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+Hyper-cores promote localization and efficient seeding in higher-order processes
+Marco Mancastroppa,1 Iacopo Iacopini,2 Giovanni Petri,3 and Alain Barrat1
+1Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT,
+Turing Center for Living Systems, Marseille, France
+2Department of Network and Data Science, Central European University, 1100 Vienna, Austria
+3CENTAI, Corso Inghilterra 3, 10138 Turin, Italy
+Going beyond networks, in order to include higher-order interactions involving groups of elements
+of arbitrary sizes, has been recognized as a major step in reaching a better description of many
+complex systems. In the resulting hypergraph representation, tools to identify particularly cohesive
+structures and central nodes are still scarce. We propose here to decompose a hypergraph in hyper-
+cores, defined as subsets of nodes connected by at least a certain number of hyperedges (groups
+of nodes) of at least a certain size. We illustrate this procedure on empirical data sets described
+by hypergraphs, showing how this suggests a novel notion of centrality for nodes in hypergraphs,
+the hyper-coreness. We assess the role of the hyper-cores and of nodes with large hyper-coreness
+values in several dynamical processes based on higher-order interactions. We show that such nodes
+have large spreading power and that spreading processes are localized in hyper-cores with large
+connectedness along groups of large sizes. In the emergence of social conventions moreover, very
+few committed individuals with high hyper-coreness can rapidly overturn a majority convention.
+Our work opens multiple research avenues, from fingerprinting and comparing empirical data sets
+to model validation and study of temporally varying hypergraphs.
+I.
+INTRODUCTION
+Network theory provides a powerful framework for de-
+scribing a wide range of complex systems composed of
+elements interacting in pairs [1–4]: over the years, this
+theory has developed numerous concepts and techniques
+to characterize the structure of complex networks at var-
+ious scales, from the single element (node or link) to
+groups of nodes to the whole system.
+Moreover, net-
+works are the support of dynamical processes of various
+types, from spreading to synchronization phenomena [3],
+thus understanding how network’s features impact such
+processes, or which parts of a network play the most im-
+portant role, is of primordial relevance. Several concepts
+and results in this respect are now well established. For
+instance, hubs, nodes with a very large number of con-
+nections (degree), are known to influence processes such
+as spreading or opinion dynamics, because of their ten-
+dency to be reached easily and their ability to transmit to
+many other nodes [1, 3]. The statistics of the individual
+number of connections of nodes are however not a rich
+enough characterization: the existence of well-connected
+groups of nodes might indeed be even more relevant. In
+this direction, the tendency of hubs –observed in real-
+world networks– to be connected to each other far above
+chance is quantified by the rich-club coefficient [5].
+A
+more systematic way to decompose a network into a hi-
+erarchy of subgraphs of increasing connectedness is given
+instead by the k-core decomposition [6–9]: the k-core of
+a network is by definition the maximal subgraph such
+that all its nodes have degree (number of neighbours in
+the subgraph) at least k. This decomposition provides a
+fingerprint of the network’s structure [8, 10, 11] and grad-
+ually focuses on more central and densely interconnected
+parts of the network that were shown to play a crucial
+role in spreading processes [12–14]. In fact, the coreness
+of a node, defined as the largest value of k such that the
+node belongs to the corresponding k-core, gives an alter-
+native measure of centrality and largely determines the
+impact of a spreading process initiated (seeded) in that
+node [12]. Given the relevance of this decomposition, it
+has also been extended to weighted networks [15], via the
+s-core decomposition [16] (s representing the strength of
+a node, i.e., the sum of the weights of its adjacent links),
+to temporally evolving networks [17, 18], and to multi-
+layer networks [19].
+Despite their convenience, network representations are
+limited to systems composed of only binary interactions.
+However, over the last few years, it has become clear that
+many real systems include interactions between groups
+of units [20, 21].
+Examples range from group conver-
+sations among friends [22] to research teams and co-
+authorship of scientific articles [23], from neural systems
+[24] to interactions between multiple species in ecologi-
+cal ones [25]. Analogously, considering a purely dyadic
+network substrate for the unfolding of processes such
+as consensus formation or (social) contagion could put
+a limit on the ability to describe key mechanisms that
+are at play. For instance, reinforcement mechanisms –
+in which two or more people can convince others in a
+group conversation– cannot be naturally accounted for
+when considering only dyadic interactions [26–29].
