ.
+(4.2)
+Notice that if θ is odd, then such a x belongs to GF iff q ≡ 1 mod 4.
+Here is the main result of this Section:
+Theorem 4.1. Assume that q > 2; G is not of type Aθ; q /∈ {3, 5, 7} if we
+are in (4.1); and OGF
+x
+intersects the split torus TF. Then O collapses.
+When q = 2, TF is trivial and the class of x could not intersect it.
+4.1. Proof of Theorem 4.1. This follows from Lemmata 4.2 and 4.3.
+Lemma 4.2. Assume that q > 2 and that we are not in the situation (4.1).
+Then O is of type C.
+Proof. Recall that x ∈ TF. We will rely on the proof of [5, Lemmata 4.1,
+4.2]. It is shown there that, for any simple root α such that sα(x) ̸= x, the
+subrack Y = xUF
+α
+� sα(x)UF
+α of OGF
+x
+is of type C. We claim that we can
+choose α such that the restriction of the projection π : Y → O is injective. If
+G is of type E8, F4, G2, then Z(G) is trivial and G = GF. Let u ̸= v ∈ Y such
+that π(u) = π(v), i.e. there is z ∈ Z(GF ), z ̸= 1, such u = zv. Hence either
+u = xxα(a) and v = sα(x)xα(a), or vice versa. In any case, x
+⋆= zsα(x).
+
+16
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Table 3. Center of some G, q odd; ζ ∈ F×
+q has order 4
+type
+q
+Z(G)
+Bθ
+⟨α∨
+θ (−1)⟩
+Cθ, θ > 2
+� �
+i odd
+α∨
+i (−1)
+�
+Dθ,
+q ≡ 1 mod 4
+�
+�
+i odd,i≤θ−2
+α∨
+i (−1)α∨
+θ−1(ζ)α∨
+θ (ζ3)
+�
+θ ∈ 2Z + 1
+q ≡ 3 mod 4
+�
+α∨
+θ−1(−1)α∨
+θ (−1)
+�
+Dθ, θ ∈ 2Z
+� �
+i odd
+α∨
+i (−1), α∨
+θ−1(−1)α∨
+θ (−1)
+�
+E7
+⟨α∨
+2 (−1)α∨
+5 (−1)α∨
+7 (−1)⟩
+Applying sα, we get z2 = 1. Thus G is not of type E6 (here Z(G) ≃ Z/3);
+and q should be odd. By ⋆, we have
+ωi(x) = ωi(zsα(x)) = ωi(z)sα(ωi)(x),
+i ∈ Iθ.
+(4.3)
+Say α = αj, j ∈ Iθ. Then sα(ωi) = ωi when i ̸= j, hence ωi(z) = 1. Now
+such z exists only in the situation (4.1), see the shape of Z(G) in Table
+3.
+□
+Lemma 4.3. If we are in situation (4.1) with q = 9, then O is of type C.
+If we are in situation (4.1) with q > 9, then O is of type D.
+Proof. We deal first with θ = 2; now B2 = C2 and G = Sp4(k).
+Let
+K ≃ GL2(q) be the subgroup of GF which is the image of j as in §3.3. Thus
+we have a monomorphism of groups GL2(q)/{±1} → G = PSp4(q).
+Let y =
+� 1 0
+0 −1
+�
+∈ GL2(q); by our assumption on x, we know that either
+x
+⋆= j(y) or x = −j(y).
+Let ̟ : GL2(q) → PGL2(q) be the canonical
+projection and y = ̟(y). Then we have a surjective map of racks
+O ∩ K/{±1} → OPGL2(q)
+y
+.
+Therefore it is enough to prove that OPGL2(q)
+y
+is of type C if q = 9 and of
+type D for q > 9.
+Let q = 9. Then PGL2(9) ≃ A6, and through this isomorphism the class
+OPGL2(q)
+y
+corresponds to the class labeled by (12, 22), which is of type C by
+Example 2.6.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+17
+Let now q > 9.
+If q ≡ 3 mod 4, then [6, Corollary 5.4 (b)] applies1.
+Assume then that q ≡ 1 mod 4. Let ζ ∈ F×
+q be a primitive 4-th root of 1
+and let u ∈ PGL2(q) be the class of
+�
+ζ
+0
+0 −ζ
+�
+. Then
+OPGL2(q)
+y
+= OPGL2(q)
+u
+= OPSL2(q)
+u
+which is of type D by [6, Corollary 5.4 (a)].
+Assume next that θ > 2. Here
+G = PΩ2θ+1(q) = GF/Z(GF ) ≃ [SO2θ+1(q), SO2θ+1(q)].
+We identify PΩ5(q) with a subgroup of PΩ2θ+1(q) via the inclusion
+SO5(q) ֒→ SO2θ+1(q),
+
+
+
+A
+e
+B
+f
+k
+g
+C
+h
+D
+
+
+ �→
+
+
+
+
+
+
+
+
+A
+0
+e
+0
+B
+0
+idθ−2
+0
+0
+0
+f
+0
+k
+0
+g
+0
+0
+0
+idθ−2
+0
+C
+0
+h
+0
+D
+
+
+
+
+
+
+
+
+,
+k ∈ Fq, A, B, C, D ∈ F2×2
+q
+, etc. Fix tθ ∈ T of the shape (4.2) and analogously
+t2 of the shape (4.2) but for type B2. If π : GF → G is the projection, then
+π(tθ) = diag (− idθ, 1, − idθ) = π(t2)γ,
+where
+γ = diag (id2, − idθ−2, 1, − idθ−2, id2) .
+Here diag refers to a diagonal of blocks. Then
+O = OG
+π(tθ) ≥ OPΩ5(q)×⟨γ⟩
+π(tθ)
+≃ OPΩ5(q)
+π(t2)
+which is of type D by the preceding argument. Hence O is of type D.
+□
+Lemma 4.4. If we are in situation (4.1) with n = 2 and q = 3, then O is
+austere, hence kthulhu.
+Proof. Indeed, PSp4(3) ≃ PSU4(2) and the semisimple class we are dealing
+with in the former group corresponds to the unipotent class of type (2, 12)
+in the latter one, which is austere by [4, Lemma 5.2].
+□
+Remark 4.5. Assume that n = 2. If we are in the situation (4.1) with q = 5,
+then calculations with GAP show that O is austere, hence kthulhu. The
+evidence obtained by performing different computations seems to indicate
+that in the case q = 7, the class is also kthulhu.
+1Notice that Corollary 5.4 (b) in loc. cit. refers implicitly to the class of involutions in
+PGL2(q) not in PSL2(q), as is transparent from the proof.
+
+18
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+4.2. Split classes in orthogonal groups. Theorem 4.1 was proved by as-
+suming that G is simply connected. For recursive arguments on the orthog-
+onal groups we need an analogous statement for the orbits of split elements
+in GF = SOn′(q) for the action of [GF, GF ] for n′ = 2n or 2n+1. Let T ≤ G
+be the subgroup of diagonal matrices
+diag(t1, . . . , tn, t−1
+n , . . . , t−1
+1 ),
+if n′ = 2n,
+diag(t1, . . . , tn, 1, t−1
+n , . . . , t−1
+1 ),
+if n′ = 2n + 1.
+Recall Remark 2.1.
+Lemma 4.6. If y ∈ TF − Z(SOn′(q)), then O[SOn′(q),SOn′(q)]
+y
+collapses,
+except in the situation (4.1), i.e., when n′ = 2n + 1 and ti = −1 for i ∈ In.
+Proof. There is always a simple root α so that the proof of [5, Lemma 4.2]
+carries over. If ti ̸= ti+1 for some i < n, then take α = αi = εi − εi+1; if,
+instead, ti = ti+1 for all i < n, then our assumptions imply tn ̸= t−1
+n
+and we
+take α = αn, i.e., εn when n′ = 2n + 1 and εn−1 + εn when n′ = 2n. The
+argument works also in types B2 and D3.
+□
+5. The special linear groups
+In this Section G = SLn(k), that is, we deal with semisimple classes in
+G = PSLn(q). As in §3.6, e ̸= x ∈ G is semisimple, x ∈ GF − Z(GF) is
+a representative of x, O = OG
+x and O = OGF
+x
+. There is an epimorphism of
+racks O ։ O.
+For inductive arguments, we will also consider classes of elements in
+GLn(q). As observed in [1, Remark 4.1], for any semisimple element y ∈
+GLn(q), we have OGLn(q)
+y
+= OSLn(q)
+y
+.
+Definition 5.1. We say that A ∈ GLn(q) is irreducible if its characteristic
+polynomial pA is irreducible; necessarily A is regular semisimple.
+From our previous work, we know:
+Remark 5.2. (i) [3, Theorem 1.1] If n = 2, and q ̸∈ {2, 3, 4, 5, 9}, then any
+O not listed in Table 1 collapses.
+(ii) [3, Props. 5.4, 5.5] If n = 3 and x is irreducible, then O is kthulhu.
+(iii) [3, Theorem 1.1] If n ≥ 3 and x is not irreducible, then O collapses.
+Remark 5.3. Let n = 2. We record information on the semisimple classes
+with q ∈ {2, 3, 4, 5, 9} for recursive arguments. Recall that PSL2(q) has two
+conjugacy classes of maximal tori: the split one, of order q − 1/(2, q − 1)
+and the Coxeter torus, of order q +1/(2, q +1), that contains the irreducible
+elements.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+19
+◦ If q = 2, then PSL2(2) ≃ S3 and the semisimple elements are the 3-
+cycles that form an abelian rack; if q = 3, then PSL2(3) ≃ A4 and the
+semisimple elements have order 2 and form an abelian rack.
+◦ If q = 4, then PSL2(4) ≃ A5. The irreducible elements have order 5 and
+form two conjugacy classes that are sober by [14, Remark 3.2 (b) and (c)].
+The split semisimple elements form the conjugacy class of 3-cycles which
+is of type C by Example 2.7.
+◦ If q = 5, then PSL2(5) ≃ A5. The irreducible elements form the con-
+jugacy class of 3-cycles which is of type C by Example 2.7.
+The split
+semisimple elements are the involutions in the class (1, 22) which is sober
+because its intersection with any subgroup of A5 is either trivial, abelian
+or indecomposable.
+◦ If q = 9, then PSL2(9) ≃ A6. The irreducible elements have order 5 and
+form two conjugacy classes that are sober by [14, Remark 3.2 (b) and
+(c)]. The split semisimple elements are the involutions in the class (12, 22)
+which is of type C by Example 2.6.
+Our main result in this Section is:
+Theorem 5.4. Let O ̸= {e} be a semisimple conjugacy class in PSLn(q).
+Then any O not listed in Table 1 collapses.
+By Remark 5.2, we will consider conjugacy classes of irreducible elements
+assuming n > 3. We will see that if n is prime, then such classes are kthulhu
+by Proposition 5.15, otherwise, they are of type C by Proposition 5.9.
+We start by a classical result whose proof we include for completeness.
+Lemma 5.5. Let n ≥ 2 and ǫ = ±1.
+If P(X) = Xn + ǫ ∈ Fp[X] is
+irreducible over Fq, then n = 2, ǫ = 1 and q ≡ 3 mod 4.
+Proof. First, ǫ = 1 and q is odd, otherwise P(1) = 0; and n = 2m is even,
+otherwise P(−1) = 0. Also, q ≡ 3 mod 4, otherwise −1 = ξ2 for some
+ξ ∈ Fq and P(X) = (Xm +ξ)(Xm −ξ) would be reducible. Let now n = 2ha
+where a is odd, and let Φd(X) be the d-th cyclotomic polynomial. We have
+the factorization over Z, hence over Fp,
+(Xn − 1)P(X) = X2n − 1 =
+�
+d|2n
+Φd(X) =⇒ P(X) =
+�
+d|2n, d∤n
+Φd(X)
+Thus Φ2h+1|P(X), hence they are equal and n = 2h. Finally, if X2h + 1 is
+irreducible over Fq for q ≡ 3 mod 4, then h = 1 by [23, Theorem 1].
+□
+
+20
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+5.1. Coxeter tori. We assume in the rest of this Section that n > 3 and
+that x is irreducible.
+In this case W = Sn and by Proposition 3.12 and
+Example 3.7 every irreducible class intersects every Coxeter torus. We fix
+the n-cycle
+w = (1, 2, . . . , n).
+By technical reasons, we fix a Coxeter torus Tw in GLn(k); then Tw∩SLn(k)
+is a Coxeter torus in SLn(k) by Remark 3.5. By [22, Example 25.4], we have
+|TF
+w| = qn − 1 = (n)q(q − 1),
+|TF
+w ∩ SLn(q)| = (n)q.
+(5.1)
+The group TF
+w is isomorphic to F×
+qn, hence it is cyclic; further, any cyclic
+subgroup of GLn(q) of order qn − 1 is conjugated to TF
+w [19, Example 1.13].
+Remark 5.6. ([21, Satz II.7.3] and [25, Theorem 2.3.5 and below]). We have
+NGLn(q)(Tw) = NGLn(q)(TF
+w) ≃ TF
+w ⋊ CW (w),
+NSLn(q)(TF
+w ∩ SLn(q)) = NGLn(q)(TF
+w) ∩ SLn(q),
+with CW(w) ≃ Z/n. Let σ be a generator of CW (w) identified as a subgroup
+of NGLn(q)(TF
+w); σ can be chosen so that σ ⊲ y = yq for any y ∈ TF
+w.
+Lemma 5.7. Let y ∈ TF
+w be irreducible in GLn(q). Then
+OGLn(q)
+y
+∩ TF
+w = O
+NGLn(q)(TF
+w)
+y
+= ⟨σ⟩ ⊲ y = {y, yq, . . . , yqj, . . . , yqn−1},
+(5.2)
+O
+NGLn(q)(TF
+w)
+y
+= OSLn(q)
+y
+∩ TF
+w = O
+NSLn(q)(TF
+w)
+y
+(5.3)
+Proof. If z ∈ TF
+w ∩ OGLn(q)
+y
+, then there is g ∈ GLn(q) such that gyg−1 = z,
+so gCGLn(q)(y)g−1 = CGLn(q)(z), that is, g ∈ NGLn(q)(Tw) since clearly z is
+also irreducible. This and Remark 5.6 imply (5.2). Since OSLn(q)
+y
+= OGLn(q)
+y
+,
+the centraliser argument as above gives (5.3).
+□
+We investigate when two elements in an irreducible class have the same
+image through the natural projection π: SLn(q) → PSLn(q). Recall that
+Gn(Fq) is the group of n-th roots of unity in Fq.
+Lemma 5.8. (Just for this Lemma, n ≥ 3). Let y, z ∈ O such that π(y) =
+π(z), i.e., y = λz for some 1 ̸= λ ∈ Gn(Fq). Then
+(i) There exists j ∈ In−1 such that λz = zqj.
+(ii) Let j ∈ In−1 be minimal satisfying zqj = λz for some 1 ̸= λ ∈ Gn(Fq).
+Then j|n and λ is a primitive n
+j -th root of 1.
+(iii) Let j ∈ In−1 be minimal satisfying zqj = λz for some 1 ̸= λ ∈ Gn(Fq)
+and let a := n
+j . Then the characteristic polynomial pz ∈ Fq[Xa]. This
+observation rectifies [3, Remark 3.1 (d)].
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+21
+(iv) Let j ∈ In be minimal satisfying π
+�
+zqj�
+= π(z). Then j ̸= 1.
+Proof. (i): The elements y and z lie in the same (unique) maximal torus,
+so y = zqj for some j ∈ I1,n by Lemma 5.7. Therefore, λz = zqj and λ ̸= 1
+implies j < n.
+(ii): If n = aj + b, with a ≥ 1 and 0 ≤ b < j, then
+z = zqn = zqaj+b = (zqaj)qb = (λaz)qb = λazqb
+that is, zqb = λ−ax. Hence b = 0 by minimality and λa = 1. Now, if λc = 1
+with c ∈ N, then zqcj = λcz = z, hence c ≥ a = n
+j .
+(iii): By assumption pz = pzqj = pλz. If pz(X) = Xn+cn−1Xn−1+· · ·+c0,
+then pλz(X) = Xn + λcn−1Xn−1 + · · · + λnc0. Thus pz(X) = pλz(X) if and
+only if ch = 0 for all h ̸∈ n
+j Z.
+(iv): If j = 1 then pz would be Xn+(−1)n by (iii). By Lemma 5.5, n = 2,
+a contradiction.
+□
+5.2. Irreducible elements of SLn(q), n not a prime. In this Subsection
+we assume that n = cd, for some c, d ∈ N≥2. Given S ∈ SLd(q) irreducible,
+we consider y = diag(S, . . . , S) ∈ SLn(q). Then CGLn(q)(y) ≃ GLc(qd).
+We claim that a Coxeter torus �T of CGLn(q)(y) remains a Coxeter torus
+in GLn(q) hence T := �T ∩ SLn(q) is a Coxeter torus in SLn(q).
+Indeed, by (5.1), we have | �T| = ((qc)d − 1) = (qn − 1). Since �T is cyclic,
+it is conjugated to TF
+w as claimed after (5.1). Thus, |T| = (n)q.
+In this subsection we will assume that x lies in a Coxeter torus T of GF
+arising from some y as above.
+Proposition 5.9. If x is irreducible, then O = OPSLn(q)
+x
+is of type C.
+Proof. Let n = cd with c prime and d ≥ 2. We set:
+�
+M := CGLn(q)(y) ≃ GLc(qd);
+M := CSLn(q)(y) = �
+M ∩ SLn(q);
+M1 := [�
+M, �
+M] ≃ SLc(qd).
+Thus M1 ≤ M ≤ �
+M. Lemma 5.7 gives
+OGLn(q)
+x
+∩ T = OSLn(q)
+x
+∩ T = {x, xq, . . . , xqj, . . . , xqn−1}
+OM
+x ∩ T = OM1
+x
+∩ T = O
+�
+M
+x ∩ T = {x, xqd, . . . , xqd(c−1)}.
+Hence xq ∈ O ∩ T but xq /∈ OM
+x . We claim that OM
+xq ̸⊂ N�
+M(T). Suppose
+the contrary.
+Then, ⟨OM
+xq ⟩ would be a non-central, normal subgroup of
+
+22
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+�
+M ≃ GLc(qd). Then SLc(qd) ≃ M1 ≤ ⟨OM
+xq ⟩ ≤ N�
+M(T), and so T ∩ M1
+would be normal in M1, a contradiction.
+We pick s ∈ OM
+xq \ N�
+M(T) and set
+s := π(s) ∈ O;
+r := π(x) ∈ O;
+H := ⟨r, s, π(M1)⟩.
+We claim that r, s and H satisfy the assumptions of Lemma 2.3. First,
+s ̸∈ N�
+M(T) implies s ⊲ r ̸= r, i.e., (2.2) holds. Indeed, s ⊲ r = r would give
+s ⊲ x ∈ Z(GF)x that combined with T = CGLn(q)(x) would force s ⊲ T = T.
+In addition, ⟨OM1
+x
+, OM1
+s
+⟩ = ⟨O �
+M
+x , O �
+M
+s ⟩ is a non-central, normal subgroup
+of �
+M ≃ GLc(qd), hence ⟨M1, x, s⟩ ≤ ⟨OM1
+x
+, OM1
+s
+⟩ and therefore
+H = ⟨π(M1), r, s⟩ ≤ ⟨Oπ(M1)
+r
+, Oπ(M1)
+s
+⟩ ≤ ⟨OH
+r , OH
+s ⟩ ≤ H.
+That is, (2.3) holds and H ≤ ⟨O⟩.
+Observe that π(M) ≃ M/Z(SLn(q)) ∩ M onto PGLc(qd), so the orbits
+Oπ(M)
+x
+and Oπ(M)
+xq
+project onto non-trivial orbits in PGLc(qd), and therefore
+|OH
+r | ≥ |Oπ(M)
+r
+| > 2 and |OH
+s | ≥ |Oπ(M)
+s
+| > 2, i.e., (2.4) holds.
+We finally analyse OH
+r ∩ OH
+s .
+First of all, π(M1) ≤ H ≤ π(M) and
+OM1
+x
+= OM
+x
+imply that OH
+r = Oπ(M)
+r
+. Similarly, OH
+s = Oπ(M)
+s
+.
+If Oπ(M)
+r
+∩ Oπ(M)
+s
+= ∅, then we are done. Otherwise,
+x ∈ Oπ(M)
+s
+∩ π(T) = Oπ(M)
+xq
+∩ π(T) = {xq, (xq)qd, . . . , (xq)qd(c−1)}.
+Therefore there exists l ∈ I0,c−1 such that xqdl+1 ∈ Gn(Fq)x. Lemma 5.8
+(iv) gives l ̸= 0. Let j ∈ In−1 be minimal satisfying xqj ∈ Gn(Fq)x. Then
+j|n by Lemma 5.8 (ii) whose argument shows that j divides also dl + 1.
+Hence, (j, d) = 1. Since j > 1 by Lemma 5.8 (iv) again, this can occur only
+if j = c and (c, d) = 1. In this case, d has a prime factor c′ different from c
+and we may repeat the whole construction replacing c by c′ and d by n
+c′ . As
+j = c ̸= c′, we get that Oπ(M)
+r
+∩ Oπ(M)
+s
+= ∅. The hypotheses of Lemma 2.3
+were verified, hence O is of type C.
+□
+5.3. Irreducible elements of SLn(q), n > 3 prime. Here n > 3 is prime.
+Recall that e ̸= x ∈ G = PSLn(q), x ∈ GF − Z(GF ) is a representative of
+x which is irreducible and belongs to the Coxeter torus T := TF
+w ∩ SLn(q);
+we set O = OG
+x and O = OGF
+x
+. There is an epimorphism of racks O ։ O.
+We will analyse all possible subgroups of GLn(q) intersecting O. We start
+by a few well-known arithmetic results instrumental for our analysis.
+Lemma 5.10. Let n be an odd prime number.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+23
+(i) If (n, q − 1) = 1, then (n, qn − 1) = (n, (n)q) = 1.
+(ii) If (n, q − 1) = n, then (n2, (n)q) = (n, (n)q) = n.
+(iii) (q − 1, (n)q) = (n, q − 1).
+(iv) (n(q − 1), (n)q) = (n, q − 1).
+Proof. (i) and (ii) are [25, Lemma 4.1.1], whilst (iii) follows from the Eu-
+clidean algorithm. We prove (iv). Combining (i), (ii) and (iii) we obtain
+(n, (n)q) = (n, q − 1) = (q − 1, (n)q),
+hence (n(q − 1), (n)q) = 1 if (n, q − 1) = 1, and n ≤ (n(q − 1), (n)q) ≤ n2 if
+(n, q − 1) = n. In this case, we discard (n(q − 1), (n)q) = n2 using (ii).
+□
+Recall σ ∈ NGLn(q)(Tw) from Remark 5.6.
+Lemma 5.11. Let n be a prime.
+(i) Let g ∈ NGLn(q)(TF
+w) \ TF
+w. Then |g| divides n(q − 1).
+(ii) O ∩ NGF (Tw) ⊂ TF
+w.
+Proof. (i) By Remark 5.6, there are k ∈ I1,n−1 and t ∈ TF
+w such that g = σkt.
+Then
+(σkt)n =
+
+ �
+τ∈⟨σk⟩
+τ ⊲ t
+
+ σnk =
+�
+τ∈⟨σ⟩
+τ ⊲ t =
+� n
+�
+i=1
+tqi
+�
+= t(n)q
+by a direct computation. Hence |g| divides n(q − 1).
+(ii) Let g ∈ O ∩ NGLn(q)(TF
+w) = O ∩ NGF (TF
+w). Recall that |x| divides
+(n)q. If g /∈ TF
+w, then |g| divides (n(q − 1), (n)q) by (i). By Lemma 5.10 (iv)
+|g| divides (n, q − 1), so g is central, contradicting its irreducibility.
+□
+We recall that a primitive prime divisor of qn − 1 is a prime number ℓ
+such that ℓ|qn − 1 and ℓ ̸ |qe − 1 for every e ∈ In−1
+Lemma 5.12. (Here n is any odd prime). Let y be an irreducible semisimple
+element in GLn(q). Then,
+(i) Either there exists a primitive prime divisor ℓ of qn − 1 dividing |y| or
+else |y| divides n(q − 1), it does not divide (q − 1) and n|q − 1.
+(ii) If y ∈ SLn(q), then there always exists a primitive prime divisor ℓ of
+qn − 1 dividing |y|.
+Proof. (i) If for every prime divisor ℓ of |y| there is an e ∈ In−1 such that
+ℓ divides qe − 1 = (q − 1)(e)q then, any such ℓ divides (q − 1)((n)q, (e)q)
+for some e < n. The latter equals (q − 1)(e, n)q = q − 1 by the Euclidean
+algorithm, so ℓ|q − 1 for any prime divisor ℓ of |y|.
+
+24
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Since y is irreducible, |y| cannot divide q − 1, so there is a prime divisor
+ℓ0 of |y| dividing q − 1 and (n)q. By Lemma 5.10 (iii) this is possible only
+if n|q − 1 and in this case ℓ0 = n. Then, Lemma 5.10 (ii) implies that |y|
+divides n(q − 1).
+(ii) If y ∈ SLn(q) then |y| divides (n)q. Assume, for a contradiction, that
+no primitive prime divisors of qn − 1 divides |y|. Then, |y| would divide
+(n(q − 1), (n)q) = (n, q − 1) by (i) and Lemma 5.10 (iii). Thus, y cannot be
+irreducible.
+□
+In the terminology of [17, Definition 1.2], Lemma 5.12 says that if n is an
+odd prime, then all irreducible elements in SLn(q) are ppd(n, q; n)-elements.
+The following result is a consequence of [17].
+Lemma 5.13. Let ℓ be a primitive prime divisor of qn − 1 dividing |x| and
+let H ≤ GLn(q) be such that x ∈ H. Then H occurs in the following list.
+(a) SLn(q0) ≤ H ≤ NGLn(q)(SLn(q0)) where q0 = pm0 with m = m0d,
+d ∈ N and (d, n) = 1.
+(b) SUn(q1/2
+0
+) ≤ H ≤ NGLn(q)(SUn(q0)) where q0 = pm0 a square with
+m = m0d, d ∈ N and (d, n) = 1.
+(c) H ≤ NGLn(q)(TF
+w) = NGLn(q)(T), and ℓ divides |H|.
+(d) H/(H ∩ Z(GLn(q)) ≃ M11, n = 5, ℓ = 11, and q5 ≡ 1 mod 11.
+(e) H/(H ∩ Z(GLn(q)) ≃ M23, or M24, n = 11, ℓ = 23, q11 ≡ 1 mod 23.
+(f) PSL2(ℓ) ≤ H/(H ∩ Z(GLn(q))) ≤ PGL2(ℓ), for ℓ ≥ 7, n = 1
+2(ℓ − 1)
+and qn ≡ 1 mod ℓ.
+Proof. The main result in [17] states that the subgroups of GLd(q) contain-
+ing a ppd(d, q; e)-element, for some 1
+2d < e ≤ d are precisely those occurring
+in the Examples 2.1, . . . , 2.9 listed therein. We extract the cases satisfying
+d = e = n an odd prime > 3.
+◦ Example 2.1 (b) and (d) and Example 2.5 are discarded because they
+occur for either d or e even.
+◦ Examples 2.1 (a) and (c) are (a) and (b) in our list.
+◦ Example 2.2 does not occur because it requires an H-stable subspace of
+the natural representation of GLn(q) and x ∈ H is irreducible.
+◦ Examples 2.3 and Examples 2.4 (a) are discarded as they require e ̸= d.
+◦ Example 2.4 (b) is the case (c) in our list.
+◦ Example 2.6 (a) is discarded because it requires the prime ℓ = n+1, which
+is impossible because n > 2.
+◦ Examples 2.6 (b) and (c) are collected in [17, Tables 2,3,4]. In Table 2, d
+is even. In Tables 3 and 4 the number e is never odd > 3.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+25
+◦ Examples 2.7 are listed in [17, Table 5]. The column with ℓ = e + 1, is
+immediately discarded, just as all rows for which d is not a prime number.
+We are left with the three possible choices for H′ ≃ H/H ∩ Z(GLn(q))
+listed in (d) and (e).
+◦ Examples 2.8 are listed in [17, Table 6] and are discarded because e ̸= d.
+◦ Examples 2.9 are listed in [17, Tables 7,8] and if d is a prime, then e is
+even. In Table 8, we discard all cases for which ℓ = e + 1 and we are left
+with the case (f) in our list.
+□
+Lemma 5.14. Let G = SLn(k), m = m0d with (d, n) = 1 and q0 = pm0.
+(i) If (n, q0 − 1) = n, then Z(GLn(q))GLn(q0) ∩ SLn(q) = SLn(q0).
+(ii) If m0 is even and (n, q1/2
+0
++ 1) = n, then
+Z(GLn(q))GUn(q1/2
+0
+) ∩ SLn(q) = SUn(q1/2
+0
+).
+Proof. (i) We prove ⊂. Let ζ idn ∈ Z(GLn(q)) and g ∈ GLn(q0) be such
+that ζg ∈ SLn(q). Now, |ζ| divides n(q0 − 1) because ζ−n = det(g) ∈ F×
+q0.
+It also divides q − 1 because ζ ∈ F×
+q . Hence it divides
+(n(q0 − 1), q − 1) = (q0 − 1) (n, (d)q0) .
+Hence, (d)q0 ≡ d mod n; since (d, n) = 1, then |ζ| divides q0 − 1, i.e.,
+ζ ∈ F×
+q0 and ζg ∈ SLn(q0).
+(ii) We prove ⊂. Let ζ idn ∈ Z(GLn(q)) and g ∈ GUn(q1/2
+0
+) be such
+that (ζ idn)g ∈ SLn(q). Now, as g = Frq1/2
+0
+φ(g), we have (det g)q1/2
+0
++1 = 1.
+Hence |ζ| divides n(q1/2
+0
++1) and also q −1 because ζ ∈ F×
+q , and so it divides
+(n(q1/2
+0
++ 1), q − 1) = (q1/2
+0
++ 1)
+�
+n, (q1/2
+0
+− 1)(d)q1/2
+0
+�
+.
+However, n is an odd prime dividing q1/2
+0
++ 1 so it does not divide q1/2
+0
+− 1;
+also, (n, (d)q1/2
+0
+) = 1 by the argument in (i) applied to q1/2
+0
+. Thus, |ζ| divides
+q1/2
+0
++ 1, that is, ζ idn ∈ Z(GUn(q1/2
+0
+)), so ζg ∈ SLn(q) ∩ GUn(q1/2
+0
+) =
+SUn(q1/2
+0
+).
+□
+In this Subsection �π: GLn(q) → PGLn(q) is the natural projection,
+whose restriction to SLn(q) is π.
+Proposition 5.15. Let x be an irreducible element in the Coxeter torus
+T = TF
+w. Then O is kthulhu.
+Proof.
+We consider all possible intersections O ∩ M for every M ≤ G
+containing x. All such groups have the form M = π(H ∩ SLn(q)) for some
+H ≤ GLn(q) containing x. We will show that either O ∩ M = OM
+x
+or else
+O ∩ M is an abelian subrack. This implies that O is kthulhu.
+
+26
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+For our analysis, we will make use of the following auxiliary facts:
+Claim 1. CPSLn(x) ∩ O = {x, xq, . . . , xqn−1}.
+Indeed, Lemma 5.7 gives
+CSLn(q)(x) ∩ O = {x, xq, . . . , xqn−1}.
+We describe CPSLn(q)(x). If z ∈ SLn(q) satisfies zxz−1 ∈ Gm(Fq)x∩O, then
+by primality of n and Lemma 5.8 (ii) and (iv) we conclude that zxz−1 = x,
+and so CPSLn(q)(x) = π(CSLn(q)(x)) = π(T), whence the claim.
+Claim 2. If �π(H) is simple, then �π(H) ≤ PSLn(q). In particular, we may
+assume H ≤ SLn(q).
+Indeed, if �π(H) is simple, then
+�π(H) = [�π(H), �π(H)] = �π([H, H]) ≤ �π([GLn(q), GLn(q)]) = π(SLn(q)).
+We set from now on H1 := H∩SLn(q) and inspect all possible M = π(H1)
+where H runs through the list of subgroups from Lemma 5.13 containing x,
+with ℓ a primitive prime divisor of qn − 1 dividing |x|. The numbering of
+items is as in Lemma 5.13.
+Case (a). Here q = pm, q0 = pm0 where m = m0d, d ∈ N and (n, d) = 1.
+Proposition 3.1 gives NGLn(q)(SLn(q0)) = Z(GLn(q))GLn(q0) so
+SLn(q0) ≤ H1 ≤ Z(GLn(q))GLn(q0) ∩ SLn(q).
+(5.4)
+We will first show that
+O ∩ Z(GLn(q))GLn(q0) ∩ SLn(q) = OSLn(q0)
+x
+.
+(5.5)
+If (n, q0 − 1) = n, then the inclusions in (5.4) are all equalities by Lemma
+5.14 (i). In this case
+O ∩ Z(GLn(q))GLn(q0) ∩ SLn(q) = O ∩ SLn(q0)
+= OSLn(k)
+x
+∩ SLn(q0) = OSLn(q0)
+x
+where the last two equalities follow from [22, Theorem 21.11] and [20, §2.11].
+Assume now that (n, q0 −1) = 1. Since x ∈ H1 ≤ Z(GLn(q))GLn(q0), there
+are z = ζ idn ∈ Z(GLn(q)) and y ∈ GLn(q0) such that
+x = zy.
+Consider x1 ∈ O ∩ Z(GLn(q))GLn(q0). Let z1 = ζ1 idn ∈ Z(GLn(q)) and
+y1 ∈ GLn(q0) be such that x1 = z1y1. By construction, |ζ| and |ζ1| divide
+n(q0 − 1) because x, x1 ∈ SLn(q).
+Since x is irreducible, y and y1 are
+again irreducible in GLn(q), whence in GLn(q0), because they are regular
+and lie in a Coxeter torus of GLn(q). Let {ηqj : j ∈ I0,n−1} ⊂ Fqn
+0 and
+{ηqj
+1 : j ∈ I0,n−1} ⊂ Fqn
+0 be the sets of eigenvalues of y and y1, respectively,
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+27
+so {ζηqj : j ∈ I0,n−1} and {ζ1ηqj
+1 : j ∈ I0,n−1} are the sets of eigenvalues of
+x and x1, respectively. Then
+{ζηqj : j ∈ I0,n−1} = {ζ1ηqj
+1 : j ∈ I0,n−1}
+and so ζη = ζ1ηqj0
+1
+for some j0.
+Therefore |ζ1ζ−1| = |ηη−qj0
+1
+| divides
+(n(q0 − 1), qn
+0 − 1) = (q0 − 1)(n, (n)q0) = q0 − 1, where the last equality
+follows from Lemma 5.10 (i). In other words, ζ1 ∈ ζF×
+q0, and z−1x1 is a
+regular semisimple matrix in GLn(q0) with the same eigenvalues as y, and
+it is therefore SLn(q0)-conjugate to y. Hence,
+O ∩ Z(GLn(q))GLn(q0) ⊂ zOSLn(q0)
+y
+= OSLn(q0)
+x
+.
+Let now x′ = π(x′) ∈ O ∩ M. Then, z′x′ ∈ O for some z′ ∈ Z(SLn(q))
+and x′ ∈ Z(SLn(q))H1, that is,
+z′x′ ∈ O ∩ Z(SLn(q))H1 ⊂ O ∩ Z(GLn(q))GLn(q0) ∩ SLn(q) = OH1
+x .
+where the equality follows from (5.4) and (5.5). Thus, x′ ∈ Z(SLn(q))OH1
+x
+and x′ ∈ Oπ(H1)
+x
+= OM
+x , showing that O ∩ M = OM
+x .
+Case (b). Here q = pm, q0 = pm0 where m0|m, m0 is even and (n, d) = 1.
+We use the same strategy as in case (a).
+Proposition 3.1 gives NGLn(q)(SUn(q1/2
+0
+)) = Z(GLn(q))GUn(q1/2
+0
+) so
+SUn(q1/2
+0
+) ≤ H1 ≤ Z(GLn(q))GUn(q1/2
+0
+) ∩ SLn(q).
+(5.6)
+We will first show that
+O ∩ Z(GLn(q))GUn(q1/2
+0
+) ∩ SLn(q) = OSUn(q1/2
+0
+)
+x
+.
+(5.7)
+If (n, q1/2
+0
++1) = n, then the inclusions in (5.6) are all equalities by Lemma
+5.14 (ii). In this case
+O ∩ Z(GLn(q))GUn(q1/2
+0
+) ∩ SLn(q) = O ∩ SUn(q1/2
+0
+)
+= OSLn(k)
+x
+∩ SUn(q1/2
+0
+) = OSUn(q1/2
+0
+)
+x
+where the last two equalities follow from [22, Theorem 21.11] and [20, §2.11].
+Assume now that (n, q1/2
+0
++1) = 1. Since x ∈ H1 ≤ Z(GLn(q))GUn(q1/2
+0
+),
+there are z = ζ idn ∈ Z(GLn(q)) and y ∈ GUn(q0) ≤ GLn(q0) such that
+x = zy.
+Consider x1 ∈ O∩Z(GLn(q))GUn(q1/2
+0
+). Let z1 = ζ1 idn ∈ Z(GLn(q)) and
+y1 ∈ GUn(q0) ≤ GLn(q0) be such that x1 = z1y1. By construction, |ζ| and
+|ζ1| divide n(q1/2
+0
++ 1) because x, x1 ∈ SLn(q) and det(g)q1/2
+0
++1 = 1 for any
+g ∈ GUn(q1/2
+0
+).
+Since x is irreducible, y and y1 are again irreducible in GLn(q), whence
+in GLn(q0). We show that |y| cannot divide n(q0 − 1). Indeed, if this were
+
+28
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+the case, then we would have yq0−1 = ξ idn for some ξ ∈ Gn(F×
+q ), with
+ξ ̸= 1.
+Since yq0 ∈ OGLn(q0)
+y
+, the characteristic polynomial py would be
+Xn − det(y) = Xn − ξ−n, which is not irreducible. Hence, by Lemma 5.12
+(i) there is a primitive prime divisor ℓ of |y| dividing qn
+0 − 1.
+Let F0 : GLn(k) → GLn(k) be given by F0(A) := Frq1/2
+0
+φ(A), for A ∈
+GLn(k), cf. Subsection 3.2. By [22, Proposition 26.6] there exists an F0-
+stable torus T′ in GLn(k) containing y. Let T = T′ ∩ GUn(q1/2
+0
+). By [22,
+Proposition 25.3 (c)] and an analysis of φ-classes in the symmetric group,
+there is a partition λ of n such that
+|T | =
+�
+λi even
+(qλi/2
+0
+− 1)
+�
+λi odd
+(qλi/2
+0
++ 1).
+The latter divides �
+λi even (qλi/2
+0
+− 1) �
+λi odd (qλi
+0 − 1) and is divisible by
+the primitive prime divisor ℓ of qn
+0 − 1. Hence, λ = (n) and |T | = (qn/2
+0
++ 1).
+Now we proceed as in case (a): considering the set of eigenvalues for x
+and x1 and of y and y1, we deduce that |ζ1ζ−1| divides
+�
+n(q1/2
+0
++ 1), qn/2
+0
++ 1
+�
+= (q1/2
+0
++ 1)(n, (n)−q1/2
+0
+) = (q1/2
+0
++ 1)
+where (n, (n)−q1/2
+0 ) = 1 because (n)−q1/2
+0
+divides qn/2
+0
++ 1 and q1/2
+0
+is not a
+root of Xn + 1 = (X + 1)n in Fn by our assumption on q0 and n.
+Hence, z1 ∈ zZ(GUn(q1/2
+0
+)), and z−1x1 is a regular semisimple matrix
+in GUn(q1/2
+0
+) with the same eigenvalues as y, and it is therefore SUn(q0)-
+conjugate to y by [20, §2.11, §8.5]. Hence,
+O ∩ Z(GLn(q))GUn(q1/2
+0
+) ⊂ zOSUn(q1/2
+0
+)
+y
+= OSUn(q1/2
+0
+)
+x
+.
+Let now x′ = π(x′) ∈ O ∩ M. Then, z′x′ ∈ O for some z′ ∈ Z(SLn(q))
+and x′ ∈ Z(SLn(q))H1, that is,
+z′x′ ∈ O ∩ Z(SLn(q))H1 ⊂ O ∩ Z(GLn(q))GUn(q1/2
+0
+) ∩ SLn(q) = OH1
+x .
+where the equality follows from (5.6) and (5.7). Thus, x′ ∈ Z(SLn(q))OH1
+x
+and x′ ∈ Oπ(H1)
+x
+= OM
+x , showing that O ∩ M = OM
+x .
+Case (c) In this case, M ≤ π(NSLn(q)(T)) = NG(π(T)), where the second
+equality follows because Z(SLn(q)) ≤ T. If y = π(y) ∈ O ∩ M then there
+is z ∈ Z(SLn(q)) such that y ∈ OSLn(q)
+zx
+∩ NSLn(q)(T), and zx is again
+irreducible. By Lemma 5.11 we see that y ∈ T, so O ∩ M ⊂ π(T) is abelian.
+Case (d) In this case n = 5 and ℓ = 11 and �π(H) = M11 = π(H1). We
+show that O ∩ M11 = OM11
+x
+. The only elements whose order is divisible
+by ℓ in M11 are of order 11, so |x| = 11. There are two classes of such
+elements in M11, say OM11
+x
+and OM11
+y
+. If O ∩ M11 = OM11
+x
+∪ OM11
+y
+, then
+⟨x⟩ − 1 ⊂ O ∩ M11 ∩ CPSLn(q)(x) = {x, xq, xq2, xq3, xq4}, a contradiction.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+29
+Case (e) In this case n = 11 and ℓ = 23. The only elements of order divisible
+by ℓ in M = M23 or M24 have order 23 and there are 2 conjugacy classes of
+such elements. We proceed as in case (c).
+Case (f) In this case ℓ|qn − 1 and M = π(H1) ≃ H1/H1 ∩ Z(SLn(q)), and
+PSL2(ℓ) ≤ H1/H1 ∩ Z(SLn(q)) ≤ PGL2(ℓ).
+As [PGL2(ℓ) : PSL2(ℓ)] ≤ 2, we have M ≃ PGL2(ℓ) or M ≃ PSL2(ℓ).
+In both cases, |x| = ℓ and we claim that O ∩ M = OM
+x .
+In PGL2(ℓ)
+all non-trivial unipotent elements are conjugate and the claim follows. Let
+y ∈ O ∩ PSL2(ℓ). By replacing y with a representative lying in the same
+Borel subgroup of PSL2(ℓ) as x, we can ensure that
+y ∈ CPSL2(ℓ)(x) ∩ O ⊂ CPSLn(q)(x) ∩ O = {x, xq, . . . , xqn−1}.
+Without loss of generality we may assume that x is the class of
+� 1 ξ
+0 1
+�
+, for
+some ξ ∈ F×
+ℓ so y is the class of
+� 1 ξ
+0 1
+�qj
+=
+�
+1 qjξ
+0
+1
+�
+for some j ∈ In−1. By
+assumption q ≡ qn+1 mod ℓ hence q is a square modulo ℓ. Therefore x and
+y are conjugate in PSL2(ℓ), whence the claim.
+□
+Remark 5.16. Consider either g ∈ M11, |g| = 11, or g ∈ M23 or M24,
+|g| = 23.
+Then the classes OM11
+g
+, OM23
+g
+or OM24
+g
+are contained either in
+OPSL5(q)
+g
+for some q, or in OPSL11(q′)
+g
+for some q′, respectively, according
+to [17] and Claim 2, see Cases (d) and (e). By Proposition 5.15, since g
+is irreducible in all cases, OG
+g
+is kthulhu, where G is either PSL5(q) or
+PSL11(q′). Hence so are OM11
+g
+, OM23
+g
+and OM24
+g
+, as was previously proved
+in [10, Teorema 3.26].
+6. Semisimple conjugacy classes represented in K
+In this section we deal with semisimple conjugacy classes intersecting the
+subgroup K which is the image of the map j : GLn(q) → GF introduced in
+§3.3. We give parallel proofs for two classes of simple groups:
+◦ G = Sp2n(k) with n ≥ 2; here G := GF /Z(GF) and π: GF → G denotes
+the standard projection.
+◦ G = SOn′(k) with n′ = 2n and n ≥ 4, or n′ = 2n + 1 and n ≥ 3; here
+G := [GF, GF ]/Z(GF ) and π: [GF , GF] → G is the standard projection.
+In the symplectic case, GF = [GF , GF ] so for brevity of the exposition we
+write [GF , GF] in both cases. We also consider such groups with smaller n
+sometimes for the sake of recursive arguments.
+We shall consider a semisimple class O in G, a class O in [GF, GF ] such
+that π(O) = O and assume that it exists A ∈ GLn(q) such that j(A) ∈ O.
+
+30
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Here are the main results of this Section:
+Theorem 6.1. Let G = Sp2n(k), n ≥ 2, and let A ∈ GLn(q) − Z(GLn(q))
+be a semisimple element, which is not an involution if n = 2 and q ≤ 7.
+Then O = OG
+π(j(A)) collapses.
+Theorem 6.2. Let G = SO2n(k) or SO2n+1(k) with n ≥ 3 in both cases
+and let A ∈ GLn(q) − Z(GLn(q)) be a semisimple element.
+Assume in addition that j(A) does not correspond to situation (4.1) if
+q ∈ {3, 5, 7}. Then O = OG
+π(j(A)) collapses.
+These theorems are proved in Subsection 6.2 after we deal in Subsection
+6.1 with the case when A is irreducible.
+In the orthogonal case, we consider the orbit O[GF ,GF ]
+j(A)
+for later applica-
+tions even if j(A) does not necessarily belong to [GF, GF ], as in Remark 2.1.
+See Lemmata 6.4 and 6.5.
+We start by some general considerations.
+Lemma 6.3. Let A ∈ GLn(q) be a semisimple element.
+(i) Oj(SLn(q))
+j(A)
+= O[K,K]
+j(A)
+= OK
+j(A) = Oj(GLn(q))
+j(A)
+.
+(ii) If A is irreducible, then either OGLn(q)
+A
+= OGLn(q)
+A−1
+or else j(A) is
+regular in GLn′(q).
+(iii) If A is irreducible, then either O[GF ,GF ]
+j(A)
+= O[GF ,GF ]
+j(A−1) , or else j(A) is
+regular.
+Proof. (i) is a consequence of the inclusions
+j(SLn(q)) ≃ [K, K] ≤ K ≃ GLn(q).
+(ii): If ζqh, h ∈ I0,n−1, are the (distinct) eigenvalues of A in k, then ζ±qh
+for h ∈ I0,n−1 (together with 1 when n′ = 2n + 1) are the eigenvalues of
+j(A). Assume that j(A) is not regular in GLn′(q); hence A and A−1 have a
+common eigenvalue. Then the sets of eigenvalues of A and A−1 coincide by
+irreducibility, that is A and A−1 are conjugate in GLn(k). Since centralisers
+in GLn(k) are connected, [20, p. 19], OGLn(q)
+A
+= OGLn(q)
+A−1
+. (iii) follows from
+(i) and (ii) and the inclusion [K, K] ≤ K ∩ [GF , GF].
+□
+6.1. A ∈ GLn(q) is irreducible. We first analyze this case.
+Lemma 6.4. (Here n ≥ 3 when G = SO2n(k) or G = SO2n+1(k)). Let
+A ∈ GLn(q) be an irreducible element such that
+◦ j(A) is not regular in GLn′(q),
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+31
+◦ pA(X) ̸= X2 + 1 when G = Sp4(k) and q ≡ 3 mod 4.
+Then
+(i) O := O[GF ,GF ]
+j(A)
+collapses.
+(ii) If j(A) ∈ O ⊆ [GF , GF], then O collapses.
+Proof.
+Observe that (ii) follows directly from (i), that we prove.
+The
+initial discussion is valid for both orthogonal and symplectic groups. Recall
+φ from (3.1). The irreduciblity assumption in A ensures that the eigenvalues
+of A are all distinct, i.e., A is regular semisimple, so we may assume that
+A is the companion matrix of its characteristic (and minimal) polynomial
+pA = Xn +an−1Xn−1 +· · · a0. That is, A, φ(A), tA, tA−1, A−1, and φ(A−1)
+have the following shape:
+A =
+� 0
+0 ···
+0
+−a0
+1
+0 ···
+0
+−a1
+0
+1 ···
+0
+−a2
+··· ··· ··· ···
+···
+0
+0 ···
+1 −an−1
+�
+,
+φ(A) =
+�
+0
+1
+0
+···
+0
+0
+0
+1
+···
+0
+···
+···
+···
+···
+···
+0
+···
+···
+0
+1
+−1/a0 −an−1/a0 ··· −a2/a0 −a1/a0
+�
+,
+tA =
+�
+0
+1
+0
+···
+0
+0
+0
+1
+···
+0
+···
+···
+···
+···
+···
+0
+0
+0
+···
+1
+−a0 −a1 −a2 ··· −an−1
+�
+,
+tA−1 =
+� −a1/a0 −a2/a0 ··· −an−1/a0 −1/a0
+1
+0
+···
+0
+0
+0
+1
+···
+0
+0
+···
+···
+···
+···
+···
+0
+0
+···
+1
+0
+�
+,
+A−1 =
+
+
+−a1/a0
+1
+0 ··· 0
+−a2/a0
+0
+1 ··· 0
+···
+··· ··· ··· ···
+−an−1/a0 0
+0 ··· 1
+−1/a0
+0
+0 ··· 0
+
+ ,
+φ(A−1) =
+� −an−1 −an−2 ··· ··· −a0
+1
+0
+0 ···
+0
+0
+1
+0 ···
+0
+···
+···
+··· ···
+···
+0
+0
+0
+1
+0
+�
+.
+Also A ̸= A−1, otherwise A would have eigenvalues ±1, contradicting irre-
+ducibility. We consider the disjoint subracks:
+R1 :=
+��
+A
+Y
+0 φ(A)
+�
+∈ O
+�
+,
+R2 :=
+��
+A−1
+Y
+0
+φ(A−1)
+�
+∈ O
+�
+,
+if n′ = 2n;
+R1 :=
+�� A 0
+Y
+0 1
+0
+0 0 φ(A)
+�
+∈ O
+�
+,
+R2 :=
+��
+A−1 0
+Y
+0
+1
+0
+0
+0 φ(A−1)
+�
+∈ O
+�
+,
+if n′ = 2n + 1.
+Now R1 ̸= ∅ by construction, and R2 ̸= ∅ by Lemma 6.3 (iii). It is easy to
+see that Ri ⊲Rj = Rj, 1 ≤ i, j ≤ 2. We continue with each group separately.
+Case 1. G = Sp2n(k). Let
+r1 :=
+�
+idn Jn
+0
+idn
+�
+⊲ j(A) =
+�
+A −AJn+tA−1Jn
+0
+Jn tA−1Jn
+�
+∈ R1,
+r2 := j(A−1) ∈ R2.
+A direct calculation shows that
+r1r2 :=
+�
+idn −A tAJn+Jn
+0
+idn
+�
+r2r1 :=
+�
+idn −Jn+A−1 tA−1Jn
+0
+idn
+�
+so r1r2 = r2r1 if and only if 2 idn = A−1 tA−1 + A tA. Let us verify that
+such an equality never holds. Comparing the diagonal entries we obtain
+a2
+i = (−1)i+1a2i
+0 (1 − a2
+0) for i > 0, whereas comparing the entries in the
+
+32
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+first row we obtain a1(a2 + a3
+0) = 0 and a1al = −a3
+0al−1 for l > 2. The
+conditions a2 + a3
+0 = 0 and a2
+2 = −a4
+0(1 − a2
+0) lead to a contradiction, hence
+necessarily a1 = 0 and so al = 0 for any l > 0 and a2
+0 = 1. In other words,
+pA(X) = Xn±1. By Lemma 5.5, this is possible only if n = 2, q ≡ 3 mod 4
+and pA(X) = X2 + 1, which is excluded by hypothesis.