+In
+these cases, systems and processes can be effectively rep-
+resented within the framework of hypergraphs, a “higher-
+order” generalization of networks in which nodes can in-
+teract in hyperedges, groups of arbitrary size [21, 30, 31].
+Higher-order interactions give rise to novel structures [32–
+34] and phenomena [20, 35], highlighting the need for new
+characterization tools able to detect hierarchies and rele-
+vant subparts of all these systems that are better repre-
+sented by hypergraphs.
+Here, we contribute to this endeavour by proposing
+the decomposition of a hypergraph in (k, m)-hyper-cores,
+which we define as a series of subhypergraphs of increas-
+ing connectivity k, ensured by hyperedges of increasing
+sizes m. We apply this decomposition to a wide range of
+data sets, representing systems of different nature, iden-
+tifying non-trivial mesoscopic higher-order structures. In
+so doing, we put forward the hyper-coreness, a new cen-
+trality measure for nodes in hypergraphs based on their
+inclusion in the hyper-cores. Finally, and crucially, we
+investigate the role of the newly defined hyper-cores
+arXiv:2301.04235v1  [physics.soc-ph]  10 Jan 2023
+
+2
+and of the nodes with largest hyper-coreness in spread-
+ing and consensus processes based on group interactions
+[28, 36, 37]. We show that spreading processes tend to
+be localized on hyper-cores associated to large k and
+m. We then study the performance of hyper-coreness-
+based strategies as opposed to both random and strength-
+based ones [16] when it comes to identifying influential
+nodes that sustain and drive higher-order processes. We
+find that hyper-coreness can be effectively used to max-
+imise the total outbreak size in non-linear spreading pro-
+cesses [36] and help committed minorities reach the tip-
+ping point leading to a systemic takeover in social con-
+vention games [38].
+II.
+RESULTS
+A.
+Hyper-core decomposition and hyper-coreness
+We define the hyper-cores, i.e. the higher-order cores
+of a hypergraph, through a systematic decomposition
+of a hypergraph in a double hierarchy of nested sub-
+hypergraphs of increasing connectedness and hyperedge
+sizes. Let us consider a (static) hypergraph H = (V, E),
+where V is the set of its N = |V| nodes and E is the
+set of its hyperedges [21].
+We recall that a hyperedge
+e = {i1, i2, ..., im} is a set of m nodes, which can thus
+represent a group interaction between these nodes. We
+denote by M = maxe∈E |e| the largest hyperedge size in
+H. Each node i ∈ V can be characterized by a vector
+of degrees d(i) = [d2(i), d3(i), ..., dm(i), ..., dM(i)] whose
+component dm(i) denotes the m-hyper-degree of the node
+i, i.e., the number of distinct hyperedges of size m to
+which it belongs.
+We denote by Dm(i) = �
+p≥m dp(i)
+the number of distinct hyperedges of size at least m to
+which i belongs.
+We define the (k, m)-hyper-core as the maximum sub-
+hypergraph J induced by the set of nodes A ⊆ V and
+with hyperedges of size at least m, such that ∀ i ∈
+A, DJ
+m(i) ≥ k, where DJ
+m(i) denotes the number of dis-
+tinct hyperedges of size at least m in which i is involved
+within the subhypergraph J . In other terms, all the nodes
+in the (k, m)-hyper-core belong to at least k hyperedges
+of size at least m, within the hyper-core itself. The set of
+hyperedges of the subhypergraph J , induced by the set
+A ⊆ V, is defined by S = {e ∩ A|e ∈ E ∧ |e ∩ A| ≥ m}
+[39] (i.e., a hyperedge of S is a subset of a hyperedge of
+E, of size at least m and containing only nodes of A).
+To obtain the (k, m)-hyper-core of a hypergraph, one
+can first remove from E all hyperedges of size smaller than
+m. One then removes recursively from V all nodes i with
+Dm(i) < k, until all the nodes in the remaining subhy-
+pergraph are involved in at least k hyperedges of size at
+least m. Note that this process does not correspond only
+to the removal of nodes with Dm(i) < k in the original
+hypergraph H: indeed, each time a node is removed, the
+sizes of the hyperedges to which it belongs decrease by
+one unit. Thus, the removal of a node can induce the
+removal of some of the hyperedges to which it belongs, if
+their size becomes less than m. In Fig. 1 we illustrate the
+process on an example hypergraph and highlight some of
+its (k, m)-hyper-cores.
+As k and m increases, the (k, m)-hyper-cores progres-
+FIG. 1. Sketch of the (k, m)-hyper-core decomposition.
+We show a hypergraph and highlight some of its (k, m)-hyper-
+cores.
+Note the inclusions as k or m increase: the (1, 2)-
+hyper-core contains the (1, 3)-hyper-core, which contains the
+(2, 3)-hyper-core; similarly the (1, 2)-hyper-core contains the
+(2, 2)-hyper-core which contains the (2, 3)-hyper-core. On the
+other hand, the (1, 3)-hyper-core and the (2, 2)-hyper-core
+share some nodes but neither is included in the other. The
+green nodes belong to the (1, 2)-hyper-core but neither to the
+(1, 3)- nor the (2, 2)- ones.
+The blue nodes belong to the
+(1, 3)-hyper-core but are excluded from the (2, 3) one.
+Or-
+ange nodes belong to the (2, 2)-hyper-core but are excluded
+from the (2, 3) one because they belong only to hyperedges
+of size 2. The (1, 4)-core and (1, 5)-core contain all the nodes
+involved respectively in at least one interaction with m ≥ 4
+and m ≥ 5 (for simplicity these cores are not highlighted).
+The (k, 2)-cores and (k, 3)-cores with k ≥ 3, and the (k, 4)-
+cores and (k, 5)-cores with k ≥ 2 are all empty. Notice that
+the node i does not belong to the (2, 3)-core even if D3(i) = 2
+because of the recursive and interaction downgrading mecha-
+nisms of the decomposition; in the (1, 3)-core and (2, 3)-core
+the pairwise interactions ei ∀i ∈ [1, 5] are excluded, thus the
+(1, 3)-core is composed of two disjoint subhypergraphs.
+sively identify groups of nodes increasingly connected
+with each other through interactions of increasing or-
+der (the (k, m)-hyper-core includes the (k, m + 1)- and
+(k + 1, m)-hyper-cores).
+We define the m-shell index
+Cm(i) of a node i as the value of k such that i belongs
+to the (k, m)-hyper-core but not to the (k + 1, m)-hyper-
+core. The (k, m)-shell S(k,m) can then be defined as the
+set of all nodes whose shell index Cm(i) at size m is k,
+and we denote by km
+max the maximum value of k such
+that the shell S(k,m) is not empty. The ratio Cm(i)/km
+max
+thus quantifies how well-connected node i is in the hy-
+pergraph. As it is a function of m, different nodes will
+have different functions with potentially different func-
+tional shapes, which makes it difficult to compare and
+rank them (see the Supplemental Material, SM, for some
+examples of Cm(i) functional shapes). We thus define for
+each node i its hyper-coreness R(i) as:
+R(i) =
+M
+�
+m=2
+g(m)Cm(i)/km
+max,
+(1)
+where g(m) is an arbitrary weight function, which can
+
+3
+weigh differently the various possible sizes of higher-order
+interactions. Hereafter, for simplicity we will fix g(m) =
+1: in this case, by definition R ∈ [0, M −1]. Other choices
+could be considered, for example to emphasise hyperedges
+of larger or smaller sizes. The R hyper-coreness gives thus
+a summary of a node centrality with respect to the various
+(k, m)-hyper-cores, taking into account how central the
+node is for all orders of interactions and making it possible
+to rank the nodes of the hypergraph.
+B.
+Hyper-core decomposition of empirical
+hypergraphs
+To illustrate the decomposition processes along (k, m)-
+hyper-cores, we rely on a number of empirical hyper-
+graphs, obtained from publicly available data sets, that
+describe a variety of systems of agents interacting in dif-
+ferent environments, both through online media and face-
+to-face. In particular, we consider data sets of face-to-face
+interactions provided by the SocioPatterns collaboration
+[40–42] and by the Contacts among Utah’s School-age
+Population (CUSP) project [43], collected in contexts
+ranging from workplaces to schools.