+Then, r1 ⊲r2 ̸= r2 and, for H := ⟨r1, r2⟩ we have OH
+r1 ∩ OH
+r2 ⊂ R1 ∩ R2 = ∅
+because A2 ̸= id.
+If p = 2, then |r1| = |r2| is odd and OSp2n(q)
+j(A)
+is of
+type C by Remark 2.4 (b). If, instead, p is odd, then r1r2 ̸= r2r1 implies
+(r1r2)2 ̸= (r2r1)2 as they are p-elements, so OSp2n(q)
+j(A)
+is of type D.
+We claim that the restriction of the projection π: Sp2n(q) → G to
+R1
+� R2 is injective.
+Indeed, this could fail only if A2 = ±1, but since
+A is irreducible, we would have A2 = −1 which would give pA(X) = X2 +1,
+with q ≡ 3 mod 4, i.e., the discarded case. Hence OG
+π(j(A)) collapses.
+Case 2. G = SO2n(k) or SO2n+1(k). For n ≥ 3 we consider the matrices:
+E :=
+
+
+
+diag(id n
+2 , − id n
+2 )
+if n is even,
+diag(id[ n
+2 ], 0, − id[ n
+2 ])
+if n is odd,
+U :=
+
+
+
+�
+idn E
+0
+idn
+�
+if G = SO2n(k),
+� idn 0 E
+0
+1
+0
+0
+0 idn
+�
+if G = SO2n+1(k).
+Then U ∈ [GF, GF ] by [22, Theorem 24.15, Proposition 24.21] and we con-
+sider the elements ri ∈ Ri, i = 1, 2:
+r1 := U ⊲ j(A) ∈ R1 =
+
+
+
+
+
+�
+A −AE+Eφ(A)
+0
+φ(A)
+�
+if G = SO2n(k),
+�
+A 0 −AE+Eφ(A)
+0 1
+0
+0 0
+φ(A)
+�
+if G = SO2n+1(k),
+r2 := j(A−1) ∈ R2.
+A direct calculation shows that r1r2 = r2r1 if and only if
+2E = AEφ(A−1) + A−1Eφ(A).
+(6.1)
+We verify that this never happens. Assume first that p is odd. By looking
+at the (1, 1)-entries we see that (6.1) never holds if n ≥ 3. Since r1r2 and
+r2r1 are p-elements, it follows that π(r1r2)2 ̸= π(r2r1)2. The restriction of
+π to R1
+� R2 is injective because A2 = − id with A irreducible would imply
+n = 2, a discarded case. Hence OG
+π(j(A)) is of type D.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+33
+Assume that p = 2, so G = SO2n(k). Then (6.1) amounts to A2E
+⋆=
+Eφ(A)2. If n ≥ 4, by looking at the first row we see that ⋆ never holds. If
+n = 3, then (6.1) holds only when a2 = a−1
+0 , a1 = a2
+0, but in this case
+pA(X) = X3 + a−1
+0 X2 + a2
+0X + a0 = (X + a0)2(X + a−1
+0 )
+is not irreducible. Since |r1| is odd and π is injective, Remark 2.4 (b) applies
+and so OG
+π(j(A)) is of type C.
+□
+Lemma 6.5. (Here n ≥ 3 for G = SO2n(k) and n ≥ 2 for G = SO2n+1(k)
+or Sp2n(k)). Let A ∈ GLn(q) be an irreducible element such that j(A) is
+regular in GLn′(q). Then
+(i) O := O[GF ,GF ]
+j(A)
+collapses.
+(ii) If j(A) ∈ O ⊆ [GF , GF], then O collapses.
+Proof. Since A is irreducible, OGLn(q)
+A
+= OGLn(q)
+Aq
+. We have O = O[GF ,GF ]
+j(Aq)
+by Lemma 6.3 (i) so we can consider the disjoint subracks:
+R1 :=
+��
+A
+Y
+0 φ(A)
+�
+∈ O
+�
+,
+R2 :=
+��
+Aq
+Y
+0 φ(Aq)
+�
+∈ O
+�
+,
+if n′ = 2n;
+R1 :=
+�� A 0
+Y
+0 1
+0
+0 0 φ(A)
+�
+∈ O
+�
+,
+R2 :=
+�� Aq 0
+Y
+0 1
+0
+0 0 φ(Aq)
+�
+∈ O
+�
+,
+if n′ = 2n + 1.
+Then Ri ⊲ Rj ⊆ Rj for 1 ≤ i, j ≤ 2.
+Let r1 = j(A) ∈ R1.
+Since j(Aq) is regular, CGLn′(q)(r1) consists of
+semisimple elements, there exists u ∈ [GF, GF ] unipotent block upper tri-
+angular, with identity diagonal blocks of size n, n if n′ = 2n and n, 1, n
+if n′ = 2n + 1, such that r2 := u ⊲ j(Aq) ∈ R2 \ {j(Aq)}. Observe that
+r2 = j(Aq)v for some non-trivial block upper triangular unipotent element
+v. Now, r1j(Aq) = j(Aq)r1 and v /∈ CGLn′(q)(r1) because the latter consists
+of semisimple elements. Hence, r1r2 ̸= r2r1.
+If p = 2, then |A| is odd and O[GF ,GF ]
+j(A)
+is of type C by Remark 2.4 (b).
+Let p be odd. Then H := ⟨r1, r2⟩ = ⟨r1, v⟩ = ⟨r2, v⟩, with v a p-element.
+Thus
+��OH
+ri
+�� ≥
+���O⟨v⟩
+ri
+��� ≥ 3 for i = 1, 2, so O[GF ,GF ]
+j(A)
+is of type C by Lemma 2.3.
+Since A is irreducible, A ̸= Aq, hence the restriction of π to R1
+� R2 is
+injective, giving (ii).
+□
+Lemma 6.6. Let G = Sp4(k), let q ≡ 3 mod 4 and let A =
+� 0 −1
+1
+0
+�
+. Then
+O = OG
+π(j(A)) is of type D, hence it collapses.
+
+34
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Proof. By assumption, π(j(A)) is an involution. Let sc,d :=
+� c d
+d −c
+�
+∈ F2×2
+q
+,
+where (c, d) ∈ F2
+q. A direct calculation shows that
+Asc,dA−1 = s−c,−d = −sc,d.
+Thus, if sc,d ∈ GL2(q), then
+π(j(A)) ⊲ π(j(sc,d)) = π(j(sc,d));
+also
+φ(sc,d) =
+−1
+c2 + d2 sc,−d.
+We pick (a, b) ∈ F2
+q such that a2 + b2 = −1. Since q ≡ 3 mod 4, we have
+ab ̸= 0. As sa,b is semisimple, with same trace and determinant as A, it lies
+in OSL2(q)
+A
+, so π (j(sa,b)) ∈ OG
+π(j(A)).
+Consider the disjoint, non-empty subracks
+R1 :=
+�
+π
+� A X
+0 −A
+�
+∈ OG
+π(j(A))
+�
+,
+R2 :=
+�
+π
+� sa,b
+X
+0
+sa,−b
+�
+∈ OG
+π(j(A))
+�
+of OG
+π(j(A)). Then Ri ⊲ Rj = Rj for i, j ∈ {1, 2}.
+We set
+r := π
+�
+id2 id2
+0
+id2
+�
+⊲ π(j(A)) = π
+� A −2A
+0 −A
+�
+∈ R1,
+s := π(j(sa,b)) ∈ R2.
+Now ab ̸= 0 implies that sa,bsa,−b is not diagonal, hence
+(rs)2 = π
+�
+id2 2(id2 +sa,bsa,−b)
+0
+id2
+�
+̸= π
+�
+id2 −2(id2 +sa,bsa,−b)
+0
+id2
+�
+= (sr)2,
+so OG
+π(j(A)) is of type D.
+□
+6.2. Proofs of Theorems 6.1 and 6.2. We now drop the irreducibility
+assumption and proceed to prove the main results of this Section.
+Proof of Theorem 6.1. For A irreducible, this is Lemmata 6.4, 6.5 and 6.6.
+If A is not irreducible, then we may assume that A is a block diagonal
+matrix diag(A1, · · · , Af) where the Ai’s are irreducible. If they are all of
+size 1, then j(A) lies in a Fq-split torus and Proposition 4.1 applies.
+If,
+instead, one of the matrices Ai has size ni ≥ 2, then n > 2 and Ai is
+non-central in GLni(q) because it is irreducible. Lemmata 6.4, 6.5 and 6.6
+imply that O
+Sp2ni(q)
+j(Ai)
+collapses. The statement follows from injectivity of the
+composition of rack maps:
+� i−1
+�
+l=1
+{j(Al)}
+�
+× O
+Sp2ni(q)
+j(Ai)
+×
+�
+f�
+m=i+1
+{j(Am)}
+�
+→ OSp2n(q)
+j(A)
+→ OG
+π(j(A)).
+□
+Proof of Theorem 6.2. For A irreducible, this is Lemmata 6.4 and 6.5.
+If A is not irreducible, then we may assume that A is a block diagonal
+matrix diag(A1, · · · , Af) where f > 1 and each Ai is an irreducible ni × ni-
+matrix.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+35
+If q = 2, then A lies in SLn(q) and is not irreducible, so the rack inclusion
+OSLn(q)
+A
+֒→ O[SOn′(q),SOn′(q)]
+j(A)
+combined with [3, Theorem 1.1] gives the claim
+because PΩ+
+n′(q) = [SOn′(q), SOn′(q)].
+If ni = 1 for all i, then j(A) lies in a Fq-split torus and Lemma 4.6 applies.
+Therefore, we assume from now on that q > 2 and ni ≥ 2 for some i.
+If ni ≥ 3 for some i, then O
+[SO2ni(q),SO2ni(q)]
+j(Ai)
+collapses by Lemmata 6.4
+and 6.5. Then the claim follows because of the injectivity of the composition
+of the rack morphisms
+�i−1
+�
+l=1
+{j(Al)}
+�
+× O
+[SO2ni(q),SO2ni(q)]
+j(Ai)
+×
+�
+f�
+l=i+1
+{j(Al)}
+�
+→ O[SOn′(q),SOn′(q)]
+j(A)
+→ OG
+π(j(A)).
+From now on we assume that n1 = 2, and ni ≤ 2 for all i.
+If ni = 1 for some i, say i = 2, then A2 = (c) for some c ∈ F×
+q . Since
+A1 is irreducible, it is regular and has no eigenvalues in Fq. Thus the block
+diagonal matrix ˜A1 = diag(A1, c) has 3 distinct eigenvalues in Fq. The ma-
+trices
+� A1 v
+0
+c
+�
+, v ∈ F2
+q, have the same eigenvalues, hence they lie in OGL3(q)
+˜
+A1
+=
+OSL3(q)
+˜
+A1
+, cf. Remark 2.1. Consider the map j : GL3(q) → SO6(q). We claim
+that j
+�
+A−1
+1
+0
+0
+c
+�
+∈ O := O[SO6(q),SO6(q)]
+j( ˜
+A1)
+.
+Indeed, there is a representative g of a suitable w ∈ W in the normaliser of
+the torus of diagonal matrices in [SO6(q), SO6(q)] that satisfies g ⊲ j( ˜
+A1) =
+j
+�
+tA−1
+1
+0
+0
+c
+�
+. Also, tA−1
+1
+∈ OSL3(q)
+A−1
+, hence j
+�
+tA−1
+1
+0
+0
+c
+�
+∈ O. Thus
+R1 =
+�
+j
+� A1 v
+0
+c
+�
+: v ∈ F2
+q
+�
+,
+R2 =
+�
+j
+�
+A−1
+1
+v
+0
+c
+�
+: v ∈ F2
+q
+�
+are subracks of O, which are disjoint because A1 is irreducible.
+Clearly,
+Ri ⊲ Rj ⊂ Rj, 1 ≤ i, j ≤ 2. Pick 0 ̸= v ∈ F2
+q and set:
+r := j
+� A1 0
+0
+c
+�
+∈ R1;
+s := j
+�
+A−1
+1
+v
+0
+c
+�
+∈ R2.
+By a direct calculation, rs = sr implies that c is an eigenvalue of A1, a
+contradiction. Similarly, (rs)2 = (sr)2 iff c2 = −1, which can occur only if
+q is even or q ≡ 1 mod 4. If q is even, then O is of type C by Remark 2.4
+(b). If q ≡ 3 mod 4 or q ≡ 1 mod 4 and c2 ̸= −1, then O is of type D.
+Assume that q ≡ 1 mod 4 and c2 = −1. We claim that
+3 = |{s, rsr−1, r2sr−2}| ≤
+���O⟨r,s⟩
+s
+��� ,
+(6.2)
+3 = |{r, srs−1, rsrs−1r−1}| ≤
+���O⟨r,s⟩
+r
+���
+(6.3)
+
+36
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+By a direct calculation, r2sr−2 = s iff A2
+1v = −v = c2v, that is, hence c
+or −c is an eigenvalue of A1, a contradiction because they both lie in Fq;
+(6.2) follows. Similarly, rsrs−1r−1 = srs−1 iff (A1 − c)2v = 0, thus c is an
+eigenvalue of A1, a contradiction. Now srs−1 ̸= r implies rsrs−1r−1 ̸= r
+and (6.3) follows. Hence O[SO3(q),SO3(q)]
+j( ˜
+A1)
+is of type C by Lemma 2.3. Since
+the composition
+(R1
+�
+R2) ×
+� f�
+l=3
+{j(Al)}
+�
+→ O[SO3(q),SO3(q)]
+j( ˜
+A1)
+×
+� f�
+l=3
+{j(Al)}
+�
+→ O[SOn′(q),SOn′(q)]
+j(A)
+→ O = OG
+π(j(A))
+is an injective morphism of racks, the statement is proved in this case.
+There remains the case ni = 2 for all i. It suffices to assume that f = 2,
+so G = SO8(q), and that A1 and A2 are the companion matrices of their
+characteristic polynomials pA1 = X2 + aX + b and pA2 = X2 + cX + d, so
+A1 :=
+� 0 −b
+1 −a
+�
+,
+A2 :=
+� 0 −d
+1 −c
+�
+,
+A :=
+�
+A1
+0
+0 A2
+�
+.
+As in the previous step, there is an element in [SO8(q), SO8(q)] mapping
+j(A) to j
+�
+A−1
+1
+0
+0
+A2
+�
+. We consider the subracks of O[SO8(q),SO8(q)]
+j(A)
+given by
+R1 :=
+�
+j
+�
+A1 M
+0 A2
+�
+∈ O[SO8(q),SO8(q)]
+j(A)
+�
+,
+R2 :=
+�
+j
+�
+A−1
+1
+M
+0
+A2
+�
+∈ O[SO8(q),SO8(q)]
+j(A)
+�
+which are disjoint since A1 is irreducible. Clearly, Ri ⊲Rj ⊂ Rj, 1 ≤ i, j ≤ 2.
+Let u :=
+�
+id2 id2
+0
+id2
+�
+∈ SL4(q) and consider
+r := j(u) ⊲ j(A) = j
+�
+A1 A2−A1
+0
+A2
+�
+∈ R1,
+s := j
+�
+A−1
+1
+0
+0
+A2
+�
+∈ R2.
+A direct calculation in GL4(q) shows that (rs)2 = (sr)2 if and only if
+(A2 − A1)A2(id2 +A2
+2) = A−1
+1 (A2 − A1)(id2 +A2
+2).
+(6.4)
+Now, det(id2 +A2
+2) = 0 implies that there exists 0 ̸= v ∈ F2
+q such that A2
+2v =
+−v. By the irreducibility of A2, we get pA2 = X2 + 1, i.e., A2 =
+� 0 −1
+1 0
+�
+.
+Assume that det(id2 +A2
+2) ̸= 0. Then (6.4) is equivalent to A2 − A1 =
+A−1
+1
+− A−1
+2 , which is equivalent to A1 = A2. Therefore, if A1 ̸= A2, possibly
+interchanging the roles of A1 and A2, we can make sure that (6.4) is not
+satisfied, hence (rs)2 ̸= (sr)2, so OSL4(q)
+A
+is of type D. If A1 = A2 ̸=
+� 0 −1
+1 0
+�
+,
+then we interchange A1 and A−1
+1
+and argue as above.
+In all cases, the
+restriction of π to R1
+� R2 is injective, so OG
+π(j(A)) is of type D.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+37
+Finally, if A1 = A2 =
+� 0 −1
+1 0
+�
+, then A ∈ SL4(q) and OPSL4(q)
+πSL4(q)(A) is of type
+D by [3, Lemmata 3.15, 3.16, 3.17], and Oπ(K)
+π(j(A)) projects onto it. Whence
+Oπ(K)
+π(j(A)) is also of type D.
+□
+7. The symplectic groups
+In this Section, G = Sp2n(k), n ≥ 2. Recall that e ̸= x ∈ G is semisimple,
+GF ∋ x �→ x, O = OGF
+x
+and O = OG
+x . Here is the main result of this Section:
+Theorem 7.1. Let x ̸∈ Z(G) be semisimple. Then O collapses unless n = 2,
+q ∈ {3, 5, 7} and x is an involution.
+Classes represented in K have been discussed in Section 6. We deal in
+Subsection 7.1 with cuspidal classes that are not Coxeter, and then with
+Coxeter classes in Subsection 7.2. Theorem 7.1 is proved in Subsection 7.3.
+7.1. Cuspidal classes. Here we discuss the semisimple classes that are
+cuspidal but not Coxeter. Below we use without further notice that a cus-
+pidal class could not meet a standard Levi subgroup by Proposition 3.12.
+We start by the following observation: two semisimple symplectic matrices
+conjugated in GL2n(q) are then conjugated in Sp2n(q).
+Lemma 7.2. In either of the following cases, O is not cuspidal: (a) Some
+eigenvalue of x lies in Fq. (b) |x| ∈ {2, 3, 4}.
+Proof. (a). Indeed, if λ ∈ Fq is an eigenvalue of x, then so is λ−1, hence
+O contains an element of the form
+�
+λ
+A′ B′
+C′ D′
+λ−1
+�
+which belongs to a Levi
+subgroup isomorphic to Sp2(n−1)(k) × k×. Thus O is not cuspidal.
+(b). By (a), we may assume that x has no eigenvalues in Fq, so ±1 are
+excluded. If |x| ∈ {2, 3, 4}, then x has at most 2 distinct eigenvalues, namely
+the two primitive roots of 1, so it is not cuspidal by Proposition 3.13.
+□
+7.1.1. Cuspidal classes in the Weyl group. As is well-known, the Weyl group
+is W = (Z/2)n⋊Sn; let (ej)j∈In be the canonical basis of (Z/2)n. We identify
+W with a subgroup of S2n as in [11]:
+W ≃ {ς ∈ S2n : ς(2n + 1 − j) = 2n + 1 − ς(j), j ∈ In},
+Sn ∋ σ �→ σ′,
+σ′(j) =
+�
+σ(j),
+if j ∈ In,
+2n + 1 − ς(2n + 1 − j),
+if j ∈ In+1,2n;
+ej �→ τj = (j
+2n + 1 − j),
+j ∈ In.
+Given h ≤ k in In, we consider the 2(k − h + 1)-cycle in S2n defined by
+ch,k = (h
+h + 1 . . . k
+2n + 1 − h
+2n − h . . . 2n + 1 − k).
+(7.1)
+
+38
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Evidently, ch,k ∈ W. Let now λλλ = (d1, . . . , dt) be a partition of n, denoted
+λλλ ⊢ n, with d1 ≥ · · · ≥ dt. Set
+cλλλ = c1,d1cd1+1,d1+d2 . . . cd1+···+dt−1+1,n ∈ W.
+(7.2)
+By the identification above, we can rephrase [16, Proposition 3.4.6]:
+Proposition 7.3. The conjugacy class of such cλλλ is cuspidal. The family
+cλλλ, λλλ ⊢ n, is a complete set of representatives of the cuspidal conjugacy
+classes of W.
+□
+For instance, if λλλ = (n), then cλλλ is a Coxeter element.
+7.1.2. Cuspidal, but not Coxeter, classes. Let λλλ = (d1, . . . , dt) ⊢ n, with
+d1 ≥ · · · ≥ dt. Let Gλλλ be the image of the injective morphism of groups
+Sp2d1(k) × Sp2d2(k) × · · · × Sp2dt(k) −→ Sp2n(k),
+��
+A1
+B1
+C1
+D1
+�
+, . . . ,
+�
+At
+Bt
+Ct
+Dt
+��
+�−→
+
+
+
+
+
+
+
+
+
+A1
+B1
+...
+...
+At
+Bt
+Ct
+Dt
+...
+...
+C1
+D1
+
+
+
+
+
+
+
+
+
+.
+Claim. cλλλ has a decomposition Γ such that GΓ = Gλλλ, cf. (3.9).
+Proof. First, wj = cd1+d2+···+dj−1+1,d1+d2+···+dj is a Coxeter element of the
+factor Sp2dj(k) of Gλλλ. Up to appropriate identifications, the union of de-
+compositions Γ1, . . . , Γt of w1, . . . , wt is a decomposition of cλλλ. This implies
+the claim.
+□
+Lemma 7.4. If the conjugacy class O in GF is cuspidal but not Coxeter,
+then it is of type C, hence it collapses.
+Proof. By Proposition 3.12, there is partition λλλ ̸= (n) such that O intersects
+GF
+λλλ = Sp2d1(q) × Sp2d2(q) × · · · × Sp2dt(q). Let x = (x1, . . . , xt) ∈ GF
+λλλ ∩ O,
+with xj ∈ Sp2dj(q) for all j, so that
+O
+GF
+λλλ
+x
+= O
+Sp2d1(q)
+x1
+× O
+Sp2d2(q)
+x2
+× · · · × O
+Sp2dt(q)
+xt
+≤ O.
+We claim that xj /∈ Z(Sp2dj(q)) for all j ∈ It. Indeed, if xj ∈ Z(Sp2dj(q)) for
+some j, then x belongs to the torus Tci
+λλλ, cf. (3.11), where cj
+λλλ ∈ W is defined
+as cλλλ in (7.2) but omitting cd1+···+dj−1+1,d1+···+dj. This is a contradiction
+because cj
+λλλ is not cuspidal proving the claim. Then the lemma follows from
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+39
+Lemma 2.9, by Remark 2.10 and Lemma 3.14. Indeed, xj ̸= xq
+j, otherwise x
+would lie in a non-cuspidal torus by Lemma 7.2
+□
+Lemma 7.5. If the conjugacy class O in GF is cuspidal but not Coxeter,
+then the conjugacy class O in G is of type C, hence it collapses.
+Proof. Let π : Sp2n(q) → PSp2n(q) be the canonical projection. We may
+assume that q is odd, thus ker π = {± id}. Keep the notation of Lemma 7.4.
+Claim 1. The Lemma holds for t > 2.
+Indeed, the Lemma 2.9 provides a subrack of O of type C with the form
+Y =
+�
+{x1} × O
+Sp2d2(q)
+x2
+× {x3}
+� � �
+{xq
+1} × O
+Sp2d2(q)
+x2
+× {x3}
+�
+and clearly the restriction of π to Y is injective.
+Claim 2. The Lemma holds for t = 2.
+If x1 ̸= −xq
+1, then the restriction of π to the subrack of type C
+Y =
+�
+{x1} × O
+Sp2d2(q)
+x2
+� � �
+{xq
+1} × O
+Sp2d2(q)
+x2
+�
+is injective; similarly if x2 ̸= −xq
+2. Thus we may assume that x1 = −xq
+1
+and x2 = −xq
+2. Now x1 lives in a Coxeter torus TF
+1 in Sp2d1(q). By [22,
+Proposition 25.3] and [16, §3.4.3], we have
+|TF
+1 | = qd1 + 1.
+Hence |x1| divides (2(q − 1), qd1 + 1) =
+
+
+
+2
+if qd1 ≡ 1 mod 4,
+4
+if qd1 ≡ 3 mod 4.
+By symmetry, we may assume that the same holds for x2.
+Hence |x|
+divides 4; this contradicts Lemma 7.2.
+□
+7.2. Coxeter classes in Sp2n(q). Let x ∈ T ′ = TF
+w be a Coxeter element
+and let O = OGF
+x
+.
+Hence x is regular and its order divides qn + 1, so
+xqn = x−1. Arguing as in [5, §2.5] we see that O ∩ T ′ = {x±qj, j ∈ I0,n−1},
+and that the action of w raises x to xq. If ξ ∈ Fq is an eigenvalue of x, then all
+other eigenvalues of x are of the form {ξqj, j ∈ I0,2n−1} = {ξ±qj, j ∈ I0,n−1},
+with ξqn = ξ−1 and they are all distinct by Proposition 3.13.
+Lemma 7.6. Assume q is odd. If x is a Coxeter element, then −x ̸∈ O.
+Proof. If −x ∈ O, then with notation as above, −ξ is an eigenvalue of x, so
+−ξ = ξqj or −ξ = ξ−qj for some j < n. In the first case ξq2j = (−ξ)qj = ξ,
+whilst in the second ξq2j = (−ξ−1)qj = −(ξqj)−1 = ξ, with 2j < 2n in both
+cases, contradicting regularity of x.
+□
+
+40
+N. ANDRUSKIEWITSCH, G. CARNOVALE, G. A. GARC´IA
+Lemma 7.7. Let x be a Coxeter element in GF. Then O is of type C.
+Proof. Let H be a subgroup of GF isomorphic to SL2(qn), which exists by
+[21, II Satz 9.24]. Any non-split torus of T ′ ≤ H has order qn + 1 by (5.1),
+hence it is a Coxeter torus in GF, [16, 3.4.3]. Therefore
+|O ∩ T ′| = |{x±qj, j ∈ I0,n−1}| = 2n,
+|OH
+x ∩ T ′| ≤ 2,
+so the intersection O ∩ H is not a single H-conjugacy class. Assume x ∈
+H. Since H ≃ SL2(qn) with n ≥ 2, the group H/Z(H) is simple, so the
+non-central normal subgroup ⟨OH
+x ⟩ coincides with H. In addition, |OH
+x | =
+qn(qn − 1) > 4, so O is of type C by Lemma 2.8. The restriction of the
+projection π: GF → G to O is injective by Lemma 7.6, so π(O) = O is of
+type C as well.
+□
+7.3. The general case. Let L be a split F-stable Levi subgroup of G.