+We also use data
+sets of email communication (email-EU, email-Enron [44–
+46]) and of other types of online interactions, namely on-
+line reviews of products (music-review [46, 47]) or online
+opinion exchange on specific topics in scientific forums
+[46, 48]. We moreover consider data describing commit-
+tees membership (house-committees, senate-committees
+[46, 49, 50]) and bills sponsorship (congress-bills, senate-
+bills [46, 49, 51, 52]) in the US Congress.
+These data
+sets cover a wide range of system sizes and have also
+very different interaction size distributions. We provide
+a detailed description of each data set in the Methods
+and in the SM. In the following, we give results on the
+music-review, email-EU, house-committees and congress-
+bills data sets while we refer to the SM for the other data
+sets.
+Figure 2 shows the results of the hyper-core decompo-
+sition on two data sets. The relative size n(k,m) of the
+(k, m)-hyper-cores exhibit distinct behaviors as a func-
+tion of k and m, identifying structural differences between
+data sets. In some cases, the decrease with k is rather
+smooth (Fig. 2a and SM), showing that most shells are
+populated. In other cases abrupt drops and plateaus can
+be observed (Fig. 2c and SM), corresponding to alter-
+natively empty and densely populated (k, m)-shells (see
+also SM for figures showing the sizes of the (k, m)-shells
+vs k and m). These differences indicate that the (k, m)-
+hyper-cores could be used to provide a fingerprint of hy-
+pergraphs, just as the k-core decomposition provides a
+fingerprint of networks [8, 10, 11]. Also the distributions
+of hyper-coreness values R differ across data sets, as illus-
+trated in the rank-order plots of Fig. 2b,d and in the SM
+for all data sets. While some data sets have an almost
+uniform distribution of values, others feature few nodes
+with high hyper-coreness and many nodes with medium
+hyper-coreness, or vice-versa –many nodes having low or
+high coreness and few with medium values (see SM). We
+also show in the SM some typical examples of the nor-
+malized m-shell index function Cm(i) as a function of m
+for various nodes, to highlight the diversity of these func-
+1
+16 31 46 61 76 91 106
+k
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+m
+n(k, m)
+a
+email-EU
+0
+250
+500
+750
+1000
+Rank(i)
+0
+5
+10
+15
+20
+R(i)
+b
+1
+6 11 16 21 26 31 36
+k
+2
+12
+22
+32
+42
+52
+62
+72
+82
+m
+n(k, m)
+c
+music-review
+0
+250
+500
+750
+1000
+Rank(i)
+0
+20
+40
+60
+80
+R(i)
+d
+10
+1
+100
+0
+50
+100
+k
+0.0
+0.5
+1.0
+n(k, m)
+m = 2.0
+m = 4.0
+m = 6.0
+m = 14.0
+0
+500
+1000
+S(i)
+0
+10
+20
+R(i)
+10
+1
+100
+0
+20
+40
+k
+0.0
+0.5
+1.0
+n(k, m)
+m = 2.0
+m = 10.0
+m = 18.0
+m = 32.0
+0
+200
+S(i)
+0
+40
+80
+R(i)
+FIG. 2. Hyper-core decomposition of empirical hyper-
+graphs. Panels a and c show colormaps giving the relative
+size n(k,m) (number of nodes in the hyper-core, divided by the
+total number of nodes N) of the (k, m)-hyper-core as a func-
+tion of k and m (white regions correspond to n(k,m) = 0). In
+the insets n(k,m) is shown as a function of k at fixed values of
+m. In panels b and d the hyper-coreness R(i) is plotted as a
+function of the corresponding node rank; the insets give scat-
+terplots of the hyper-coreness R(i) vs. the s-coreness, S(i),
+for all nodes. In panels a and b we consider the email-EU
+data set: R(i) and S(i) have a Pearson correlation coefficient
+of ρ = 0.90 (p-value p ≪ 0.001) and the corresponding rank-
+ings have a Kendall’s τ coefficient of τ = 0.85 (p ≪ 0.001); in
+panels c and d we consider the music-review data set: R(i)
+and S(i) have a Pearson correlation coefficient of ρ = 0.74
+(p ≪ 0.001) and the corresponding rankings a Kendall’s τ
+coefficient of τ = 0.58 (p ≪ 0.001).