+Then, there exist f > 0, m ≥ 0 and ni for i ∈ If satisfying n = e + �f
+i=1 ni
+such that L is isomorphic the image of the injective morphism of groups
+�j : GLn1(k) × · · · × GLnf (k) × Sp2e(k) → Sp2n(k)
+(7.3)
+(A1, · · · , Af, A) �→ diag(A1, . . . , Ar, A, φ(Af), . . . , φ(A1))
+(7.4)
+Proof of Theorem 7.1. If n = 2, q = 3 and x is a non-central involution,
+then O is kthulhu by Lemma 4.4. Assume that q /∈ {5, 7} if n = 2 and
+x is a non-central involution. If x is cuspidal but not Coxeter, we invoke
+Lemma 7.5, whilst if x is Coxeter, then the claim follows from Lemma 7.7.
+If x is not cuspidal, then by Proposition 3.12 we may assume that x ∈ LF
+for a proper standard Levi subgroup L of G. Let �j be as in (7.3) and let
+x = �j(x1, . . . , xf, y). Taking ni, for i ∈ If and e to be minimal, and possibly
+increasing f, we assume that each xi is irreducible in GLni(q) and y is
+cuspidal in Sp2e(q). Under these assumptions xi ∈ Z(GLni(q)) if and only
+if ni = 1. If e = 0 the statement follows from Theorem 6.1. If e ≥ 2, then
+we consider the rack embedding
+{x1} × · · · × {xf} × OSp2e(q)
+y
+→ OSp2n(q)
+x
+→ OPSp2n(q)
+x
+(7.5)
+and invoke either Lemma 7.5 or Lemma 7.7.
+Assume from now on that e = 1, i.e., y is irreducible in Sp2(q) ≃ SL2(q).
+If there exists and ni such that ni > 1, then we consider the rack embedding
+{x1} × · · · × O
+Sp2ni(q)
+xi
+× · · · × {xf} × {y} → OSp2n(q)
+x
+→ OPSp2n(q)
+x
+(7.6)
+and invoke Theorem 6.1.
+There remains the case in which ni = 1 for every i. We assume that
+f = 1, for if f > 1 we can use the rack injection
+{x1} × · · · × {xf−1} × O
+Sp2(nf +1)(q)
+˜j(xf ,y)
+→ OSp2n(q)
+x
+→ OPSp2n(q)
+x
+.
+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+41
+Since y is irreducible, it lies in a non-split maximal torus, so its order divides
+q + 1 and so yq = y−1 ∈ OSL2(q)
+y
+. Also, if py = X2 − zX + 1, then z ̸= ±2.
+We may assume that y =
+� 0 1
+−1 z
+�
+so y−1 =
+� z −1
+1 0
+�
+and x = diag(λ, y, λ−1).
+We consider the following subracks of O = OSp4(q)
+x
+:
+R :=
+�
+x′ =
+�
+λ ∗
+∗
+0 y
+∗
+0 0 λ−1
+�
+: x′ ∈ O
+�
+,
+S :=
+�
+x′ =
+�
+λ−1
+∗
+∗
+0
+y−1 ∗
+0
+0
+λ
+�
+: x′ ∈ O
+�
+.
+By construction, R ⊲ S ⊂ S and S ⊲ R ⊂ R.
+Observe that R ∩ S = ∅;
+otherwise y = y−1 and p = 2, but in this case y would not be semisimple.
+Let M ∈ SL2(q) be such that M ⊲ y = y−1 and let
+r :=
+� 1 1 0 0
+0 1 0 0
+0 0 1 −1
+0 0 0 1
+�
+⊲ x =
+� λ −λ 1
+1
+0
+0
+1
+1
+0 −1 z z−λ−1
+0
+0
+0
+λ−1
+�
+∈ R,
+s :=
+� 0
+0 1
+0 M 0
+−1 0 0
+�
+⊲ x = diag(λ−1, y−1, λ) ∈ S.
+A direct calculation shows that rs = sr only if λ2 = 1 and z = ±2, a
+discarded case. Taking H := ⟨r, s⟩, we see that OH
+r ∩ OH
+s
+⊂ R ∩ S = ∅.
+Thus, if p = 2, then |x| is odd so OSp4(q)
+x
+is of type C by Remark 2.4. If p
+is odd, then (rs)2 ̸= (sr)2 because rs and sr are upper triangular unipotent
+matrices by construction, and so OSp4(q)
+x
+is of type D. We claim that the
+restriction of π to R � S is injective: indeed injectivity could fail only for
+λ2 + 1 = 0 and z = 0 but in this case, λ ∈ Fq would be a root of py which is
+irreducible, a contradiction.
+□
+References
+[1] N. Andruskiewitsch, G. Carnovale, G. A. Garc´ıa. Finite-dimensional pointed Hopf
+algebras over finite simple groups of Lie type I. Unipotent classes in PSLn(q). J.
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+[2]
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+[3]
+Finite-dimensional pointed Hopf algebras over finite simple groups of Lie
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+Finite-dimensional pointed Hopf algebras over finite simple groups of Lie
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+[5]
+Finite-dimensional pointed Hopf algebras over finite simple groups of Lie
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+associated to simple racks, Contemp. Math. 537 (2011), 31–56.
+
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+[8]
+Pointed Hopf algebras over the sporadic simple groups. J. Algebra 325
+(2011), 305–320.
+[9] N. Andruskiewitsch, M. Gra˜na. From racks to pointed Hopf algebras, Adv. Math.
+178 (2003), 177–243.
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+Birkh¨auser Boston, Inc., Boston, MA, 2000. xii+251
+[12] G. Carnovale, M. Costantini. Finite-dimensional pointed Hopf algebras over finite
+simple groups of Lie type VI. Suzuki and Ree groups, J. Pure Appl. Alg. 225,
+106568 (2021).
+[13] G. Carnovale, A. Garc´ıa Iglesias, θ-semisimple twisted conjugacy classes of type D
+in PSL(n,q), Journal of Lie Theory 26(1), 193–218 (2016).
+[14] F. Fantino. Conjugacy classes of p-cycles of type D in alternating groups. Commun.
+Algebra 42 4426–4434 (2014).
+[15] F. Fantino, L. Vendramin. On twisted conjugacy classes of type D in sporadic simple
+groups. Contemp. Math. 585 (2013) 247–259.
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+[18] I. Heckenberger and L. Vendramin. The classification of Nichols algebras with finite
+root system of rank two, J. Europ. Math. Soc. 19 (2017), 1977–2017.
+[19] G. Hiss. Finite groups of Lie type and their representations, Lond. Math. Soc. Lect.
+Note Ser. 387 (2011), 1–40.
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+Soc., Providence, RI, 1995.
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+Finite fields and their applications 2, 439–442 (1996).
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+Math. Soc. 13, 907–913 (1962).
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+Lecture Notes in Mathematics 1519, Springer (1992).
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+
+NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES
+43
+N. A.: FaMAF-Universidad Nacional de C´ordoba, CIEM (CONICET), Medina
+Allende s/n, Ciudad Universitaria, 5000 C´ordoba, Argentina.
+Email address: nicolas.andruskiewitsch@unc.edu.ar
+G. C.: Dipartimento di Matematica Tullio Levi-Civita, Universit`a degli Studi
+di Padova, via Trieste 63, 3512,1 Padova, Italia.
+Email address: carnoval@math.unipd.it, +39-049-8271354
+G. A. G.: Departamento de Matem´atica, Facultad de Ciencias Exactas, Uni-
+versidad Nacional de La Plata. CMaLP-CIC-CONICET. Calle 47 y Calle 115,
+1900 La Plata, Argentina.
+Email address: ggarcia@mate.unlp.edu.ar
+

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+PatentsView-Evaluation: Evaluation Datasets and
+Tools to Advance Research on Inventor Name
+Disambiguation
+Olivier Binette1,2, Sarvo Madhavan1, Jack Butler1, Beth Anne Card1, Emily Melluso1, and Christina Jones1
+1Duke University
+2American Institutes for Research
+Abstract—We present PatentsView-Evaluation, a Python pack-
+age that enables researchers to evaluate the performance of inven-
+tor name disambiguation systems such as PatentsView.org. The
+package includes benchmark datasets and evaluation tools, and
+aims to advance research on inventor name disambiguation by
+providing access to high-quality evaluation data and improving
+evaluation standards.
+Index Terms—Digital libraries, Inventor name disambiguation,
+PatentsView, Statistical Evaluation, Open-source software
+I. INTRODUCTION
+Inventor name disambiguation is the task of identifying
+unique inventors in patent datasets (Li et al., 2014; Toole
+et al., 2021). This requires using contextual information to
+distinguish between different inventors with the same name
+and to resolve name variations. Since there are no unique
+identifiers for inventors on U.S. patents, disambiguation is
+done using statistical algorithms which provide approximate
+solutions. The task is closely related to author name disam-
+biguation in digital libraries (Ferreira et al., 2012; Smalheiser
+et al., 2009; Subramanian et al., 2021) and is a particular case
+of entity resolution (Binette and Steorts, 2022; Christen, 2012;
+Christophides et al., 2021).
+Unfortunately, progress in the field has been hindered
+by misleading evaluation methodology and a lack of repre-
+sentative benchmark datasets (Wang et al., 2022). Naively
+computing performance metrics (i.e., precision and F-score)
+on benchmark datasets leads to biased estimates and flipped
+rankings of competing algorithms in many cases (Binette et al.,
+2022). This is due to the non-trivial scaling of entity resolution
+performance: while it is easy to disambiguate small benchmark
+datasets, the opportunity for error grows quadratically as
+a function of dataset size. Furthermore, some benchmark
+datasets are outdated or unavailable to the general public.
+To address these challenges, we have released PatentsView-
+Evaluation, a Python package that is available at github.com/
+patentsView/patentsView-Evaluation/ and that can be installed
+from PyPI (PyPI Authors, 2022) using:
+pip install pv-evaluation
+This is an open-source Python package which contains a suite
+of benchmark datasets and evaluation tools for representative
+performance evaluation. The package includes datasets used in
+the U.S. Patents and Trademarks Office (USPTO) 2015 dis-
+ambiguation competition, Azoulay’s Academic Life Sciences
+dataset which was previously unavailable to the general public,
+as well as a novel dataset extending what was developed by
+PatentsView in Binette et al. (2022) specifically for evaluation
+purposes. To facilitate performance evaluation, the package
+also includes representative precision and recall estimators as
+well as a suite of summary statistics and visualizations.
+The rest of the paper is structured as follows. In section II,
+we provide an overview of the package’s modules, including
+the available data and performance estimators. Section III
+summarizes our contributions and outlines our vision for future
+research.
+II. OVERVIEW OF THE PACKAGE
+PatentsView-Evaluation is built on top of the ER-evaluation
+Python package (Binette, 2022) which provides its core entity
+resolution evaluation functionality. It contains two main sub-
+modules. The benchmark module provides data, summary
+statistics, and visualizations. The template module provides
+templated reports that can be compiled to html using the
+Quarto publishing system (quarto.org).
+A. Benchmark Datasets
+Inventor disambiguation associates inventor mentions to
+unique inventor identifiers. Here, an inventor mention is the
+combination of a patent number and an authorship sequence
+number, resulting in a mention ID. For instance, the mention
+ID “US11379060-0” refers to the first inventor listed on U.S.
+patent number 11379060.
+Our benchmark datasets are pandas Series (Wes McKin-
+ney, 2010) indexed by inventor mentions and with values
+corresponding to a unique inventor identifier. Note that, while
+benchmark datasets aim to provide a ground truth disambigua-
+tion of a set of inventors, they may still contain errors resulting
+from the inherent uncertainty and difficulty of disambiguating
+inventors.
+The inventors benchmarks which we provide are listed
+below. These are available in the package through functions
+named load_*_inventors_benchmark().
+1) The Academic Life Sciences (ALS) dataset from the
+file named “patents 2005 12” was graciously shared by
+arXiv:2301.03591v1  [cs.DL]  9 Jan 2023
+
+2018
+2019
+2020
+2021
+2022
+0.4
+0.6
+0.8
+1
+estimator
+pairwise precision
+pairwise recall
+Pairwise precision and Recall
+value
+Fig. 1. Pairwise precision and recall estimates over PatentsView’s disambiguation history.
+Pierre Azoulay (personal communication) with permis-
+sion to release the corresponding clustering of inventor
+mentions. This dataset and variations of it were referred
+to in Azoulay et al. (2007, 2011); Ventura et al. (2015).
+We prepared the data by associating mention IDs to each
+record based on patent numbers and inventor mention
+names.
+2) The Israeli inventors benchmark from Trajtenberg and
+Shiff (2008).
+3) Li’s 2011 inventors benchmark from Li et al. (2014).
+4) The Engineer and Scientist inventors benchmark
+from PatentsView’s 2015 disambiguation competition
+(PatentsView, 2015).
+5) PatentsView’s
+2021
+inventors
+benchmark
+from
+Monath et al. (2021), which contains a set of particularly
+ambiguous inventor mentions.
+6) Binette’s 2022 inventors benchmark which extends
+Binette et al. (2022) and covers U.S. patents granted
+between 1976 and December 31, 2021. This is a random
+sample of inventors with sampling probabilities propor-
+tional to an inventor’s number of patents.
+B. Performance Estimators
+As previously noted, naively computing precision and re-
+call on benchmark datasets results in misleading figures. As
+such, PatentsView-Evaluation borrows from Binette et al.
+(2022) methodology for representative performance estima-
+tion. Given a set of inventor disambiguations for U.S. patents
+granted between 1976 and December 31, 2021, the function
+inventor_estimates_trend_plot() provides a plot
+of estimated precision and recall for each disambiguation
+with uncertainty quantification (± one standard deviation). By
+default, these estimates are based on Binette’s 2022 inventors
+benchmark. Estimates corresponding to the use of other bench-
+mark datasets can be obtained by passing them as additional
+arguments. Figure 1 showcases the resulting plot with default
+arguments.
+C. Summary Statistics and Visualizations
+In addition to performance metric estimators, PatentsView-
+Evaluation provides a suite of summary statistics visual-
+izations based on the ER-Evaluation package. This allows
+monitoring metrics such as the matching rate, the name
+variation rate, name homonymy rate, and the cluster size
+distribution entropy. More information on the definition of
+these metrics is provided in Binette (2022). The function
+inventor_summary_trend_plot() provides one entry
+point to visualizing these metrics for PatentsView’s disam-
+biguation history. Figure 2 showcases its output. Notice how,
+around 2021, the homonymy rate changes from around 0.2 to
+nearly 0.4 before going back down close to 0.05. These are
+major differences to the disambiguation which are not reflected
+in the matching rate.
+D. Templated HTML Reports
+The last component of PatentsView-Evaluation is a tem-
+plated report which can be compiled to HTML using
+Quarto. It allows the comparison of a set of inventor dis-
+ambiguations and through summary statistics, evaluation met-
+rics, and error visualization. The entry point is the func-
+tion render_inventor_disambiguation_report()
+which takes as arguments a set of disambiguation files.
+III. DISCUSSION
+In this paper, we presented PatentsView-Evaluation, a
+Python package with evaluation data and tools to advance
+inventor name disambiguation. We provided an overview of
+the package as well as a few examples of its capabilities.
+PatentsView’s vision for improved inventor name disam-
+biguation builds upon its experience and the success of its ex-
+isting system. We aim to improve the maintainability, modular-
+ity, and performance of PatentsView’s system through separate
+innovation within its three main components: (1) the feature
+engineering component which defines pairwise comparison
+metrics for given patent attributes, (2) the similarity modeling
+component which estimates pairwise match probabilities, and
+(3) the clustering component which resolves transitive inven-
+tor clusters. For (1), we aim to develop additional features
+
+2018
+2019
+2020
+2021
+2022
+0
+0.2
+0.4
+0.6
+0.8
+1
+metric
+Matching rate
+Homonimy rate
+Name variation rate
+Summary Statistics
+date
+value
+Fig. 2. Evolution of summary statistics over PatentsView’s disambiguation history.
+through the use of modern text analysis and natural language
+processing methods. For (2), we aim to develop flexible
+semi-supervised methods which can account for dependencies
+between features and biases in the benchmark datasets. Finally,
+for (3), we aim to better tune clustering algorithms to opti-
+mize key performance metrics. Through the use of principled
+performance evaluation tools available in the PatentsView-
+Evaluation package, new methodological developments can
+now be rigorously tested.
+REFERENCES
+Azoulay, P., W. Ding, and T. Stuart (2007). The determinants
+of faculty patenting behavior: Demographics or opportuni-
+ties? Journal of economic behavior & organization 63(4),
+599–623.
+Azoulay, P., J. S. Graff Zivin, and G. Manso (2011). Incentives
+and creativity: evidence from the academic life sciences.
+The RAND Journal of Economics 42(3), 527–554.
+Binette, O. (2022). ER-Evaluation: An end-to-end evaluation
+framework for entity resolution systems.
+Available on
+GitHub at https://github.com/OlivierBinette/ER-Evaluation.
+Binette, O. and R. C. Steorts (2022). (Almost) all of entity
+resolution. Science Advances 8(12), eabi8021.
+Binette, O., S. A. York, E. Hickerson, Y. Baek, S. Madha-
+van, and C. Jones (2022).
+Estimating the performance
+of entity resolution algorithms: Lessons learned through
+patentsview.org. arXiv e-prints. arXiv:2210.01230.
+Christen, P. (2012).
+Data Matching: Concepts and Tech-
+niques for Record Linkage, Entity Resolution, and Duplicate
+Detection. Data-Centric Systems and Applications. Berlin
+Heidelberg: Springer-Verlag.
+Christophides, V., V. Efthymiou, T. Palpanas, G. Papadakis,
+and K. Stefanidis (2021). An overview of end-to-end entity
+resolution for big data. ACM Computing Surveys 53(6), 1–2.
+Ferreira, A. A., M. A. Gonc¸alves, and A. H. Laender (2012).
+A brief survey of automatic methods for author name
+disambiguation. ACM Sigmod Record 41(2), 15–26.
+Li, G. C., R. Lai, A. D’Amour, D. M. Doolin, Y. Sun, V. I.
+Torvik, A. Z. Yu, and F. Lee (2014). Disambiguation and
+co-authorship networks of the U.S. patent inventor database
+(1975-2010). Research Policy 43(6), 941–955.
+Monath, N., C. Jones, and S. Madhavan (2021). PatentsView:
+Disambiguating Inventors, Assigness, and Locations. Tech-
+nical report, American Institutes for Research, Arlington,
+Virginia.
+PatentsView (2015). Inventor disambiguation workshop sum-
+mary.
+Available online at https://patentsview.org/events/
+workshop-2015.
+PyPI Authors (2022). Python package index - PyPI.
+Smalheiser, N. R., V. I. Torvik, et al. (2009). Author name
+disambiguation. Annual review of information science and
+technology 43(1), 1.
+Subramanian, S., D. King, D. Downey, and S. Feldman (2021).
+S2and: A benchmark and evaluation system for author name
+disambiguation. In 2021 ACM/IEEE Joint Conference on
+Digital Libraries (JCDL), pp. 170–179.
+Toole, A., C. Jones, and S. Madhavan (2021). PatentsView:
+An open data platform to advance science and technology
+policy. Technical report, USPTO Economic Working Paper.
+Trajtenberg, M. and G. Shiff (2008).
+Identification and
+mobility of israeli patenting inventors.
+Technical report,
+Pinhas Sapir.
+Ventura, S. L., R. Nugent, and E. R. Fuchs (2015). Seeing
+the non-stars: (Some) sources of bias in past disambiguation
+approaches and a new public tool leveraging labeled records.
+Research Policy 44(9), 1672–1701.
+Wang, T., H. Lin, C. Fu, X. Han, L. Sun, F. Xiong, H. Chen,
+M. Lu, and X. Zhu (2022). Bridging the gap between reality
+and ideality of entity matching: A revisiting and benchmark
+re-construction. arXiv e-prins. arXiv:2205.05889.
+Wes McKinney (2010). Data Structures for Statistical Com-
+puting in Python.
+In St´efan van der Walt and Jarrod
+Millman (Eds.), Proceedings of the 9th Python in Science
+Conference, pp. 56 – 61.
+

KNE2T4oBgHgl3EQfAgYc/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf,len=263
+page_content='PatentsView-Evaluation: Evaluation Datasets and  Tools to Advance Research on Inventor Name  Disambiguation  Olivier Binette1,2, Sarvo Madhavan1, Jack Butler1, Beth Anne Card1, Emily Melluso1, and Christina Jones1  1Duke University  2American Institutes for Research  Abstract—We present PatentsView-Evaluation, a Python pack-  age that enables researchers to evaluate the performance of inven-  tor name disambiguation systems such as PatentsView.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The  package includes benchmark datasets and evaluation tools, and  aims to advance research on inventor name disambiguation by  providing access to high-quality evaluation data and improving  evaluation standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  Index Terms—Digital libraries, Inventor name disambiguation,  PatentsView, Statistical Evaluation, Open-source software  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' INTRODUCTION  Inventor name disambiguation is the task of identifying  unique inventors in patent datasets (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Toole  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' This requires using contextual information to  distinguish between different inventors with the same name  and to resolve name variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Since there are no unique  identifiers for inventors on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' patents, disambiguation is  done using statistical algorithms which provide approximate  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The task is closely related to author name disam-  biguation in digital libraries (Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Smalheiser  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2021) and is a particular case  of entity resolution (Binette and Steorts, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Christen, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  Christophides et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  Unfortunately, progress in the field has been hindered  by misleading evaluation methodology and a lack of repre-  sentative benchmark datasets (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Naively  computing performance metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=', precision and F-score)  on benchmark datasets leads to biased estimates and flipped  rankings of competing algorithms in many cases (Binette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' This is due to the non-trivial scaling of entity resolution  performance: while it is easy to disambiguate small benchmark  datasets, the opportunity for error grows quadratically as  a function of dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Furthermore, some benchmark  datasets are outdated or unavailable to the general public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  To address these challenges, we have released PatentsView-  Evaluation, a Python package that is available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='com/  patentsView/patentsView-Evaluation/ and that can be installed  from PyPI (PyPI Authors, 2022) using:  pip install pv-evaluation  This is an open-source Python package which contains a suite  of benchmark datasets and evaluation tools for representative  performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The package includes datasets used in  the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Patents and Trademarks Office (USPTO) 2015 dis-  ambiguation competition, Azoulay’s Academic Life Sciences  dataset which was previously unavailable to the general public,  as well as a novel dataset extending what was developed by  PatentsView in Binette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' (2022) specifically for evaluation  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' To facilitate performance evaluation, the package  also includes representative precision and recall estimators as  well as a suite of summary statistics and visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' In section II,  we provide an overview of the package’s modules, including  the available data and performance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Section III  summarizes our contributions and outlines our vision for future  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' OVERVIEW OF THE PACKAGE  PatentsView-Evaluation is built on top of the ER-evaluation  Python package (Binette, 2022) which provides its core entity  resolution evaluation functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' It contains two main sub-  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The benchmark module provides data, summary  statistics, and visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The template module provides  templated reports that can be compiled to html using the  Quarto publishing system (quarto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Benchmark Datasets  Inventor disambiguation associates inventor mentions to  unique inventor identi��ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Here, an inventor mention is the  combination of a patent number and an authorship sequence  number, resulting in a mention ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' For instance, the mention  ID “US11379060-0” refers to the first inventor listed on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  patent number 11379060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  Our benchmark datasets are pandas Series (Wes McKin-  ney, 2010) indexed by inventor mentions and with values  corresponding to a unique inventor identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Note that, while  benchmark datasets aim to provide a ground truth disambigua-  tion of a set of inventors, they may still contain errors resulting  from the inherent uncertainty and difficulty of disambiguating  inventors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  The inventors benchmarks which we provide are listed  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' These are available in the package through functions  named load_*_inventors_benchmark().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  1) The Academic Life Sciences (ALS) dataset from the  file named “patents 2005 12” was graciously shared by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='03591v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='DL]  9 Jan 2023  2018  2019  2020  2021  2022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='8  1  estimator  pairwise precision  pairwise recall  Pairwise precision and Recall  value  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Pairwise precision and recall estimates over PatentsView’s disambiguation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  Pierre Azoulay (personal communication) with permis-  sion to release the corresponding clustering of inventor  mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' This dataset and variations of it were referred  to in Azoulay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' (2007, 2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Ventura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  We prepared the data by associating mention IDs to each  record based on patent numbers and inventor mention  names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  2) The Israeli inventors benchmark from Trajtenberg and  Shiff (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  3) Li’s 2011 inventors benchmark from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  4) The Engineer and Scientist inventors benchmark  from PatentsView’s 2015 disambiguation competition  (PatentsView, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  5) PatentsView’s  2021  inventors  benchmark  from  Monath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' (2021), which contains a set of particularly  ambiguous inventor mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  6) Binette’s 2022 inventors benchmark which extends  Binette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' (2022) and covers U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' patents granted  between 1976 and December 31, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' This is a random  sample of inventors with sampling probabilities propor-  tional to an inventor’s number of patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Performance Estimators  As previously noted, naively computing precision and re-  call on benchmark datasets results in misleading figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' As  such, PatentsView-Evaluation borrows from Binette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  (2022) methodology for representative performance estima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Given a set of inventor disambiguations for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' patents  granted between 1976 and December 31, 2021, the function  inventor_estimates_trend_plot() provides a plot  of estimated precision and recall for each disambiguation  with uncertainty quantification (± one standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' By  default, these estimates are based on Binette’s 2022 inventors  benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Estimates corresponding to the use of other bench-  mark datasets can be obtained by passing them as additional  arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Figure 1 showcases the resulting plot with default  arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Summary Statistics and Visualizations  In addition to performance metric estimators, PatentsView-  Evaluation provides a suite of summary statistics visual-  izations based on the ER-Evaluation package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' This allows  monitoring metrics such as the matching rate, the name  variation rate, name homonymy rate, and the cluster size  distribution entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' More information on the definition of  these metrics is provided in Binette (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The function  inventor_summary_trend_plot() provides one entry  point to visualizing these metrics for PatentsView’s disam-  biguation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Figure 2 showcases its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Notice how,  around 2021, the homonymy rate changes from around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='2 to  nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='4 before going back down close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' These are  major differences to the disambiguation which are not reflected  in the matching rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Templated HTML Reports  The last component of PatentsView-Evaluation is a tem-  plated report which can be compiled to HTML using  Quarto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' It allows the comparison of a set of inventor dis-  ambiguations and through summary statistics, evaluation met-  rics, and error visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' The entry point is the func-  tion render_inventor_disambiguation_report()  which takes as arguments a set of disambiguation files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' DISCUSSION  In this paper, we presented PatentsView-Evaluation, a  Python package with evaluation data and tools to advance  inventor name disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' We provided an overview of  the package as well as a few examples of its capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  PatentsView’s vision for improved inventor name disam-  biguation builds upon its experience and the success of its ex-  isting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' We aim to improve the maintainability, modular-  ity, and performance of PatentsView’s system through separate  innovation within its three main components: (1) the feature  engineering component which defines pairwise comparison  metrics for given patent attributes, (2) the similarity modeling  component which estimates pairwise match probabilities, and  (3) the clustering component which resolves transitive inven-  tor clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' For (1), we aim to develop additional features  2018  2019  2020  2021  2022  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='8  1  metric  Matching rate  Homonimy rate  Name variation rate  Summary Statistics  date  value  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Evolution of summary statistics over PatentsView’s disambiguation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content='  through the use of modern text analysis and natural language  processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' For (2), we aim to develop flexible  semi-supervised methods which can account for dependencies  between features and biases in the benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Finally,  for (3), we aim to better tune clustering algorithms to opti-  mize key performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
+page_content=' Through the use of principled  performance evaluation tools available in the PatentsView-  Evaluation package, new methodological developments can  now be rigorously tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE2T4oBgHgl3EQfAgYc/content/2301.03591v1.pdf'}
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KNE4T4oBgHgl3EQfJAzg/content/tmp_files/2301.04918v1.pdf.txt

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+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 13, 2023
+Merging binary black holes formed through double-core evolution
+Y. Qin1, 2, R.-C. Hu2, G. Meynet3, 4, Y. Z. Wang5, J.-P. Zhu6, H. F. Song7, X. W. Shu1, and S. C. Wu8, 9
+1 Department of Physics, Anhui Normal University, Wuhu, Anhui, 241000, China
+e-mail: yingqin2013@hotmail.com
+2 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning
+530004, China
+3 Département d’Astronomie, Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
+4 Gravitational Wave Science Center (GWSC), Université de Genève, CH-1211 Geneva, Switzerland
+e-mail: Georges.Meynet@unige.ch
+5 Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing,
+210033, People’s Republic of China
+6 Department of Astronomy, School of Physics, Peking University, Beijing 100871, China
+7 College of Physics, Guizhou University, Guiyang city, Guizhou Province, 550025, P.R. China
+8 Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), D-30167 Hannover, Germany
+9 Leibniz Universität Hannover, D-30167 Hannover, Germany
+January 13, 2023
+ABSTRACT
+Context. To date, various formation channels of merging events have been heavily explored with the detection of nearly 100 double
+black hole (BH) merger events reported by the LIGO-Virgo-KAGRA (LVK) Collaboration. We here systematically investigate an
+alternative formation scenario, i.e., binary BHs (BBHs) formed through double helium stars (hereafter double-core evolution channel).
+In this scenario, the two helium stars (He-rich stars) could be the outcome of the classical isolated binary evolution scenario involving
+with and without common-envelope phase (i.e., CE channel and stable mass transfer channel), or alternatively of massive close
+binaries evolving chemically homogeneously (i.e., CHE channel).
+Aims. We study the properties (i.e., the chirp masses and the effective spins) of binary BHs (BBHs) formed through the double-
+core evolution, and investigate the impact of different efficiencies of angular momentum transport within massive He-rich stars on
+double-core evolution.
+Methods. We perform detailed stellar structure and binary evolution calculations that take into account internal differential rotation
+and mass loss of He-rich stars, as well as tidal interactions in binaries. We systematically study the parameter space of initial binary
+He-rich stars, including initial mass and metallicity of He-rich stars, as well as initial orbital periods. Apart from direct core collapse
+with mass and angular momentum conserved, we also follow the framework in Batta & Ramirez-Ruiz (2019) to estimate the mass
+and spin of the resulting BHs.
+Results. We show that the radii of massive He-rich stars decrease as a function of time, which comes mainly from mass loss and
+mixing in high metallicity and from mixing in low metallicity. For double He-rich stars with equal masses in binaries, we find that
+tides start to be at work on the Zero Age Helium Main Sequence (ZAHeMS: the time when a He-rich star starts to burn helium in the
+core, which is analogous to ZAMS for core hydrogen burning) for initial orbital periods not longer than 1.0 day, depending on the
+initial metallicities. Besides the stellar mass loss rate and tidal interactions in binaries, we find that the role of the angular momentum
+transport efficiency in determining the resulting BH spins, becomes stronger when considering BH progenitors originated from a
+higher metal-metallicity environment. We highlight that double-core evolution scenario does not always produce fast-spinning BBHs
+and compare the properties of the BBHs reported from the LVK with our modeling.
+Conclusions. After detailed binary calculations of double-core evolution, we have confirmed that the spin of the BH is not only
+determined by the interplay of the binary’s different initial conditions (metallicity, mass and orbital period), but also dependent on the
+angular momentum transport efficiency within its progenitor. We predict that, with the sensitivity improvements to the LVK’s next
+observing run (O4), the sample of merging BBHs will contain more sources with positive but moderate (even high) χeff and part of
+the events are likely formed through the double-core evolution channel.
+Key words. binaries: close – stars: Wolf-Rayet – stars: black holes – stars: rotation
+1. Introduction
+The LIGO-Virgo-KAGRA (LVK) Collaboration has released the Gravitational Wave Transient Catalog 3 (GWTC-3, The LIGO
+Scientific Collaboration et al. 2021b), consisting of 69 confident binary black hole (BBH) merger events with the detection threshold
+to count events with false alarm rate (FAR) < 1 yr−1. With the targeted sample of BBHs in GWTC-3, the LVK Collaboration has
+also inferred their intrinsic properties (e.g., merger rates, masses and effective inspiral spins), among which the effective inspiral
+Article number, page 1 of 16
+arXiv:2301.04918v1  [astro-ph.HE]  12 Jan 2023
+
+A&A proofs: manuscript no. main
+spin χeff 1 has been widely considered as a probe to distinguish the formation channels of merging BBH events (Abbott et al. 2016b;
+Farr et al. 2017, 2018; The LIGO Scientific Collaboration et al. 2021c; Roulet et al. 2021). The majority of the BBHs reported by the
+LVK Collaboration have low χeff, while several BBH mergers 2 show definitely high positive χeff, e.g., 0.28+0.26
+−0.29, 0.31+0.20
+−0.22, 0.33+0.22
+−0.25,
+0.37+0.21
+−0.25, 0.52+0.19
+−0.19, for GW190706, GW190519, GW190620, GW170729, GW190517, respectively (Abbott et al. 2021). We also
+note that these high values of χeff are heavily under debate (see e.g., Callister et al. 2022; Vitale et al. 2022, references therein).
+Substantial progress for understanding the origin of BBHs has been made in the field over the last 7 years since the discovery
+of the first GW event GW150914 (Abbott et al. 2016a). However, the formation process of BBH merger events remains an open
+scientific question. Leading models of BBH formation include isolated binary evolution via either common envelope (CE, e.g.,
+Phinney 1991; Tutukov & Yungelson 1973; Belczynski et al. 2007; Ivanova et al. 2013; Postnov & Yungelson 2014; Belczynski
+et al. 2016; Vigna-Gómez et al. 2018; Qin et al. 2018; Bavera et al. 2020; Hu et al. 2022), stable Roche-lobe overflow (RLOF, e.g.,
+van den Heuvel et al. 2017; Inayoshi et al. 2017; Bavera et al. 2021; Olejak et al. 2021; Olejak & Belczynski 2021; Gallegos-Garcia
+et al. 2021; Marchant et al. 2021; Tanikawa et al. 2022; Shao & Li 2022; van Son et al. 2022a,b), or chemical mixing (Marchant
+et al. 2016; Mandel & de Mink 2016; de Mink & Mandel 2016; Song et al. 2016; du Buisson et al. 2020; Riley et al. 2021), as
+well as dynamical assembly in globular clusters and galactic nuclear clusters (e.g., Rodriguez et al. 2015; Antonini et al. 2016;
+Safarzadeh et al. 2020; Mapelli et al. 2021; Fragione et al. 2022), or efficient migration assisted in active galactic nuclei (AGN)
+disks (Secunda et al. 2019; McKernan et al. 2020; Tagawa et al. 2020; Saavik Ford & McKernan 2022). Alternatively, two BHs can
+be the occurrence of hierarchical stellar-mass BH mergers (Doctor et al. 2020; Kimball et al. 2020, 2021; Gerosa & Fishbach 2021).
+Zevin et al. (2021) recently investigated multiple formation pathways (isolated binary evolution channels and dynamical assem-
+bly channels) and found that neither channel can contribute more than ≃ 70% of the BBHs reported in GWTC-2. Moreover, it was
+pointed out in Mandel & Farmer (2022) (also see Mapelli (2020); Mandel & Broekgaarden (2022)) that the merger rates for BBHs
+can vary by orders of magnitude for different formation scenarios. So far, it is still a challenge to quantitatively predict the properties
+of merging BBHs due to uncertain physics involved in single and/or binary evolution (Abadie et al. 2010; Dominik et al. 2015; de
+Mink & Belczynski 2015; Giacobbo & Mapelli 2018; Tang et al. 2020; Broekgaarden et al. 2022; Belczynski et al. 2022; Peng et al.
+2022).
+Alternatively, merging BBHs could be formed through the double-core evolution. This scenario involving the CE phase has
+been recently investigated, focusing on low-mass He-rich stars leading to form double NSs (Dewi et al. 2006; Hwang et al. 2015;
+Vigna-Gómez et al. 2018). More massive stars with mass-ratio close to one at low metallicities evolving from ZAMS (Zero Age
+Main Sequence) in close binaries can undergo several stable mass transfer phases during core hydrogen burning (Case A mass
+transfer phase) and thus form double He-rich stars as potential progenitors of BBHs (see Figure 3 in Marchant et al. 2016). On the
+other hand, two massive stars could first evolve to form a close binary system of a He-star and a main-sequence companion star after
+the first mass transfer, and subsequently the second mass transfer from MS/giant star to the He-star leads to form massive He-rich
+binary stars in a short orbit.
+For now most BBH systems reported by the LVK Collaboration are still consistent with zero BH spins. Recently, by employing
+a variety of complementary methods to measure the distribution of spin magnitudes and orientations for BBH mergers, Callister
+et al. (2022) found that the existence of a subpopulation of BHs with vanishing spins is not required by current data. The fact at
+the moment no event necessarily requires a high spin does of course not mean that there are not among those already detected any
+that may present a high spin. High BH spins may indicate that the inefficient AM transport mechanism within the BH progenitor is
+preferred (Qin et al. 2019a,b, 2022b). This finding can be reached given the assumption that BBHs are formed through the classical
+isolated binary evolution channel involving CE phase, before which the initially more massive star collapses to form the first-born
+BH. Accordingly, the progenitor of the first-born BH is in a wide orbit in which the tides from its companion are too weak to change
+the spin AM of both components. Therefore, the resultant BH spin, inherited from the AM content of its progenitor, is exclusively
+determined by the AM transport efficiency within the progenitor star during post main sequence expansion. In case of an efficient
+transport, any removal of the outer layers (at the time of CE phase) slows the whole star, even its core. In case of a less efficient
+coupling, the core spins faster than the envelope and removing the envelope will make appear a faster rotating core than in the case
+of the efficient AM transport. Alternatively, it is shown in Olejak & Belczynski (2021) that fast-spinning BHs in merging BBHs can
+be formed by tidal spin-up through either a stable mass transfer phase leading to the mass ratio reversal, or the CE phase forming
+equal-mass BH components. For the case of stable mass transfer (see their Figure 1 in Olejak & Belczynski 2021), the initially more
+massive star evolves first to become a BH, and then its companion obtains enough mass via the first RLOF to become a massive
+He-rich star due to losing its hydrogen envelope onto the first-born BH in the second RLOF. The He-rich star subsequently evolves
+to become a fast-spinning BH by the tides (Qin et al. 2018). As for the other case (see their Figure 2 in Olejak & Belczynski 2021)
+the two stars initially with equal-mass instead form twin-mass He-rich stars following the RLOF mass transfer and subsequent CE
+phase, after which two fast-spinning BHs are formed via Wolf-Rayet tides. More recently, under the assumption of the Eddington-
+limited accretion onto BHs and efficient AM transport within massive stars, Zevin & Bavera (2022) investigated the isolated binary
+evolution regarding forming highly-spinning BHs and concluded that it is difficult to form systems with moderate or high spins in
+the primary BH component. However, the BH can be efficiently spun up by highly super-Eddington accretion (Bavera et al. 2021;
+van Son et al. 2020; Qin et al. 2022a; Shao & Li 2022).
+The Tayler-Spruit dynamo (Spruit 2002), produced by differential rotation in the radiative layers, is considered as one of potential
+mechanisms responsible for the efficient transport of AM between the stellar core and its radiative envelope. In brief, the TS dynamo
+starts for a small radial magnetic field component (its precise initial value has no importance since it is rapidly enhanced by the
+1 χeff = (M1χ1z + M2χ2z)/(M1 + M2), where M1 and M2 are the component masses of the two BHs, χ1z and χ2z are dimensionless BH spin
+magnitudes aligned to the direction of the orbital angular momentum (AM).
+2 GW190403 and GW190805 with high χeff were reported from deeper searches in GWTC-2.1 (The LIGO Scientific Collaboration et al. 2021a),
+but a low-significance FAR threshold of 2 per day.
+Article number, page 2 of 16
+
+Y. Qin et al.: Merging binary black holes formed through double-core evolution
+dynamo mechanism). This component is wounded up per differential rotation and an azimuthal component field is formed. An
+azimuthal field is unstable by the Tayler instability, i.e., an nonaxisymmetric pinch type instability, which has consequence to
+amplify the azimuthal field and the radial one. The new radial component is wounded up and the instability starts again. This
+amplification mechanism lasts until the growth timescale of the magnetic field is equal to its damping timescale. Assuming that
+stationary situation is reached at every time step and that the length over which the instability can develop is small enough for
+allowing the excess energy in the differential rotation to overcome the stabilizing entropy gradient and large enough for the magnetic
+field to not decay too fast, it is possible to deduce the diffusion and viscosity coeffiecients. The revised TS dynamo (Fuller et al.
+2019) is based on the fact that the damping timescale can be much longer than the one assumed in the original Tayler-Spruit dynamo.
+In that case larger magnetic fields can be reached and stronger coupling achieved (see the discussion in Eggenberger et al. 2022).
+Stellar models with the original Tayler-Spruit dynamo (TS dynamo) can well reproduce the rotation rates for the Sun (Eggen-
+berger et al. 2005), white dwarfs and NSs (Heger et al. 2005; Suijs et al. 2008). However the TS dynamo is currently challenged
+for explaining the slow rotation rates of cores in red giants (Eggenberger et al. 2012; Cantiello et al. 2014). Recently, the revised
+TS dynamo (Fuller et al. 2019), which was proposed to better match lower core rotation rates for sub-giant and red giant stars in
+better agreement with observed values, faces a challenge to reproduce the observational constraints on asteroseismic data of evolved
+stars (Eggenberger et al. 2019; den Hartogh et al. 2020). Applying the revised TS dynamo to massive He-rich stars in close binary
+systems predicts lower BH spins when compared with the original TS dynamo (Fuller & Lu 2022). More recently, Eggenberger
+et al. (2022) derived a new calibrated version of the original TS dynamo to better account for the evolution of the core rotation rates
+along the red giant branch stars when compared with the revised dynamo version. There was a theoretical debate on the existence
+of the dynamo (Zahn et al. 2007). Ji et al. (2022) recently performed three-dimentional magnetohydrodynamic simulations of the
+Tayler instability in rotating stellar interiors, and claimed to observe dynamo action via the amplification of poloidal magnetic field,
+indicating the TS instability could be important for magnetic field generation and AM transport in the radiative regions of evolving
+stars. The detailed comparisons between different versions of TS dynamo are beyond the scope of this work. Therefore, we are
+focused on the impact of the original TS dynamo within massive He-rich stars on the spin of resultant BH and its comparison when
+the TS dynamo is not included.
+In this paper, we systematically investigate an alternative evolutionary scenario to form BBHs from double He-rich stars, i.e.,
+double-core evolution first proposed by Brown (1995) who studied the formation of double NSss. In Section 2, we introduce the
+main methods used in the stellar and binary evolution models. We present our detailed results in Section 3. The conclusions and
+discussion are summarized in Section 4.
+2. Methods
+We use release 15140 of MESA stellar evolution code (Paxton et al. 2011, 2013, 2015, 2018, 2019) to perform all of the binary
+evolution calculations in this work. We adopt three different kinds of metallicities, Z = Z⊙, 0.1Z⊙, 0.01Z⊙, where the solar metallicity
+is Z⊙ = 0.0142 (Asplund et al. 2009). We create He-rich stars at zero-age helium main sequence following the same method as in
+Qin et al. (2018); Bavera et al. (2020); Hu et al. (2022); Fragos et al. (2022), and then relax the created He-rich stars to reach the
+thermal equilibrium when the ratio of the He-burning luminosity to the total luminosity ≥ 99%. We model convection using the
+standard mixing-length theory (Böhm-Vitense 1958) with a parameter α = 1.5 and semiconvection according to Langer et al. (1983)
+with an efficiency parameter αsc = 1.0. We adopt Ledoux convection criterion to treat the boundaries of the convective zones and
+consider the step overshooting as an extension given by αp = 0.1Hp, where Hp is the pressure scale height at the Ledoux boundary
+limit. The network of approx12.net is adopted for nucleosynthesis.
+We treat rotational mixing and AM transport as diffusive processes Heger & Langer (2000), including the effects of Eddington-
+Sweet circulations, the Goldreich–Schubert–Fricke instability, as well as secular and dynamical shear mixing. We include diffusive
+element mixing from these processes with an efficiency parameter fc = 1/30 (Chaboyer & Zahn 1992; Heger & Langer 2000). We
+use the standard efficient AM transport mechanism (e.g., Spruit 1999, 2002). Stellar winds of He-rich stars are modeled with the
+standard “Dutch” scheme, multiplied with a scaling factor of 2/3 to match the recently updated modeling of helium stars’ winds
+(Higgins et al. 2021).
+He-rich stars are modeled to reach the carbon exhaustion in the center. The baryonic remnant mass is calculated following
+the “delayed” supernova prescription as in Fryer et al. (2012). In order to calculate the mass and spin of the BH, we follow the
+framework in Batta & Ramirez-Ruiz (2019), which has been recently implemented in recent work (Bavera et al. 2020; Hu et al.
+2022). We take into account the neutrino loss as in Zevin et al. (2020). We adopt 2.5 M⊙ as the maximum NS mass. As a comparison
+(see appendix A), we also considered BHs formed through direct core collapse without receiving any mass loss or natal kicks (Fryer
+1999; Belczynski et al. 2008). Very recently, it was reported on VFTS 243 that an X-ray quiet BH was born with a negligible kick
+in a massive binary within the Large Magellanic Cloud (Shenar et al. 2022).
+Tidal interaction in close binary systems plays a critical role in the evolution of the orbit and the internal AM for the two stellar
+components. In this work, we use the dynamical tides model (Zahn 1975; Hut 1981) to calculate the synchronization timescale
+(Tsync), which is dependent on the tidal coefficient E2. The two He-rich stars are assumed to be non-rotating at ZAHeMS. The main
+reason is that He-rich stars can be quickly spun up in close orbits. For both He-rich (also H-rich) stars, Qin et al. (2018) recently
+updated an approximate expression of E2, mainly depending the convective core radius and the star’s radius for a wide range of
+initial masses and evolutionary stages at different metallicities.
+In this study, we are focused on detailed investigations of a parameter space study with various initial conditions of close double
+He-rich stars. We cover the initial masses of He-rich stars from 5 - 65 M⊙, the initial orbital periods in a range of 0.1 - 6 days. We
+evolve two He-rich stars with equal mass at different initial metallicities assuming two different AM transport mechanisms.
+Article number, page 3 of 16
+
+A&A proofs: manuscript no. main
+3. Results
+3.1. Hertzsprung-Russell diagrams of single He-rich stars
+Here we present the Hertzsprung-Russell (HR) diagram of single He-rich stars from the onset of the core helium burning (i.e.,
+ZAHeMS) to the exhaustion of their central carbon. All of the He-rich stars are assumed to be non-rotating with different metallicities
+(1.0 Z⊙, 0.1 Z⊙ and 1.0 Z⊙), in the mass range of 5 - 60 M⊙ at a step of 5 M⊙. In Fig. 1, the core helium burning phase begins on the
+right ends of the different curves labelled by the core He-mass, evolution then brings the stars to the left (the effective temperature
+increases). The evolution of the luminosity is different depending on the initial metallicity, rapidly decreasing at the beginning at 1.0
+Z⊙ in the high mass range, and increasing at 0.01 Z⊙ in this same mass domain. This is an effect of the different mass loss rates at
+different metallicities. At high metallicities the strong mass loss rate decreases rapidly the luminoisity. At a low metallicity the mass
+is much less decreased and the main effect comes from the fact that the mean molecular weight increases increasing the luminosity,
+overcoming the effect due to the weak mass loss. These stars evolves towards to bluer regions of the HR diagram, which is similar
+to H-rich stars evolving chemically homogeneously on the main sequence. The main difference, however, is that for more massive
+He-rich stars their mass decreases and as a consequence the radius shrinks.
+5.0
+5.05
+5.1
+5.15
+5.2
+5.25
+log[Teff/K]
+4.5
+5.0
+5.5
+6.0
+6.5
+log[L/L ]
+5 M
+10 M
+15 M
+20 M
+25 M
+30 M
+35 M
+40 M
+45 M
+50 M
+55 M
+60 M
+1.0 Z
+5.0
+5.05
+5.1
+5.15
+5.2
+log[Teff/K]
+5 M
+10 M
+15 M
+20 M
+25 M
+30 M
+35 M
+40 M
+45 M
+50 M
+55 M
+60 M
+0.1 Z
+5.0
+5.05
+5.1
+5.15
+5.2
+log[Teff/K]
+5 M
+10 M
+15 M
+20 M
+25 M
+30 M
+35 M
+40 M
+45 M
+50 M
+55 M
+60 M
+0.01 Z
+Fig. 1. Hertzsprung-Russell diagrams of various single non-rotating He-rich stars with different initial metallicities (Left panel: 1.0 Z⊙, middle
+panel: 0.1 Z⊙. bottom panel: 0.01 Z⊙.) evolving from Zero Age Helium Main Sequence (ZAHeMS) to the central helium exhaustion. The blue
+dashed lines refer to contours of constant radii.
+3.2. Spin of BHs formed from double-core evolution
+3.2.1. Impact of TS dynamo on BH spins
+Let us first show how different efficiencies of AM transport within He-rich stars change their rotation frequency at different evolu-
+tionary stages and thus the resulting spin parameters of BHs. As a case study, we evolve a binary system of two equal-mass He-rich
+stars, with initial mass MZamsHe = 39.80 M⊙ at the initial orbital period Pinit. = 0.63 days, until the end of their central carbon
+depletion.
+First of all, we show in Fig. 2 that the AM of the star and its core increases rapidly at the beginning due to the tidal interaction
+that spun up the star. Under the assumption that the wind mass lost is carrying the specific AM of the mass-losing star (Jeans mass
+loss), the He-rich star and its inner core will thus be slowed down. This situation, however, can be reversed for He-rich stars in
+a close binary system, in which tides are efficient to spin up the outer layers of the stars and their cores through strong coupling
+within the stars. We present the impact of the TS dynamo on the evolution of the internal rotation frequency of He-rich stars at
+different evolutionary stages, which are shown in the top two panels in Fig. 2. On the top left panel, we present models with the TS
+dynamo included (hereafter, TS on) given the solar metallicity, while we leave the discussion of a lower-metallicity model for the
+next section. First of all, with the TS dynamo the model in top left panel (see blue line) shows for the He-rich star in the middle of
+the core helium burning a flat distribution of a constant rotation frequency. This is because the star evolves with TS on like a solid
+body during core He burning phase. The whole star then gets spun up by the tidal interaction from its companion, as the star evolves
+off its core helium burning phase, from which on the star has rotation frequency of its outer layers slightly decreasing towards
+the surface due to the occurrence of increasing chemical gradient in the late evolutionary stage. The rotation frequency of the star
+continues increasing after the middle of the core helium burning, which is due to the tidal spun-up from its companion. In contrast,
+similar models without including TS dynamo (hereafter, TS off) on the top right panel in Fig. 2, show clear differences. The first
+difference is that the whole star due to less efficient coupling (e.g., TS off) between the outer layers and the stellar core is not a solid
+body from the early evolutionary stage (i.e., middle of core helium burning) to the late stages (see yellow sold line for the model at
+Article number, page 4 of 16
+
+Y. Qin et al.: Merging binary black holes formed through double-core evolution
+the central carbon ignition and red line for the central carbon depletion, respectively). Additionally, we also note that the star with
+TS off has a much larger rotational frequency throughout the whole evolutionary phase when compared with TS-on models. This is
+because inefficient coupling (TS off) between the outer layers and the stellar core allows the star to retain more AM and can instead
+be spun up if tides are strong.
+In this section, we present the impact of TS dynamo on the evolution of the AM of the He-rich stars and their inner cores at
+different evolutionary stages. We show in Fig. 3 that two binary evolutionary sequences of the same initial orbital period Pinit. =
+0.63 days and different initial masses MZamsHe = 10.00 M⊙ (top row) and MZamsHe = 39.80 M⊙ (bottom row), assuming TS on (solid
+lines) and off (dashed lines). Here we only show the solar-metallicity models, and leave the discussion of the rest in the following
+section. In the top left panel, we can see clear differences of the total AM of He-rich stars with TS on (black solid line) and off
+(black dashed line) starting before the middle of core helium (He) burning stage. For the model with TS on, during the core He
+burning phase the total AM of the star slowly decreases and then reaches the lower limit at the central He depletion. Additionally,
+the total AM keeps almost constant during the whole carbon (C) burning phase. Nevertheless, the model with the TS off shows that
+the star’s total AM slightly decreases from the core He burning phase and then keeps constant until the central C depletion. The
+AM of the carbon-oxygen (CO) core of the He-rich star shows a similar trend, but with a shallow decay after igniting its the central
+carbon. The resulting BHs calculated using the prescription in Batta & Ramirez-Ruiz (2019) show the spins 0.08 (TS on) and 0.29
+(TS off), respectively. For more massive He-rich binaries (MZamsHe = 39.80 M⊙), the bottom left panel presents a similar finding,
+but with a much higher difference on the BH spin value, 0.07 (TS on) and 0.49 (TS off). First, He-rich stars in a very close binary
+are synchronised with their orbit due to strong tides, which allows more massive star to carry more AM given the same initial orbit
+when compared to binary systems of less components. On top of that, more massive He-rich stars are expected to have less lifetime
+before the core-collapse, resulting in more AM content within the progenitors and thus high resultant BH spins.
+3.2.2. Impact of the metallicity on BH spins
+As shown in the previous section, the TS dynamo has a significant impact on the evolution of the AM of the He-rich star and its core
+at later evolutionary stages, which further determines the spin values of the BH at birth. We here describe how the initial metallicity
+of He-rich stars can play a role in determining the spins of the resulting BHs.
+It is well known that the stellar winds are strongly dependent on the metallicity of the mass-losing He-rich stars (Vink et al. 2001;
+Vink & de Koter 2005; Eldridge & Vink 2006; Sander et al. 2020). We can see that the He-rich star at its central carbon depletion
+has a much larger mass (around 37.5 M⊙) at 0.01 Z⊙ when compared with a solar metallicity (around 20 M⊙). We show in the
+two bottom panels of Fig. 2 that, the He-rich star and its inner core have a similar and higher rotation rate at different evolutionary
+stages when compared with corresponding TS-on models at Z⊙. Additionally, the bottom right panel shows that the He-rich star
+evolves deviating from a solid body, slightly decreasing rotation rate from the stellar core to its outer layers. We also see that at
+solar metallicity the TS-off models retain more AM, but the difference is much less marked than at sub-solar metallicity (e.g., 0.01
+Z⊙, see two bottom panels in Fig. 2). This difference is caused by the effect of the metallicity-depend wind mass loss which plays a
+critical in determining the final AM content of the progenitor star.
+In Fig. 3, we can see that at 0.1 Z⊙, both He-rich stars and their cores have a higher AM at different evolutionary stages when
+compared with 1.0 Z⊙ (see the top left panel in Fig. 3). Interestingly, we note that the TS dynamo plays a small role in determining
+the evolution of the AM of He-rich star and its core at a low metallicity (see the top middle and right panel). This is expected as
+the wind carrying the specific AM of the mass-losing He-rich star is weaker at lower metallicities, which weakens the effect of AM
+transport within stars. Therefore, the spin values of resultant BHs formed at lower metallicities are accordingly higher, i.e., 0.1 Z⊙
+and 0.14 at 0.01 Z⊙ for TS on (TS off: 0.43 Z⊙ and 0.49 at 0.01 Z⊙).
+3.2.3. Parameter space analysis
+First of all, we show in the top left panel of Fig. 4, BH masses as a function of the He-rich initial mass and initial orbital period at
+solar metallicity. First, the binary systems either start to overflow their Roche lobes at the first model for Pinit. ∼ 0.1 days (0.2 days
+for MZamsHe ∼ 30 M⊙) or undergo the second lagrangian point (L2) overflowing for MZamsHe ∼ 38 M⊙ and Pinit. ∼ 0.1 days. Second,
+given the “delayed” supernova prescription (Fryer et al. 2012) the lower mass limit of the He-rich star that can collapse to form a
+BH (given solar metallicity) is around 12 M⊙, below which a NS is formed instead (The study of NS formation is not considered
+in this work). Third, a He-rich star can form a BH with the maximum mass of around 26 M⊙. At 0.1 Z⊙ (see top middle panel), we
+note that ∼ 40 M⊙ BH can be formed. Notably, the initial orbital period starts to have an impact on the mass of the resulting BH
+when its immediate progenitor (He-rich star) has an initial mass ≳ 40 M⊙. This is because He-rich stars tend to lose more masses
+at a higher rotation rate in a closer binary system, which is due to the rotationally-enhanced mass loss (Langer 1997; Maeder &
+Meynet 2000). It is clearly shown at 0.01 Z⊙ that more massive BHs (> 55 M⊙ see top right panel) can be formed. It is worth noting
+that the efficient AM transport within He-rich stars plays a negligible role in determining the BH mass.
+We then present in Fig. 5 the spin parameters a∗ of BHs formed from collapsing He-rich stars in close binaries with various
+conditions and assumed AM transport processes. Let us first show the spins of resultant BHs assuming efficient AM transport
+within He-rich stars. We note that the tides start to play a role when the initial orbital period Pinit. is not longer than 1.0 day for all
+metallicities. As demonstrated in recent studies (Qin et al. 2018; Fuller & Lu 2022), the interplay between the tides and wind mass
+loss of He-rich stars determines the AM of resultant BHs at birth and thus their spin magnitudes. At solar metallicity, the formed
+BHs are found to have low spin values (i.e., a∗ ≲ 0.4, see top left panel). This is because the wind mass loss of He-rich stars at a
+high metallicity is dominant over the tides. The spin magnitudes of BHs formed at lower metal poor environments are shown in the
+middle (0.1 Z⊙) and right panel (0.01 Z⊙). Therefore, at a given initial orbital period (Pinit. ≲ 1.0 day), high BH spins can be reached
+Article number, page 5 of 16
+
+A&A proofs: manuscript no. main
+0
+5
+10
+15
+20
+25
+30
+Enclosed Mass [M ]
+10 5
+10 4
+10 3
+10 2
+ [rad/s]
+Z = 1.0 Z
+TS on
+0
+5
+10
+15
+20
+25
+30
+Enclosed Mass [M ]
+Z = 1.0 Z
+TS off
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Enclosed Mass [M ]
+10 4
+10 3
+10 2
+ [rad/s]
+Z = 0.01 Z
+TS on
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Enclosed Mass [M ]
+Z = 0.01 Z
+TS off
+Middle of Core He burning
+Central C Ignition
+Central C Depletion
+Fig. 2. As a function of mass coordinate, we plot the angular velocity ω of He-rich stars at three evolutionary stages, i.e., middle of core helium
+burning (blue), Central carbon ignition (yellow), and central carbon depletion (red). The initial mass of He-rich star is MZamsHe = 39.80 M⊙ and
+the initial orbital period Pinit. = 0.63 days. Different efficiencies of AM transport mechanism and metallicities are assumed. Left column: TS on,
+right column: TS off. Top row: 1.0 Z⊙, bottom row: 0.01 Z⊙.
+for models at 0.1 and 0.01 Z⊙. At 0.01 Z⊙, we clearly see in the top right panel that, the spin magnitudes continue increasing with
+initial orbital period for all different initial masses of He-rich star. This is because the progenitor of the BH has very weak winds
+at 0.01 Z⊙, and thus loses negligible mass and AM. Furthermore, it is clear to see that the spins of BHs, originated from He-rich
+stars with initial mass ≲ 20 M⊙, slightly increase with initial mass. This is because the wind of low-mass He-rich stars at very
+low metallicity (0.01 Z⊙) is significantly weak. Accordingly, for initially higher mass of He-rich stars, more infalling mass with its
+corresponding AM can be accreted to the newly-formed BHs (Batta & Ramirez-Ruiz 2019), resulting in higher final BH spins.
+Assuming inefficient AM transport within He-rich stars, we show the spins of resultant BHs in the second row of Fig. 5. As
+shown clearly in the bottom left panel, high BH spins (> 0.9) can be reached at solar metallicity. Additionally, the spin covers
+the whole range, i.e., from minimum to maximum. For initial orbital periods Pinit. ≲ 1.0 day, we note that the BH spin gradually
+decreases with increasing initial mass of He-rich stars, which is because massive He-rich stars are prone to be slowed down due to
+their strong winds at high metallicity. This is in contrast to the results of models at very low metallicity (see the top right panel),
+where the wind mass loss of He-rich stars is significantly weak at 0.01 Z⊙. Notably, we can see that He-rich stars tend to form
+higher-spinning BHs at lower metallicities which correspond to weaker wind mass loss (see bottom middle and right panel).
+3.3. Merging timescales and comparisons with observed merging BBHs
+After two BHs form from the core-collapse of He-rich stars, gravitational wave (GW) emission shrinks the separation by removing
+the orbital AM, and eventually leads to the merger the compact objects. The timescale for two point masses to spiral in through GW
+emission from an initial eccentricity being zero (circular orbit) is given by Peters (1964)
+Tmerger =
+5
+512
+c5
+G3M3
+2q−2
+1 + q−1 a4,
+(1)
+where M is the BH mass, q the mass ratio of the two BHs (q = 1 for our case) and a is the orbital separation.
+We show the color bar in Fig. 6 corresponding to Tmerger of merging BBHs due to GW emission. First, comparing the two rows
+of different AM transport mechanisms shows negligible impact on the merging timescale. This is because significant differences are
+only expected for the AM content of the BH progenitors, rather than the properties (two component masses and the final separation)
+of the binary system just after the birth of two BHs. Second, the parameter space of systems that are able to merge within a Hubble
+time is extended in lower metallicities. This is because BH progenitors at a higher metallicity tend to lose more mass, and the BBHs
+at birth thus have larger separations (Tmerger ∝ a4). Third, given a specific initial orbital period and metallcity, BBHs with initially
+higher mass have shorter merging timescales (Tmerger ∝ M−3).
+Figure 7 presents the merging timescales Tmerger as a function of the effective inspiral spin χeff and the chirp mass Mchirp. 69
+high-confidence BBH events (false alarm rate < 1 per year) officially reported from the LVK are also shown in grey for comparison
+Article number, page 6 of 16
+
+Y. Qin et al.: Merging binary black holes formed through double-core evolution
+0
+100
+200
+300
+Model number
+1049
+1050
+J (cm2 g/s)
+TS on: a ,final = 0.08
+TS off: a ,final = 0.29
+Pinit.  = 0.63 days
+Minit.  = 10.00 M
+Z = 1.0 Z
+0
+50
+100
+150
+200
+250
+Model number
+TS on: a ,final = 0.10
+TS off: a ,final = 0.43
+Z = 0.1 Z
+0
+50
+100
+150
+200
+Model number
+TS on: a ,final = 0.14
+TS off: a ,final = 0.49
+Z = 0.01 Z
+0
+200
+400
+600
+800
+Model number
+1049
+1050
+1051
+1052
+J (cm2 g/s)
+TS on: a ,final = 0.07
+TS off: a ,final = 0.49
+Pinit.  = 0.63 days
+Minit.  = 39.80 M
+Z = 1.0 Z
+0
+100
+200
+300
+Model number
+TS on: a ,final = 0.21
+TS off: a ,final = 0.52
+Z = 0.1 Z
+0
+50
+100
+150
+200
+Model number
+TS on: a ,final = 0.50
+TS off: a ,final = 0.56
+Z = 0.01 Z
+Jtotal (TS on)
+JCO (TS on)
+Jtotal (TS off)
+JCO (TS off)
+Middle of Core He burning
+Central He Depletion
+Central C ignition
+Fig. 3. AM of the He-rich star and its Carbon-Oxygen (CO) core (black lines: Jtotal, blue lines: JCO) as a function of model number for two binary
+sequences (Top row: two equal-mass He-rich stars with initial helium star mass MZamsHe = 10.0 M⊙, initial orbital period Pinit. = 0.63 days; bottom
+row: MZamsHe = 39.8 M⊙, Pinit. = 0.63 days). Similar to Fig. 2, we assume two different efficiencies of AM transport mechanism, i.e., solid lines:
+TS on, dashed lines: TS off. Left column: 1.0 Z⊙, middle column: 0.1 Z⊙, right column: 0.01 Z⊙. Three evolutionary stages are marked in different
+symbols: square: middle of core helium burning, circle: central helium depletion, filled circle: central carbon ignition. The spin parameters of BHs
+formed from He-rich stars are presented.
+in each panel. We present systems formed from the same initial orbital period in dashed line. We assume that the formed BHs
+have spin components perfectly aligned to the direction of the orbital AM. First of all, we note that the initial metallicity plays an
+important role in forming systems with the observable properties (χeff and Mchirp). i.e., lower metallicities corresponding to formed
+systems with higher χeff and larger Mchirp. More specifically, the Mchirp can be reached around 26 M⊙ at solar metallicity (40 and
+58 M⊙ at 0.1 and 0.01 Z⊙, respectively). Furthermore, the magnitude of χeff can vary from 0.0 (Pinit. = 1.0 day) and 1.0 (Pinit. = 0.2
+days). We note that so far no BBHs with χeff = 1 has been reported from the LVK collaboration. Third, the AM transport mechanism
+in these observable properties of BBHs starts to play a more important role at higher metallicities (solar metallicity, see the two
+left panels in Fig. 7). Additionally, under the assumption of inefficient AM transport, double He-rich stars can form observable
+BBHs with χeff > 0.80 (< 0.5 with TS on) with initially Pinit. = 0.4 days and χeff > 0.5 (< 0.25 with TS on) with initially Pinit. =
+0.6 days, respectively. It is also clearly shown in the left panels of Fig. 7 that more BBHs formed from double He-rich stars at
+solar metallicity will not be merged within a Hubble time (see black triangles) when compared with low-metallicity models. We
+note the trend that the observed BBHs with higher values of both χeff and Mchirp, can be better explained in our modeling at lower
+metallicites. In particular, the fraction of the BBHs, which have χeff higher than that of GW190517 (it has the highest χeff reported
+in the LVK), is 8.9% at 1.0 Z⊙, 20.3% at 0.1 Z⊙, and 26.9% at 0.01 Z⊙ (For TS off: 18.1% at 1.0 Z⊙, 23.2% at 0.1 Z⊙, and 28.7%
+at 0.01 Z⊙), respectively. GW190521 was reported from the LVK to have the highest Mchirp (Abbott et al. 2020), which might be a
+straddling binary using a population informed prior (Fishbach & Holz 2020). This event is an outlier in our models, as the upper
+limit of the BH mass in this study is assumed not to be higher than ∼ 65 M⊙ due to (pulsational) pair-instability supernovae (see
+discussion in the next section). Additionally, GW190517−055101 has the largest χeff reported in the second Gravitational-Wave
+Transient Catalog (GWTC-2) (Abbott et al. 2021), which can be explained with our models at lower metallicities (see the second
+and third rows in Fig. 7), regardless of the assumed efficiencies of AM transport. Therefore, the BBH progenitor of this event might
+have gone through the double-core evolution at low metallicities (e.g., Z < 0.1 Z⊙.)
+4. Conclusions and Discussion
+In this work, we first present the Hertzsprung-Russel diagrams of single non-rotating He-rich stars in a mass range of 5 - 60 M⊙ at
+different metallicities, evolving from ZAHeMS to the central helium exhaustion. We then systematically study an alternative forma-
+tion scenario of BBHs (i.e., the double-core evolution) by modeling double He-rich stars in various parameter spaces (metallicity
+and initial mass of He-rich stars, as well as the orbital period). Furthermore, we also investigate the impact of the different AM
+transport mechanism on the evolution of He-rich stars in different evolutionary stages, the properties of resulting BBHs at birth, as
+well as the merging timescale.
+Article number, page 7 of 16
+
+A&A proofs: manuscript no. main
+101
+MZamsHe[M ]
+10 1
+100
+Pinit. [day]
+1.0 Z
+101
+MZamsHe[M ]
+0.1 Z
+101
+MZamsHe[M ]
+TS on
+0.01 Z
+10
+20
+40
+MZamsHe[M ]
+10 1
+100
+Pinit. [day]
+10
+20
+40
+MZamsHe[M ]
+10
+20
+40
+MZamsHe[M ]
+TS off
+10
+20
+30
+40
+50
+60
+MBH[M ]
+Initial Overflow
+L2 Overflow
+NS
+Fig. 4. BH mass MBH as a function of the He-rich star’s initial mass and orbital period, are marked with the color of the filled circles while gray
+squares represent that the compact objects formed through direct core-collapse of the He-rich star are NSs. Left column: 1.0 Z⊙, middle column:
+0.1 Z⊙, right column: 0.01 Z⊙. Top row: TS on; bottom row: TS off. The cross symbols represent the systems overflowing their Roche lobes at their
+initial models, while the plus symbols refer to the systems overflowing the second Legrangian point (L2) at the initial models.
+We calculate the baryonic remnant mass following the “delayed” supernova prescription shown in Fryer et al. (2012), and
+taking into account the impact of accretion feedback onto the newly-formed BHs. The upper limit of the BH mass is around 26,
+40 and 58 M⊙ at 1.0 Z⊙, 0.1 Z⊙ and 0.01 Z⊙, respectively. We find that tides for double He-rich stars can only be important when
+the initial orbital periods are less than 1.0 day, which is similar to previous studies of a He-rich star accompanied by a BH/NS
+(Qin et al. 2018; Bavera et al. 2020; Fuller & Lu 2022). We note that the initial metallicity of He-rich stars should be high for the
+efficient AM transport to play a significant role in determining the spin magnitude of the newly-formed BHs, since their progenitors
+(massive He-rich He-stars) are more inclined to be slowed down by stronger winds mass-loss especially when rotating like a solid
+body. The χeff for BBHs formed through the double-core evolution is not always high, but it can cover the whole range of BH spin,
+i.e., from minimum (0.0) to maximum (1.0), depending on the initial orbital period of the binary systems. The chirp mass Mchirp
+of BBH is strongly dependent on the initial metallicity of He-rich stars (e.g., Belczynski et al. 2010; Stevenson et al. 2017, 2019).
+More specifically, the chirp mass Mchirp of the BBH from double-core evolution at 1.0 Z⊙ can not be larger than 26 M⊙, regardless
+of the efficiency of the AM transport within He-rich stars.
+After detailed investigations of the double-core evolution, we would expect that this channel could predict a certain fraction of
+BBH populations with high χeff and Mchirp. More events with the above features are expected to be captured by the LVK with its
+improving sensitivity in the upcoming fourth observing run. The quantitative merger rate from this channel is beyond the scope of
+the current work. Therefore we plan to investigate the quantitative contribution of this channel to the intrinsic BBH population with
+the population synthesis study and the impact of different physical processes on the outcomes in the near future.
+The formation of massive He-rich binary stars might not involve the CE phase, in which the criteria for its occurrence are
+still under development. Recent investigations suggest that the BBHs merger rate from the CE channel might be overestimated in
+rapid population synthesis studies (e.g., Pavlovskii et al. 2017; Marchant et al. 2021; Klencki et al. 2021; Gallegos-Garcia et al.
+2021; Olejak et al. 2021). Their studies indicate that stable mass transfer channel could be a dominant channel for the formation
+of merging BBHs (e.g., Shao & Li 2022; Briel et al. 2022). van Son et al. (2022a) recently found that stable mass transfer channel
+preferentially form BBH systems with more massive component BH masses. Furthermore, by varying the metallicity-dependent
+cosmic star formation history, van Son et al. (2022b) found the variations affect the slope of the high mass end of the BBH mass
+distribution, but have a slight impact on the CE channel. In addition, massive He-rich binary stars could be formed through the CHE
+channel. In this channel, the two massive stars initially evolve in a close orbit and thus have strong chemical mixing due to strong
+tides.
+Here we briefly summarise some main uncertainties in our binary modeling. First, stellar wind mass loss is one of key uncertain
+physical processes in the evolution of massive stars, which can have a significant impact on the mass and the spin of resultant BHs.
+Second, it is unclear whether supernova kicks (natal kicks) are imparted onto BHs during the core-collapse process. BHs formed
+from direct core-collapse of massive stars were considered to receive no natal kick (Belczynski et al. 2008). Nevertheless, we note
+Article number, page 8 of 16
+
+Y. Qin et al.: Merging binary black holes formed through double-core evolution
+Fig. 5. As in Fig. 4, but the color denotes the BH spin parameter a⋆.
+that a recent work by Farr et al. (2011); Tauris (2022) which argued that, rather than dynamical formation, isolated binary evolution
+can still explain the observed BBHs if BHs have spin-axis tossed due to the supernova kicks during their formation process in the
+core collapse of massive stars. The stellar evolution theory predicts a mass “gap” in the BH birth function caused by the (pulsational)
+pair-instability supernovae (Fowler & Hoyle 1964; Rakavy & Shaviv 1967; Barkat et al. 1967; Fraley 1968; Heger et al. 2003), which
+is still uncertain and thus plays a critical role in determining the upper limit of the BH mass below the “gap” (see Woosley & Heger
+2021, and references therein). The constraints from current observations of BBHs reported from the LVK are still weak due to a
+statistically small sample. Therefore, we expect the sample of BBH events with higher χeff and Mchirp will be significantly expanded
+in the upcoming fourth run, which will be used to make stronger constraints on the supernova kicks during the formation process of
+BHs from massive stars.
+Acknowledgements. Y.Q. acknowledges the support from the Doctoral research start-up funding of Anhui Normal University and from Key Laboratory for Rel-
+ativistic Astrophysics in Guangxi University. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12003002, 12192220,
+12192221, 11863003, 12173010) and the Natural Science Foundation of Universities in Anhui Province (Grant No. KJ2021A0106). G.M. has received funding
+from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 833925, project
+STAREX). All figures were made with the free Python module Matplotlib (Hunter 2007).
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+Article number, page 12 of 16
+
+Y. Qin et al.: Merging binary black holes formed through double-core evolution
+Appendix A: Direct core collapse with mass and angular momentum conserved
+In this section, we present the results of BBH formation through double-core evolution channel, assuming that the mass and AM are
+conserved during the formation process in the core collapse of He-rich stars. As shown earlier, tides can be only important for tidal
+interaction of double He-rich stars if the initial orbital periods are less 1 days. We first show in Fig. A.1 the evolution of three cases
+(MZamsHe = 12, 20, and 40 M⊙ for the same Pinit. = 0.6 days) from the beginning of core helium burning to their carbon depletion in
+the center. We adopt efficient (TS on) and inefficient (TS off) AM transport within He-rich stars and three initial metallicities (1.0
+Z⊙, 0.1 Z⊙ and 0.01 Z⊙).
+We show in the top left panel of Fig. A.1 the evolution of BH spin as a function of the He-rich star mass and its orbital period,
+under the assumption that He-rich stars at any time can directly collapse to form BHs without losing mass and corresponding AM.
+Let us begin with a case of an 40 M⊙ double He-rich stars, it was efficiently spun up and thus formed a fast-spinning BH at the
+beginning of core helium burning (see the star symbol). The orbital separation slightly expands during the core helium burning,
+which however makes BH spin (∼ 0.4 at the middle of core helium burning, see the square symbol) gradually decrease and end
+up with being close to zero at the central helium depletion. We note that there is a negligible discrepancy of BH spin calculated at
+between the central helium depletion (the triangle symbol) and the central carbon depletion (the circle symbol). The other two cases
+could form lower-mass binary BHs being slowly rotating in a closer binaries due to weaker winds mass loss. At lower metallcities,
+the same binaries will form faster-spinning BHs in shorter orbits (see middle left panel for 0.1 Z⊙ and bottom left panel for 0.01
+Z⊙). When the inefficient AM transport (TS off) is adopted, we can clearly see the formed BBHs spinning faster when compared
+with the same metallicity.
+With the same parameter space, we also compute for each binary system the evolution of the BH spin under different metallicities
+and efficiencies of AM transport. We first present results assuming efficient AM transport. As shown in Fig. A.2, all He-rich stars
+with an initial mass of less than 12 M⊙, at 1.0 Z⊙, form NSs. He-rich stars with initial orbital period of longer than 1.0 day end up
+with being non-spinning BHs. We find that BHs can have moderate spin magnitudes with Pinit. in a range of 0.3 - 1.0 days, below
+which fast-spinning BHs are formed. Similar to Fig. A.2, we can see for lower metallicities (see Fig. A.2 and Fig. A.3) that the
+formed BHs with spins decreasing as the orbit slowly expands. We then show the results with different metallicities of the inefficient
+AM transport in Fig. A.5, Fig. A.6 and Fig. A.7. The mass and spin of the newly-formed BHs calculated using direct core-collapse
+with mass and AM conserved are slightly larger when compared with those by taking into account the accretion feedback during
+core-collapse modeling (see details in Batta & Ramirez-Ruiz 2019).
+Fig. A.1. Evolution of the spin parameter a⋆ as a function of the orbital period and mass of He-rich stars at different metallicities and AM transport
+mechanisms. Top left panel: TS on and 1.0 Z⊙, middle left panel: TS on and 0.1 Z⊙, bottom left panel: TS on and 0.01 Z⊙; Top right panel: TS
+off and 1.0 Z⊙, middle right panel: TS off and 0.1 Z⊙, bottom right panel: TS off and 0.01 Z⊙. The spin a⋆ at different evolutionary stages are
+marked with symbols, star: beginning of core He burning, square: middle of core He burning, triangle: central He depletion, circle: central carbon
+depletion.
+Article number, page 13 of 16
+
+a*
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+☆ Beginning of Core He burning
+ Middle of Core He burning
+ Central He Depletion
+o Central Carbon Depletion
+2
+TS off, 1.0 Zo
+TS on, 1.0 Zo
+2.0
+2.0
+Porb[day]
+[day]
+1.0
+1.0
+☆一
+☆
+0.6
+0.6 
+10
+20
+40
+10
+20
+40
+TS off, 0.1 Zo
+TS on, 0.1 Zo
+1.0
+1.0
+P
+P
+0.7
+0.7
+10
+20
+40
+10
+20
+40
+0.7
+0.7
+TS on, 0.01 Zo
+TS off, 0.01 Zo
+0.68
+0.68
+[day]
+[day]
+?
+0.66
+?
+0.66
+Porbl
+0.64
+0.64
+0★
+中
+0.62
+0.62
+10
+20
+40
+10
+20
+40
+MHe[Mo]
+MHe[Mo]A&A proofs: manuscript no. main
+Fig. A.2. Evolution of the spin parameter a⋆ (the color bar) as a function of the orbital period and mass of He-rich stars with TS included at solar
+metallicity. The colored lines linking the two symbols show the evolution of the binary. The color along the line gives BH spins a⋆ along the
+evolution (from the ZAHeMS to the central carbon depletion), assuming that their progenitors (He-rich stars) directly collapse to form
+BHs with mass and AM conserved.
+Fig. A.3. As in Fig . A.2, but for the metallicity Z = 0.1 Z⊙.
+Article number, page 14 of 16
+
+a*
+0.4
+0.6
+0.2
+0.8
+1.0
+TS on, 0.1 Zo
+X Initial Overflow
+NS
++ L2 Overflow
+10
+. [day]
+Porb.
+100
+C
+X
+X
++
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+10-
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
+X
+X
+X
+X
+X
+X
+X
++
++
++
+X
+X
+40
+5
+10
+20
+MHe[Mo]a*
+0.4
+0.2
+0.6
+0.8
+1.0
+TS on, 1.0 Zo
+NS
+Initial Overflow
+X
++ L2 Overflow
+101
+Porb. [day]
+100
+X
+X
+G
+G
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
++
++
+10-
+X
+X
+X
+X
++
++
++
+X
+X
+5
+10
+20
+40
+MHe[Mo]Y. Qin et al.: Merging binary black holes formed through double-core evolution
+Fig. A.4. As in Fig . A.2, but for the metallicity Z = 0.01 Z⊙.
+Fig. A.5. As in Fig . A.2, but without TS included.
+Article number, page 15 of 16
+
+a*
+0.4
+0.6
+0.8
+0.2
+1.0
+TS on, 0.01 Zo
+NS
+ Initial Overflow
++
+L2 Overflow
+101
+口
+口
+口
+口
+口
+口
+口
+口
+[day]
+C
+C
+C
+100
+Q
+Porb.
+G
+C
+C
+口
+口
+6
+6
+6
+口
+X
+X
+X
+6
+6
+口
+G
+X
+X
+X
+X
+X
+5
+X
+X
+X
+X
+10-1
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
++
++
++
+5
+10
+20
+40
+MHe[Mo]a*
+0.4
+0.2
+0.6
+0.8
+1.0
+TS off, 1.0 Zo
+NS
+Initial Overflow
+X
++ L2 Overflow
+101
+Porb. [day]
+100
+X
+X
+G
+G
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
++
++
+10-
+X
+X
+X
+X
++
++
++
+5
+10
+20
+40
+MHe[Mo]A&A proofs: manuscript no. main
+Fig. A.6. As in Fig . A.5, but for the metallicity Z = 0.1 Z⊙.
+Fig. A.7. As in Fig . A.5, but for the metallicity Z = 0.01 Z⊙.
+Article number, page 16 of 16
+
+a*
+0.4
+0.6
+0.2
+0.8
+1.0
+TS off, 0.1 Zo
+X Initial Overflow
+NS
++ L2 Overflow
+10
+. [day]
+Porb.
+100
+C
+X
+=
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+10-
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
+X
+X
+X
+X
+X
+X
+X
++
++
++
+X
+X
+40
+5
+10
+20
+MHe[Mo]a*
+0.4
+0.6
+0.8
+0.2
+1.0
+TS off, 0.01 Zo
+NS
+ Initial Overflow
++
+ L2 Overflow
+101
+口
+口
+口
+口
+口
+口
+口
+[day]
+C
+C
+C
+C
+100
+Porb.
+口
+6
+6
+6
+6
+X
+口
+6
+6
+口
+6
+5
+X
+X
+X
+X
+口
+6
+X
+X
+X
+X
+X
+5
+X
+X
+X
+X
+10-1
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
+X
++
++
++
++
+5
+10
+20
+40
+MHe[Mo]

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+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+Yuhao Zhang 1 Aws Albarghouthi 1 Loris D’Antoni 1
+Abstract
+Neural networks are vulnerable to backdoor poi-
+soning attacks, where the attackers maliciously
+poison the training set and insert triggers into the
+test input to change the prediction of the victim
+model. Existing defenses for backdoor attacks
+either provide no formal guarantees or come with
+expensive-to-compute and ineffective probabilis-
+tic guarantees. We present PECAN, an efficient
+and certified approach for defending against back-
+door attacks. The key insight powering PECAN
+is to apply off-the-shelf test-time evasion certi-
+fication techniques on a set of neural networks
+trained on disjoint partitions of the data. We
+evaluate PECAN on image classification and mal-
+ware detection datasets. Our results demonstrate
+that PECAN can (1) significantly outperform the
+state-of-the-art certified backdoor defense, both
+in defense strength and efficiency, and (2) on real
+backdoor attacks, PECAN can reduce attack suc-
+cess rate by order of magnitude when compared
+to a range of baselines from the literature.
+1. Introduction
+Deep learning models are vulnerable to backdoor poisoning
+attacks (Saha et al., 2020; Turner et al., 2019), which assume
+that the attackers can maliciously poison a small fragment
+of the training set before model training and add triggers to
+inputs at test time. As a result, the prediction of the victim
+model that was trained on the poisoned training set will
+diverge in the presence of a trigger in the test input.
+Effective backdoor attacks have been proposed for various
+domains, such as image recognition (Gu et al., 2017), senti-
+ment analysis (Qi et al., 2021), and malware detection (Sev-
+eri et al., 2021). For example, Severi et al. (2021) can break
+malware detection models as follows: The attacker poisons
+a small portion of benign software in the training set by
+modifying the values of the most important features so that
+1Department of Computer Science, University of Wisconsin-
+Madison, Madison, USA. Correspondence to: Yuhao Zhang
+<yuhaoz@cs.wisc.edu>.
+the victim model recognizes these values as evidence of the
+benign prediction. At test time, the attacker inserts a trig-
+ger by changing the corresponding features of malware to
+camouflage it as benign software and thus making it bypass
+the examination of the victim model. Thus, backdoor at-
+tacks are of great concern to the safety and security of deep
+learning models and systems, particularly as training data is
+gathered from different sources, e.g., via web scraping.
+Several works have studied defenses against various types
+of attacks. We identify two limitations with these defenses.
+First, many existing approaches only provide empirical
+defenses that are specific to certain attacks and do not
+generalize to all backdoor attacks. Second, existing cer-
+tified defenses—i.e., approaches that produce robustness
+certificates—are either unable to handle backdoor attacks,
+or are probabilistic (instead of deterministic), and therefore
+expensive and ineffective in practice.
+Why certification?
+A defense against backdoor attacks
+should construct effective certificates (proofs) that the
+learned model can indeed defend against backdoor attacks.
+Empirical defenses (Geiping et al., 2021a; Liu et al., 2018)
+do not provide certificates, can only defend against specific
+attacks, and can be bypassed by new unaccounted-for at-
+tacks (Wang et al., 2020b; Koh et al., 2022). Certification
+has been successful at building models that are provably
+robust to trigger-less poisoning attacks and evasion attacks,
+but models trained to withstand such attacks are still weak
+against backdoor attacks. The trigger-less attack (Zhu et al.,
+2019; Shafahi et al., 2018; Aghakhani et al., 2021; Geiping
+et al., 2021b) assumes the attacker can poison the training set
+but cannot modify the test inputs, e.g., adding triggers, while
+the evasion attack (Madry et al., 2018) assumes the attacker
+modifies the test inputs but cannot poison the training set.
+Existing certified defenses against trigger-less and evasion
+attacks, e.g., DPA (Levine & Feizi, 2021) and CROWN-
+IBP (Zhang et al., 2020), cannot defend against backdoor
+attacks as they can either defend against the poison in the
+training data or the triggers at test time, but not both. As
+we show in the experiments, we can break these certified
+defenses using a backdoor attack (Section 5.3).
+Why determinism?
+It is desirable for a certified defense
+to be deterministic because probabilistic defenses (Zhang
+et al., 2022b; Weber et al., 2020) typically require one to
+arXiv:2301.11824v1  [cs.CR]  27 Jan 2023
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+D
+D1
+D2
+...
+Dn
+Step 1: Dataset Partitioning
+AD1
+AD2
+...
+ADn
+x
+7
+cert
+1
+abstain
+...
+...
+7
+cert
+Step 2: Evasion Certification
+Step 3: Aggregation
+y∗: the top label, y′: runner-up label,
+N1: # of certified y∗, N2: # of certified y′,
+N3: # of abstain.
+y∗: prediction of PECAN,
+⌊
+N1−N2−N3−1y∗>y′
+2
+⌋: certified radius.
+Training
+Testing
+Figure 1. An overview of our approach PECAN.
+retrain thousands of models when performing predictions
+for a single test input. Retraining can be mitigated by Bon-
+ferroni correction, which allows reusing the trained models
+for a fixed number of predictions. However, retraining is
+still necessary after a short period, making it hard to deploy
+these defenses in practice. On the other hand, determin-
+istic defenses (Levine & Feizi, 2021; Wang et al., 2022b)
+can reuse the trained models an arbitrary number of times
+when producing certificates for different test inputs. Fur-
+thermore, probabilistic defenses for backdoor attacks, e.g.,
+BagFlip (Zhang et al., 2022b), need to add noise to the train-
+ing data, resulting in low accuracy for datasets that cannot
+tolerate too much noise when training (Section 5.2).
+PECAN
+In this paper, we propose PECAN (Partitioning
+data and Ensembling of Certified neurAl Networks), a de-
+terministic certified defense against backdoor attacks for
+neural networks. The key insight underlying PECAN is that
+we can take any off-the-shelf technique for evasion certifi-
+cation and use it to construct a certified backdoor defense.
+This insight results in a simple implementation and allows
+us to seamlessly leverage future advances in evasion cer-
+tification algorithms. Specifically, PECAN trains a set of
+neural networks on disjoint partitions of the dataset, and
+then applies evasion certification to the neural networks. By
+partitioning the dataset, we analytically bound the number
+of poisoned data seen per neural network; by employing eva-
+sion certification, we bound the number of neural networks
+that are robust in the face of triggers. Using this information,
+we efficiently derive a backdoor-robustness guarantee.
+Figure 1 illustrates the workflow of PECAN. In Step 1, in-
+spired by deep partition aggregation (Levine & Feizi, 2021),
+PECAN deterministically partitions a dataset into multiple
+disjoint subsets. This step ensures that a poisoned data item
+only affects a single partition. In Step 2, PECAN trains an
+ensemble of neural networks, one on each partition. At test
+time, PECAN performs evasion certification to check which
+neural networks are immune to triggers; those that are not
+immune (or that cannot be proven immune) abstain from
+performing a prediction. Finally, in Step 3, PECAN aggre-
+gates the results of the ensemble and produces a prediction
+together with a robustness certificate: the percentage of the
+poisoned data in the training set that the training process
+can tolerate, the certified radius.
+We evaluate PECAN on two three datasets, MNIST, CI-
+FAR10, and EMBER. First, we show that PECAN outper-
+forms or competes with BagFlip, the state-of-the-art prob-
+abilistic certified defense against backdoor attacks. Fur-
+thermore, BagFlip takes hours to compute the certificate,
+while PECAN only takes a few seconds. Second, when
+we evaluate PECAN against a concrete known backdoor
+attack (Severi et al., 2021), PECAN reduces the attack suc-
+cess rate to 1.85%, while DPA and CROWN-IBP fail to
+defend against the backdoor attack on 18.05% and 15.24%
+of the cases, respectively. The results show that PECAN
+can defend against a known backdoor attack while other
+baselines, such as DPA and CROWN-IBP, cannot.
+2. Related Work
+Deep learning models are vulnerable to backdoor at-
+tacks (Saha et al., 2020; Turner et al., 2019). Although
+many empirical defenses (Geiping et al., 2021a; Liu et al.,
+2018) have been proposed, recent works (Wang et al., 2020b;
+Koh et al., 2022) show that new attacks can break these em-
+pirical defenses. Therefore, certified defense is crucial for
+defending against backdoor attacks.
+Certified defenses against backdoor attacks
+Existing
+certification approaches provide probabilistic certificates by
+extending randomized smoothing (Cohen et al., 2019; Dvi-
+jotham et al., 2020; Lee et al., 2019), originally proposed to
+defend against adversarial evasion attacks, to defend against
+backdoor attacks. BagFlip (Zhang et al., 2022b) is the
+state-of-the-art model-agnostic probabilistic defense against
+feature-flipping backdoor attacks. Wang et al. (2020a); We-
+ber et al. (2020) proposed backdoor-attack defenses that
+are also model-agnostic, but are less effective than BagFlip.
+PECAN is deterministic and therefore less expensive and
+more effective than these defenses. Probabilistic defenses
+are model-agnostic; while PECAN is evaluated on neural
+networks, it can work for any machine learning model as
+long as a deterministic evasion certification approach of the
+model is available. Weber et al. (2020) proposed a determin-
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+istic de-randomized smoothing approach for kNN classifiers.
+Their approach computes the certificates using an expen-
+sive dynamic programming algorithm, whereas PECAN’s
+certification algorithm has constant time complexity.
+Certified defenses against trigger-less attacks
+Many
+approaches provide certificates for trigger-less attacks. Jia
+et al. (2021) use bootstrap aggregating (Bagging). Chen
+et al. (2020) extended Bagging with new selection strate-
+gies. Rosenfeld et al. (2020) defend against label-flipping
+attacks on linear classifiers. Differential privacy (Ma et al.,
+2019) can also provide probabilistic certificates for trigger-
+less attacks. DPA (Levine & Feizi, 2021) is a deterministic
+defense that partitions the training set and ensembles the
+trained classifiers. Wang et al. (2022b) proposed FA, an
+extension of DPA, by introducing a spread stage. A conjec-
+ture proposed by Wang et al. (2022a) implies that DPA and
+FA are asymptotically optimal defenses against trigger-less
+attacks. Chen et al. (2022) proposed to compute collective
+certificates, while PECAN computes sample-wise certifi-
+cates. Jia et al. (2020); Meyer et al. (2021); Drews et al.
+(2020) provide certificates for nearest neighborhood classi-
+fiers and decision trees. The approaches listed above only
+defend against trigger-less attacks, while PECAN is a deter-
+ministic approach for backdoor attacks.
+Certified defenses against evasion attacks
+There are
+two lines of certified defense against evasion attacks: com-
+plete certification (Wang et al., 2021; Zhang et al., 2022a;
+Katz et al., 2019) and incomplete certification (Xu et al.,
+2020; Zhang et al., 2021; Singh et al., 2019). The com-
+plete certified defenses either find an adversarial example
+or generate proof that all inputs in the given perturbation
+space will be correctly classified. Compared to the complete
+certified defenses, the incomplete ones will abstain from pre-
+dicting if they cannot prove the correctness of the prediction
+because their techniques will introduce over-approximation.
+The complete approaches do not have over-approximation
+issues but require expensive verification algorithms such as
+branch and bound. Our implementation of PECAN uses an
+incomplete certified approach CROWN-IBP (Zhang et al.,
+2020) because it is the best incomplete approach, trading off
+between efficiency and the degree of over-approximation.
+3. Problem Definition
+Given a dataset D = {(x1, y1), . . . , (xn, yn)}, a (test) input
+x, and a machine learning algorithm A, we write AD to
+denote the machine learning model learned on dataset D
+by the algorithm A, and AD(x) to denote the output label
+predicted by the model AD on input x. We assume the
+algorithm will behave the same if trained on the same dataset
+across multiple runs. This assumption can be guaranteed by
+fixing the random seeds during training.
+We are interested in certifying that if an attacker has poi-
+soned the dataset, the model we have trained on the dataset
+will still behave “well” on the test input with maliciously
+added triggers. Before describing what “well” means, we
+need to define the perturbation spaces of the dataset and
+the test input, i.e., what possible changes the attacker could
+make to the dataset and the test input.
+Perturbation space of the dataset
+Following Levine &
+Feizi (2021), we define a general perturbation space over
+the dataset, allowing attackers to delete, insert, or modify
+training examples in the dataset. Given a dataset D and a
+radius r ≥ 0, we define the perturbation space as the set of
+datasets that can be obtained by deleting or inserting up to r
+examples in D:
+Sr(D) =
+�
+�D | |D ⊖ �D| ≤ r
+�
+,
+where A ⊖ B is the symmetric difference of sets A and
+B. Intuitively, r quantifies how many examples need to be
+deleted or inserted to transform from D to �D.
+Example 3.1. If the attacker modifies one training example
+x ∈ D to another training example �x to form a poisoned
+dataset �D = (D \ {x}) ∪ {�x}. Then �D ∈ S2(D) but
+�D /∈ S1(D) because Sr(D) considers one modification as
+one deletion and one insertion.
+Note that we assume a more general perturbation space
+of the training set than the one considered by Zhang et al.
+(2022b); Weber et al. (2020); Wang et al. (2020a); our work
+allows inserting and deleting examples instead of just modi-
+fying existing training examples.
+Perturbation space of the test input
+We write π(x) to
+denote the set of perturbed examples that an attacker can
+transform the example x into. Formally, the perturbation
+space π(x) can be defined as the lp norm ball with radius s
+around the test input x,
+π(x) = {�x | ∥x − �x∥p ≤ s}
+Example 3.2. BagFlip (Zhang et al., 2022b) considers the
+l0 feature-flip perturbation Fs(x), which allows the attacker
+to modify up to s features in an input x,
+Fs(x) = {�x | ∥x − �x∥0 ≤ s}
+Threat models
+Next, we define what type of guarantees
+we are interested in our learning algorithm and model. We
+consider backdoor attacks, where the attacker can perturb
+both the training set and the test input. For the training set,
+we assume we are given a perturbation space Sr(D) of the
+training set D with a radius r ≥ 0. For the test input, we
+assume a perturbation space π(x) of the test input x with a
+given lp norm and the radius s.
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+We say that an algorithm A is robust to a backdoor attack
+on a backdoored test input �x if the algorithm trained on any
+perturbed dataset �D would predict the backdoored input �x
+the same as AD(x). Formally,
+∀ �D ∈ Sr(D), �x ∈ π(x). A �
+D(�x) = AD(x)
+(1)
+Remark 3.1. When r = 0, Eq 1 degenerates to evasion
+robustness, i.e., ∀�x ∈ π(x). AD(�x) = AD(x), because
+S0(D) = {D}.
+Given a large enough radius r, an attacker can always
+change enough inputs and succeed at breaking robustness.
+Therefore, we will typically focus on computing the max-
+imal radius r for which we can prove that Eq 1 for given
+perturbation spaces Sr(D) and π(x). We refer to this quan-
+tity as the certified radius.
+Certified guarantees
+This paper aims to design a cer-
+tifiable algorithm A, which can defend against backdoor
+attacks, and to compute the certified radius of A. In our ex-
+periments (Section 5.2), we suppose a given benign dataset
+D and a benign test input x, and we certifiably quantify the
+robustness of the algorithm A against backdoor attacks by
+computing the certified radius.
+In Section 5.3, we also experiment with how the certifiable
+algorithm A defends the backdoor attacks if a poisoned
+dataset �D and a test input �x with malicious triggers are
+given, but the clean data is unknown. We theoretically show
+that we can still compute the certified radius if the clean
+data D and x are unknown in Section 4.3.
+4. The PECAN Certification Technique
+Our approach, which we call PECAN (Partitioning data and
+Ensembling of Certified neurAl Networks), is a determin-
+istic certification technique that defends against backdoor
+attacks. Given a learning algorithm A, we show how to
+automatically construct a new learning algorithm ¯A with
+certified backdoor-robustness guarantees (Equation (1)) in
+Section 4.1. In Section 4.2, we prove the certified backdoor-
+robustness guarantees (Equation (1)) provided by ¯A. We
+further discuss how ¯A can defend against a backdoored
+dataset and formally justify our discussion in Section 4.3.
+4.1. Constructing Certifiable Algorithm ¯A
+The key idea of PECAN is that we can take any off-the-shelf
+technique for evasion certification and use it to construct
+a certified backdoor defense. Intuitively, PECAN uses the
+evasion certification to defend against the possible triggers
+at test time, and it encapsulates the evasion certification in
+deep partition aggregation (DPA) (Levine & Feizi, 2021) to
+defend against training set poisoning.
+Given a dataset D, a test input x, and a machine learning
+algorithm A, PECAN produce a new learning algorithm ¯A
+as described in the following steps (shown in Figure 1),
+Dataset Partitioning
+We partition the dataset D into n
+disjoint sub-datasets, denoted as D1, . . . , Dn, using a hash
+function that deterministically maps each training example
+into a sub-dataset Di. Train n classifiers AD1, . . . , ADn on
+these sub-datasets.
+Evasion Certification
+We certify whether the prediction
+of each classifier ADi is robust under the perturbation space
+π(x) by any evasion certification approach for the learn-
+ing algorithm, e.g., CROWN-IBP for neural networks (Xu
+et al., 2020). Formally, the certification approach determines
+whether the following equation holds,
+∀�x ∈ π(x). ADi(x) = ADi(�x)
+(2)
+We denote the output of each certification as Aπ
+Di(x), which
+can either be Aπ
+Di(x) = cert, meaning Eq 2 is certified.
+Otherwise, Aπ
+Di(x) = abstain, meaning the certification
+approach cannot certify Eq 2.
+Aggregation
+We compute the top label y∗ by aggre-
+gating all predictions from ADi(x).
+Concretely, y∗ ≜
+argmax
+y∈C
+�n
+i=1 1ADi(x)=y, where C = {0, 1, . . .} is the set
+of possible labels. Note that if a tie happens when taking
+the argmax, we break ties deterministically by setting the
+smaller label index as y∗. We denote the runner-up label
+as y′ as argmax
+y∈C∧y̸=y∗
+�n
+i=1 1ADi(x)=y. We count the number
+of certified predictions equal to y∗ as N1, the number of
+certified predictions equal to y′ as N2, and the number of
+abstentions as N3,
+N1 =
+n
+�
+i=1
+1ADi(x)=y∗∧Aπ
+Di(x)=cert,
+N2 =
+n
+�
+i=1
+1ADi(x)=y′∧Aπ
+Di(x)=cert,
+N3 =
+n
+�
+i=1
+1Aπ
+Di(x)=abstain.
+We set the prediction ¯AD(x) as y∗. We compute the cer-
+tified radius r in the following two cases. If N1 − N2 −
+N3 − 1y∗>y′ < 0, we set r as ⋄, i.e., a value denoting no
+certification. In this case, PECAN cannot certify that ¯A is
+robust to evasion attacks even if the dataset is not poisoned.
+Otherwise, we compute r as ⌊
+N1−N2−N3−1y∗>y′
+2
+⌋. A spe-
+cial case is r = 0, when PECAN can certify ¯A is robust
+to evasion attacks, but cannot certify that it is robust if the
+dataset is poisoned.
+We note that the computation of the certified radius is equiv-
+alent to DPA when no classifier abstains, i.e., N3 = 0,
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+D
+�
+D
+x
+�x
+D1
+D2
+...
+...
+...
+Dn
+�
+D1
+�
+D2
+...
+...
+...
+�
+Dn
+7
+abstain
+5
+abstain
+7
+cert
+7
+cert
+7
+cert
+1
+cert
+7 → y′
+5 → y′
+7 → y′
+7 → y′
+7
+1
+Dabs
+Attacked by �x
+Dbd
+Attacked by �
+D
+Dsafe
+Clean
+Figure 2. An illustration of the proof of Theorem 4.1. It shows
+the worst case for PECAN, where the attacker can change all
+predictions in Dabs and Dbd to the runner-up label y′. Note
+that we group Dabs, Dbd, and Dsafe together to ease illustration.
+4.2. Proving the Soundness of PECAN
+In this section, we show that the prediction ¯AD(x) and the
+certified radius r satisfy the certified backdoor-robustness
+guarantees (Equation (1)) by proving the following theorem.
+Theorem 4.1 (Soundness of PECAN). Given a dataset D
+and a test input x, PECAN computes the prediction ¯AD(x)
+and the certified radius as r. Then, either r = ⋄ or
+∀ �D ∈ Sr(D), �x ∈ π(x). ¯A �
+D(�x) = ¯AD(x)
+(3)
+Proof. For any poisoned dataset �D, we partition �D into
+n sub-datasets { �D1, . . . , �Dn} according to {D1, . . . , Dn}
+from the clean dataset D. Note that we can determine such
+a correspondence between Di and �Di because our hash
+function is deterministic and only depends on each train-
+ing example. We further divide {D1, . . . , Dn} into three
+disjoint parts Dabs, Dbd, and Dsafe in the following way,
+• Dabs = {Di | Aπ
+Di(x) = abstain} are the sub-
+datasets, on which A abstains from making the pre-
+diction on x. From the definition of N3, we have
+|Dabs| = N3.
+Intuitively, Dabs contains the sub-
+datasets that can possibly be attacked by the test input
+�x with malicious triggers.
+• Dbd are the sub-datasets on which A does not abstain
+and are also poisoned, i.e., each of them has at least one
+training example removed or inserted. Even though we
+do not know the exact sub-datasets in Dbd, we know
+|Dbd| ≤ r because �D ∈ Sr(D) constrains that there
+are at most r such poisoned sub-datasets.
+• Dsafe = {Di | Di = �Di ∧ Aπ
+Di(x) = cert} contains
+the clean sub-datasets, on which A does not abstain.
+We denote the numbers of the original top prediction y∗ and
+the original runner-up prediction y′ on the backdoored data
+�D and �x as �
+Ny∗ and �
+Ny′, respectively. Formally,
+�
+Ny∗ =
+n
+�
+i=1
+1A�
+Di(�x)=y∗,
+�
+Ny′ =
+n
+�
+i=1
+1A�
+Di(�x)=y′
+Next, we prove Eq 3 for any backdoored data �D and �x by
+showing that
+�
+Ny∗ ≥ �
+Ny′ + 1y∗>y′
+(4)
+We prove Eq 4 by showing a lower bound of �
+Ny∗ is N1 − r
+and an upper bound of �
+Ny′ is N2 + r + N3. Together with
+the definition of r, we can prove Eq 4 because we have,
+�
+Ny∗ − �
+Ny′ − 1y∗>y′
+≥N1 − r − (N2 + r + N3) − 1y∗>y′
+=N1 − N2 − 2r − N3 − 1y∗>y′
+=N1 − N2 − 2⌊N1 − N2 − N3 − 1y∗>y′
+2
+⌋ − N3 − 1y∗>y′
+≥N1 − N2 − (N1 − N2 − N3 − 1y∗>y′) − N3 − 1y∗>y′
+=0.
+Note that the second last line holds iff N1 − N2 − N3 −
+1y∗>y′ ≥ 0. Otherwise, we have r = ⋄.
+As shown in Figure 2, the lower bound of �
+Ny∗ can be com-
+puted by noticing that 1) the attacker can change any predic-
+tion in Dbd from y∗ to another label because these datasets
+are poisoned, 2) the attacker can change any prediction in
+Dabs to another label because CROWN-IBP cannot certify
+the prediction under the evasion attacks, and 3) the attacker
+cannot change anything in Dsafe because of the guarantee
+of CROWN-IBP and Dsafe is not poisoned,
+∀Di ∈ Dsafe, �x ∈ π(x). ADi(x) = ADi(�x) = A �
+Di(�x)
+The upper bound of �
+Ny′ can be computed by noticing that
+1) the attacker can change any prediction in Dbd to y′, 2)
+the attacker can change any prediction in Dabs to y′, and 3)
+the attacker cannot change anything in Dsafe.
+We complete the proof by showing that the best attack strat-
+egy of the attacker is to change the prediction of ¯A to the
+runner-up label y′. If the attacker chooses to change the
+prediction of ¯A to another label y′′, denoted the counts as
+�
+Ny′′, then the upper bound of �
+Ny′′ will be always smaller
+or equal to �
+Ny′.
+4.3. PECAN under the Backdoored Data
+The above algorithm and proof of PECAN assume that a
+clean dataset D and a clean test example x are already given.
+However, we may be interested in another scenario where
+the poisoned dataset �D ∈ Sr(D) and the input example
+
+Pattern BackdoorPECAN: A Deterministic Certified Defense Against Backdoor Attacks
+�x ∈ π(x) with malicious triggers are given, and the clean
+data D and x are unknown. In other words, we want to find
+the maximal radius r such that ¯A �
+D(�x) = ¯AD(x) for any D
+and x that can be perturbed to �D and �x by the perturbation
+Sr and π, respectively. Formally,
+∀D, x. �D ∈ Sr(D) ∧ �x ∈ π(x) =⇒
+¯A �
+D(�x) = ¯AD(x)
+(5)
+Intuitively, Eq 5 is the symmetrical version of Eq 1. Ow-
+ing to the symmetrical definition of Sr and π, if we apply
+PECAN to the given poisoned data �D, �x, then the predic-
+tion ¯A �
+D(�x) and the certified radius r satisfy the certified
+backdoor-robustness guarantee (Eq 5). The following the-
+orem formally states the soundness of PECAN under the
+backdoored data. We prove Theorem 4.2 in Appendix A.
+Theorem 4.2 (Soundness of PECAN under the backdoored
+data). Given a dataset �D and a test input �x, PECAN com-
+putes the prediction ¯A �
+D(�x) and the certified radius as r.
+Then, either r = ⋄ or Eq 5 holds.
+5. Experiments
+We implemented PECAN in Python and provided the im-
+plementation in the supplementary materials. In our evalua-
+tion, we use CROWN-IBP, implemented in auto-LiRPA (Xu
+et al., 2020), as the evasion defense approach for neural
+networks. We also use CROWN-IBP to train the classifiers
+in the dataset partitioning step since the classifiers trained
+by CROWN-IBP can improve the certification rate in the
+evasion certification step.
+In Section 5.2, we evaluate the effectiveness and efficiency
+of PECAN by comparing it to BagFlip (Zhang et al., 2022b),
+the state-of-the-art probabilistic certified defense against
+backdoor attacks. In Section 5.3, we evaluate the effective-
+ness of PECAN under the backdoor attack (Severi et al.,
+2021) for malware detection and compare PECAN to other
+baselines, DPA and CROWN-IBP.
+5.1. Experimental Setup
+Datasets
+We conduct experiments on MNIST, CIFAR10,
+and EMBER (Anderson & Roth, 2018) datasets. MNIST is
+an image classification dataset containing 60,000 training
+and 10,000 test examples. CIFAR10 is an image classifica-
+tion dataset containing 50,000 training and 10,000 test ex-
+amples. EMBER is a malware detection dataset containing
+600,000 training and 200,000 test examples. Each example
+is a vector containing 2,351 features of the software.
+Models
+For image classification datasets MNIST and CI-
+FAR10, we train fully-connected neural networks with four
+layers for PECAN, while BagFlip uses CNN and ResNet for
+MNIST and CIFAR10, respectively. We do not use CNN
+and ResNet because CROWN-IBP used in PECAN has a
+higher abstention rate for deeper and more complex neu-
+ral network structures. We use the same fully-connected
+neural network for EMBER as in related works (Zhang
+et al., 2022b; Severi et al., 2021). We use the same data
+augmentation for PECAN and other baselines.
+Metrics
+For each test input xi, yi, the algorithm ¯A will
+predict a label and the certified radius ri. In this section, we
+assume that the attacker had modified R% examples in the
+training set. We denote R as the modification amount. We
+summarize all the metrics used as follows,
+Certified Accuracy denotes the percentage of test examples
+that are correctly classified and whose certified radii are
+no less than R, i.e., 1
+m
+�m
+i=1 1 ¯
+AD(xi)=yi∧ ri
+|D| ≥2R%, where
+m and |D| are the sizes of test set and training set, respec-
+tively. Notice that there is a factor of 2 on the modification
+amount R because Sr(D) considers one modification as one
+insertion and one deletion, as illustrated in Example 3.1.
+Normal Accuracy denotes the percentage of test examples
+that are correctly classified by the algorithm without certifi-
+cation, i.e., 1
+m
+�m
+i=1 1 ¯
+AD(xi)=yi.
+Attack Success Rate (ASR). In Section 5.3, we are interested
+in how many test examples are certified but wrongly clas-
+sified by the classifier, i.e., 1
+m
+�m
+i=1 1 ¯
+AD(xi)̸=yi∧ ri
+|D| ≥2R%.
+We denote the above quantity as the attack success rate. We
+note that a prediction can still be incorrect even if it is cer-
+tified by PECAN because the classifier can have incorrect
+predictions even when the data is clean.
+Abstention Rate is computed as 1
+m
+�m
+i=1 1 ri
+|D| <2R%.
+5.2. Effectiveness and Efficiency of PECAN
+We evaluate the effectiveness and efficiency of PECAN on
+MNIST, CIFAR10, and EMBER under the backdoor attack
+with the l0 feature-flip perturbation F1, which allows the
+attacker to modify up to one feature in an example. We
+compare PECAN to BagFlip, the state-of-the-art probabilis-
+tic certified defense against l0 feature-flip backdoor attacks.
+Moreover, we note that PECAN needs to construct harder
+proofs than BagFlip because their definitions of perturbation
+space are different, as discussed in Appendix B.1.
+In Appendix B.2, we evaluate the effectiveness of PECAN
+against the perturbation space with the l∞ norm.
+Summary of the results
+PECAN achieves significantly
+higher certified accuracy than BagFlip on CIFAR10 and
+EMBER. PECAN achieves competitive results on MNIST
+compared to BagFlip. PECAN has similar normal accu-
+racy as BagFlip for all datasets. PECAN is more efficient
+than BagFlip at computing the certified radius.
+Setup
+We use the same hyper-parameters for BagFlip as
+reported in their paper for all datasets. For PECAN, we vary
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+0
+2
+4
+·10−2
+0
+20
+40
+60
+80
+100
+Modification Amount R (%)
+Certifiable Accuracy
+0
+0.5
+1
+1.5
+2
+·10−2
+BagFlip
+PECAN-a
+PECAN-b
+(a) CIFAR10 F1
+(b) EMBER F1
+Figure 3. Comparison to BagFlip on CIFAR10 and EMBER, show-
+ing the normal accuracy (dotted lines) and the certified accuracy
+(solid lines) at different modification amounts R. For CIFAR10:
+a = 50 and b = 100. For EMBER: a = 200 and b = 400.
+0
+0.5
+1
+1.5
+0
+20
+40
+60
+80
+100
+Modification Amount R (%)
+Certifiable Accuracy
+BagFlip
+PECAN-600
+PECAN-1200
+PECAN-2000
+Figure 4. Comparison to BagFlip on MNIST, showing the normal
+accuracy (dotted lines) and the certified accuracy (solid lines) at
+different modification amounts R.
+n, the number of partitions, to ensure a fair comparison be-
+tween BagFlip. Appendix B.1 presents a detailed discussion
+of hyper-parameter settings for BagFlip and PECAN. We
+denote PECAN with different settings of n as PECAN-n.
+BagFlip achieves meaningful results only on MNIST, where
+we also tune the parameter n of PECAN to 2000 to achieve
+the same certified accuracy of BagFlip at R = 0 and com-
+pare their results following the practice in related works (Jia
+et al., 2021; 2020).
+Results
+Figure 3 shows the comparison between PECAN
+and BagFlip on CIFAR10 and EMBER. PECAN achieves
+significantly higher certified accuracy than BagFlip
+across all modification amounts R and the similar nor-
+mal accuracy as BagFlip for both datasets.
+BagFlip performs poorly on CIFAR10 and EMBER because
+these two datasets cannot tolerate the high level of noise that
+the BagFlip algorithm adds to the training data. Specifically,
+BagFlip can add 20% noise to the training data of MNIST,
+i.e., a feature (pixel) in a training example will be flipped to
+another value with 20% probability. However, for CIFAR10
+and EMBER, this probability has to be decreased to 5% to
+maintain normal accuracy.
+Figure 4 shows the comparison between PECAN and
+BagFlip on MNIST. PECAN achieves competitive results
+compared to BagFlip. We find that two approaches have
+similar normal accuracy.
+Comparing PECAN-600 and
+PECAN-1200 with BagFlip, we find that 1) PECAN-600
+and PECAN-1200 achieves higher certified accuracy than
+BagFlip when R ∈ [0, 0.25] and R ∈ [0, 0.17], respec-
+tively, and 2) BagFlip has non-zero certified accuracy when
+R ∈ [0.5, 1.5], where the certified accuracy of PECAN-600
+and PECAN-1200 is zero. Comparing PECAN-2000 with
+BagFlip, we find that BagFlip outperforms PECAN-2000
+across all modification amounts R.
+We argue that the gap of certified accuracy between PECAN-
+2000 and BagFlip mainly comes from the different def-
+initions of the perturbation spaces as discussed in Ap-
+pendix B.1. Moreover, the root cause of this difference
+is owing to the probabilistic nature of BagFlip.
+PECAN is more efficient than BagFlip at computing the
+certified radius. PECAN computes the certified radius in a
+constant time complexity via the closed-form solution in the
+aggregation step. However, in our experiment of the MNIST
+dataset, BagFlip requires 8 hours to prepare a lookup table
+because BagFlip does not have a closed-form solution for
+computing the certified radius.
+5.3. PECAN under the Backdoored Data
+We evaluate the effectiveness of PECAN under the back-
+door attack (Severi et al., 2021) for malware detection on the
+EMBER dataset. We do not compare PECAN to BagFlip
+because BagFlip has poor certified accuracy on EMBER, as
+shown in Figure 3. We also evaluate other baselines, DPA
+and CROWN-IBP, which do not aim to defend against back-
+door attacks. DPA is the certified defense against trigger-less
+attacks, and CROWN-IBP is the certified defense against
+evasion attacks. We also present the results of the victim
+classifiers without any defense. Appendix B.4 shows that
+the empirical defense spectral signatures (Tran et al., 2018)
+cannot defend against the backdoor attack.
+Summary of the results
+PECAN reduces the ASR of
+the victim model on the test set with malicious triggers
+from 41.33% to 1.85%, while the other baselines fail
+to defend against the backdoor attack. Being the most
+conservative, PECAN has the highest abstention rate.
+Setup
+We use Severi et al. (2021) to generate backdoored
+data by modifying 0.1% training examples and adding trig-
+gers into the test inputs that should be labeled as malware to
+fool the victim model to predict the malware with malicious
+triggers as benign software (non-malware). We generate
+three poisoned datasets �D1, �D2, �D3 and their correspond-
+ing test sets with triggers by perturbations F1, F2, and F3,
+which allow the attacker to modify up to one, two, and three
+features in an example, respectively.
+We report the results of all approaches on the malware test
+sets with triggers and the malware test sets without trig-
+gers, i.e., the original malware test set. The results on the
+non-malware test sets without triggers can be found in Ap-
+pendix B.3. ASR on malware is a much more critical metric
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+0
+20
+40
+60
+80
+100
+ASR
+Correct
+Abstain
+PECAN
+DPA
+C-IBP
+NoDef
+Figure 5. Results of PECAN, DPA, CROWN-IBP (C-IBP), and
+vanilla model without defense (NoDef) trained on three poisoned
+EMBER datasets when evaluated on the malware test set with
+malicious triggers. We note that NoDef does not have abstention
+rates because it does not use any defense.
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+0
+20
+40
+60
+80
+100
+ASR
+Correct
+Abstain
+PECAN
+DPA
+C-IBP
+NoDef
+Figure 6. Results of PECAN, DPA, C-IBP, and NoDef when evalu-
+ated on the (original) malware test set without malicious triggers.
+than the ASR on non-malware, because the former shows
+how many pieces of malware can bypass the classifier.
+For PECAN and DPA, we show their results at modification
+amount R = 0.1%. We show CROWN-IBP results against
+the perturbations F1, F2, and F3 regardless of R because
+CROWN-IBP does not consider R.
+Results
+Figures 5 and 6 show the ASR, accuracy, and ab-
+stention rate of all the approaches on the malware test set
+with and without triggers, respectively. Table 1 in the ap-
+pendix shows the detailed numbers. Note that PECAN is
+the only certified approach for backdoor attacks. The
+results of other baselines can be seen as empirical be-
+cause DPA and CROWN-IBP certify a different goal,
+and NoDef has no defense.
+PECAN can defend against the backdoor attack on the
+EMBER dataset. Figures 5 and 6 show that PECAN has
+the lowest ASR 1.85% and 1.03% on both malware test
+sets with and without triggers on average, compared to
+DPA (18.05%, 1.98%), CROWN-IBP (15.24%, 6.82%),
+and NoDef (41.33%, 2.12%).
+0
+5 · 10−2
+0.1
+0
+20
+40
+60
+80
+100
+Modification Amount R (%)
+Test Set Percentage
+0
+5 · 10−2
+0.1
+Correct
+ASR
+Abstain
+(a) PECAN
+(b) DPA
+Figure 7. Comparison between PECAN and DPA trained on �D3
+across all modification amount R when evaluated on the malware
+test set with triggers.
+DPA and CROWN-IBP fail to defend against the back-
+door attack. The average ASR of DPA and CROWN-IBP
+on the malware test set with triggers are 18.05% and 15.24%
+in Figure 5, respectively, meaning that many malware with
+triggers can bypass their defenses. The average ASR of
+DPA on the malware test set without triggers, 1.98%, is
+much lower than its ASR on the one with triggers, 18.05%,
+which shows that DPA successfully defends against trigger-
+less attacks when the test input does not have any trigger.
+CROWN-IBP has high ASR on both the malware test sets
+with and without triggers, as CROWN-IBP cannot defend
+against the poison in the training sets.
+PECAN has higher abstention rates than other ap-
+proaches.
+On average, PECAN abstains from 50.41%
+predictions compared to DPA (34.73%) and CROWN-IBP
+(26.44%). We further compare the accuracy, ASR, and ab-
+stention rate of PECAN and DPA across all modification
+amount R when trained on �D3 in Figure 7. The results on
+�D1 and �D2 are shown in Appendix B.5. We can observe
+that PECAN has a much lower ASR than DPA across all
+modification amounts. Meanwhile, Figure 7 shows that
+the certification of PECAN might be over-conservative be-
+cause the ASR is low (3.17%) even when we regard �D3 as
+non-poisoned (when R = 0), yet �D3 is actually poisoned.
+6. Conclusion, Limitations, and Future Work
+We presented PECAN, a deterministic certified approach
+to effectively and efficiently defend against backdoor at-
+tacks. We foresee many future improvements to PECAN.
+First, we implemented PECAN as a certified defense special-
+ized for neural networks because the evasion certification
+step, CROWN-IBP, is limited to neural networks. However,
+we can replace CROWN-IBP with an evasion certification
+approach for another machine learning model to get a cor-
+responding backdoor defense for that model. Second, we
+adopt the idea of deep partition aggregation (DPA) to de-
+sign the partition and aggregation steps in PECAN. We can
+improve these steps by using finite aggregation (FA) (Wang
+et al., 2022b), which extends DPA and gives higher certified
+accuracy. Third, during the certification of evasion attacks,
+we need to propagate the abstraction of the same test input
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+through thousands of neural networks that have different
+weights but the same architecture. Sharing the propaga-
+tion results among different neural networks (Fischer et al.,
+2022) can greatly improve the efficiency of PECAN and
+may enable using complete certification methods.
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
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+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+A. Proof of Theorem 4.2
+Proof. Theorem 4.1 tells that either r = ⋄ or the following
+equation holds,
+∀D′ ∈ Sr( �D), x′ ∈ π(�x). ¯A �
+D(�x) = ¯AD′(x′)
+(6)
+By the symmetrical definition of Sr and π, we have
+∀D. �D ∈ Sr(D) =⇒ D ∈ Sr( �D)
+(7)
+∀x. �x ∈ π(x) =⇒ x ∈ π(�x).
+(8)
+Then, for all possible clean data D and x, we have
+�D ∈ Sr(D) ∧ �x ∈ π(x)
+=⇒ D ∈ Sr( �D) ∧ x ∈ π(�x)
+(By Eq 7 and Eq 8)
+=⇒ ¯A �
+D(�x) = ¯AD(x)
+(By Eq 6)
+B. Experiment
+B.1. Detailed Setup of Section 5.2
+Following the BagFlip paper (Zhang et al., 2022b), we set k,
+the number of training examples in a bag used in BagFlip,
+as 100, 1000, and 3000 for the MNIST, CIFAR10, and
+EMBER dataset, respectively. For PECAN, we vary n, the
+number of partitions, according to the value of k in BagFlip
+by setting n = |D|
+k .
+BagFlip defines their perturbation space S′
+r(D) that is dif-
+ferent from PECAN,
+S′
+r(D) =
+�
+�D | max(|D \ �D|, | �D \ D|) ≤ r
+�
+,
+where A \ B is the set difference, i.e., the elements in A
+but not in B. Notice that with the same radius r, the above
+definition gives a larger S′
+r(D) than Sr(D) as the following
+example shows.
+Example B.1. If the attacker modifies one training example
+x ∈ D to another training example �x to form a poisoned
+dataset �D = D \ {x} ∪ {�x}. Then �D ∈ S2(D) but �D /∈
+S1(D) because Sr(D) considers one modification as one
+deletion and one insertion. However, we have �D ∈ S′
+1(D).
+Chen et al. (2022) show that S′
+r(D) works when the ap-
+proach uses non-deterministic sub-sampling (Jia et al., 2021;
+Zhang et al., 2022b). However, the certification of determin-
+istic approaches only works under the definition of Sr(D).
+We adjust the computation of certified accuracy for BagFlip
+as 1
+m
+�m
+i=1 1 ¯
+AD(xi)=yi∧ ri
+|D| ≥R% by removing the factor 2
+on R. Thus, we are also interested in the performance of
+PECAN when n = |D|
+2k to compensate the removed factor
+2.
+0
+0.2
+0.4
+0
+20
+40
+60
+80
+100
+Modification Amount R (%)
+Certifiable Accuracy
+0
+2
+4
+·10−2
+PECAN-a
+PECAN-b
+(a) MNIST s = 0.1
+(b) CIFAR10 s = 2/255
+Figure 8. Results of PECAN on CIFAR10 and EMBER, showing
+the normal accuracy (dotted lines) and the certified accuracy (solid
+lines) at different modification amounts R. For MNIST: a = 600
+and b = 1200. For CIFAR10: a = 50 and b = 100.
+B.2. Evaluation on the l∞ Perturbation Space
+Setup
+As the CROWN-IBP used in PECAN can handle
+π with different lp norms, PECAN can handle different lp
+norms as well. We evaluate PECAN on the l∞ norm with
+distance s = 0.1 and s = 2/255 on MNIST and CIFAR10,
+respectively, because the l∞ norm is widely applied to eval-
+uate the robustness of image classifiers. In this experimental
+setting, we use two CNN models for MNIST and CIFAR10
+because CROWN-IBP works better for CNN on l∞ norm
+than on l0 norm. For training on MNIST and CIFAR10, we
+train on s = 0.2 and s = 5/255 but test on s = 0.1 and
+s = 2/255 to overcome the overfitting issue when s is small,
+following the practice in the original paper of CROWN-IBP.
+For the experiments on l0 (Sections 5.2 and 5.3), we set
+the κstart = 0 and κend = 0 for CROWN-IBP. For the ex-
+periments on l∞, we set the κstart = 1 and κend = 0 for
+CROWN-IBP.
+Results
+Figure 8 shows the results of PECAN against l∞
+perturbation space. The results show that PECAN achieves
+certified accuracy similar to F1 as shown in Figures 3 and 4.
+B.3. Comparison to DPA, CROWN-IBP, and NoDef on
+the Non-Malware Test Set without Trigger
+Figure 9 shows that NoDef has the lowest ASR of 2.70% on
+the non-malware set without trigger than all three defenses
+because the backdoor attack does not aim to attack the
+prediction of non-malware. However, PECAN still achieves
+the lowest ASR of 5.73% compared to DPA (7.85%) and
+CROWN-IBP (6.68%).
+B.4. Comparison to Spectral Signatures
+We followed the experiment in Severi et al. (2021) to filter
+out poisoned examples in the training dataset �D3. After
+removing the top 15% outliers in the non-malware training
+set, we observe that only 14% (84 out of 600) of the poison
+is removed. Then we train a new model using the filtered
+training set. We find the ASR of the new model on the
+
+PECAN: A Deterministic Certified Defense Against Backdoor Attacks
+Table 1. Results of PECAN, DPA, CROWN-IBP (C-IBP) and vanilla model without defense (NoDef) trained on three backdoored EMBER
+datasets. Malware with triggers is the backdoored test data that should be labeled as malware. Malware w/o triggers is the original test
+data that should be labeled as malware. Non-malware w/o triggers is the original test data that should be labeled as non-malware.
+Test sets
+Malware with triggers
+Malware w/o triggers
+Non-Malware w/o triggers
+Approaches
+PECAN
+DPA
+C-IBP
+NoDef PECAN
+DPA
+C-IBP
+NoDef PECAN
+DPA
+C-IBP
+NoDef
+�D1
+ASR. (↓)
+2.38%
+4.68%
+2.72% 21.15%
+1.27%
+2.00%
+2.10%
+1.92%
+6.48%
+7.81%
+9.41%
+2.94%
+Correct Pred. (↑) 38.42% 33.57% 67.86% 78.85% 58.01% 64.34% 77.21% 98.08% 73.82% 79.29% 83.66% 97.06%
+Abstention Rate
+59.20% 61.75% 29.42%
+N/A 40.72% 33.66% 20.69%
+N/A 19.70% 12.90%
+6.93%
+N/A
+�D2
+ASR. (↓)
+1.98% 24.61% 28.57% 41.17%
+1.12%
+1.96%
+8.05%
+2.16%
+5.55%
+7.83%
+6.11%
+2.64%
+Correct Pred. (↑) 29.33% 20.12% 34.78% 58.83% 44.62% 64.34% 65.01% 97.84% 65.95% 79.03% 90.68% 97.36%
+Abstention Rate
+68.69% 55.27% 36.64%
+N/A 54.27% 33.70% 26.95%
+N/A 28.50% 13.14%
+3.21%
+N/A
+�D3
+ASR. (↓)
+1.19% 24.87% 14.42% 61.67%
+0.71%
+1.97% 10.32%
+2.28%
+5.16%
+7.91%
+4.51%
+2.51%
+Correct Pred. (↑) 21.48% 19.59% 27.59% 38.33% 34.45% 64.58% 41.10% 97.72% 54.40% 78.96% 87.91% 97.49%
+Abstention Rate
+77.33% 55.54% 57.99%
+N/A 64.84% 33.45% 48.58%
+N/A 40.44% 13.14%
+7.59%
+N/A
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+�
+D1 �
+D2 �
+D3
+0
+20
+40
+60
+80
+100
+ASR
+Correct
+Abstain
+PECAN
+DPA
+C-IBP
+NoDef
+Figure 9. Results of PECAN, DPA, CROWN-IBP (C-IBP), and
+vanilla model without defense (NoDef) trained on three poisoned
+EMBER datasets when evaluated on the (original) non-malware
+test set without triggers.
+malware set with triggers, the malware set without triggers,
+and the non-malware set without triggers are 48.55%, 1.38%,
+and 8.91%, respectively. These ASRs are all higher than
+PECAN’s 1.19%, 0.71%, and 5.16% on the three parts of
+the test set.
+B.5. Comparison to DPA on �D1 and �D2
+Figures 10 and 11 show the comparison between PECAN
+and DPA on �D1 and �D2. We can observe that PECAN
+has much higher ASRs than DPA across all modification
+amounts on �D1 and �D2.
+0
+5 · 10−2
+0.1
+0
+20
+40
+60
+80
+100
+Modification Amount R (%)
+Test Set Percentage
+0
+5 · 10−2
+0.1
+Correct
+ASR
+Abstain
+(a) PECAN
+(b) DPA
+Figure 10. Comparison between PECAN and DPA trained on �D1
+across all modification amount R when evaluated on the malware
+test set with triggers.
+0
+5 · 10−2
+0.1
+0
+20
+40
+60
+80
+100
+Modification Amount R (%)
+Test Set Percentage
+0
+5 · 10−2
+0.1
+Correct
+ASR
+Abstain
+(a) PECAN
+(b) DPA
+Figure 11. Comparison between PECAN and DPA trained on �D2
+across all modification amount R when evaluated on the malware
+test set with triggers.
+

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