+tions and the need to define a summary index such as the
+hyper-coreness.
+We finally compare in the insets of Fig. 2b,d the hyper-
+coreness with the centrality of nodes obtained by disre-
+garding the higher-order character of the interactions and
+projecting the hypergraph H onto a network.
+To this
+aim, we transform each hyperedge in a network clique,
+and each edge (i, j) of the resulting network is weighted
+by the number of distinct hyperedges in H involving both
+i and j. We then perform the s-core decomposition of this
+weighted network and assign its s-coreness S(i) to each
+node i [16]. S(i) and R(i) are positively correlated, but
+they do not provide exactly the same information.
+In
+particular the hyper-coreness enhances the information
+given by the s-coreness by providing an internal hierar-
+chy within the nodes of maximal s-coreness. This is evi-
+dent from the scatter plots, as nodes presenting the same
+s-coreness values correspond to values of hyper-coreness
+that can span across a broad range (y axis).
+Having illustrated the relevance of the newly defined
+cores on empirical hypergraphs, we now move to study
+the role of these substructures in dynamical processes
+taking place on hypergraphs. In particular, we are go-
+
+4
+ing to investigate whether the (k, m)-hyper-cores and the
+hyper-coreness centrality can be used to identify nodes
+and structures relevant for spreading and consensus pro-
+cesses whose mechanisms are explicitly defined on hyper-
+edges.
+C.
+Higher-order contagion processes localize in
+hyper-cores
+Networks have been widely used to describe the sub-
+strate on which contagion processes take place, such as
+the spread of a pathogen or information.
+In standard
+diffusion modeling approaches, nodes represent individ-
+uals that at any time can be in one of several possible
+states, such as S (susceptible), I (infectious) or R (re-
+covered); S nodes are typically thought to become I at
+rate β when they share a link with an infectious (I) in-
+dividual, while infected (I) nodes recover spontaneously
+at rate µ, either becoming again susceptible (S), in what
+is usually called the SIS model [53], or becoming recov-
+ered (R) in the so-called SIR model. Recently, several
+models have been proposed to take into account possible
+higher-order mechanisms, that amount to reinforcement
+mechanisms affecting the contagion probability due to the
+simultaneous exposure to multiple sources of infections in
+group interactions [26, 36, 54, 55]. For instance, in a so-
+cial contagion process, the probability that an individual
+is convinced upon separate exposures to two “infectious”
+neighbours can be reinforced if these exposures occur dur-
+ing a group discussion featuring the three individuals al-
+together.
+Here, we show that hyper-cores play a crucial role in
+the dynamics of higher-order spreading processes. In or-
+der to do this, we consider the recently proposed higher-
+order non-linear contagion [36]. In this model, each sus-
+ceptible node in a hyperedge of size m in which there
+are i infected individuals becomes infectious with rate
+λiν, where ν controls the non-linearity of the process (for
+ν = 1 the usual linear contagion is recovered, while for
+ν > 1 non-linearities are introduced) and λ ∈ [0, 1] (see
+Methods for more details on the model).
+Infected in-
+dividuals (I) recover independently at constant rate µ,
+becoming either susceptible S (SIS model) or R (SIR).
+The higher-order nature of contagion produces novel
+effects on the epidemic phenomenology, including abrupt
+transitions with bistability in the SIS phase diagram, in-
+termittent regimes [37, 54], and a mesoscopic localization
+of the infection on large hyperedges [36].
+Against this
+background, the connectivity properties of hyper-cores
+we highlighted so far suggest that cores might play an
+even stronger role in such localization.
+To investigate this point, we perform numerical simula-
+tions of the higher-order non-linear SIS model on empiri-
+cal hypergraphs. The system is initialized with one single
+seed of infection (a randomly chosen node in state I) in an
+otherwise fully susceptible population. We let the process
+evolve (see Methods for simulation details) until a steady
+state is reached in which the number of infectious individ-
+uals fluctuates (we consider parameter values such that
+the system remains active and the epidemic does not die
+out rapidly). We then consider a finite time-window T
+and measure for each node j the time τ(j) that it spends
+1
+6 11 16 21 26 31 36
+k
+2
+12
+22
+32
+42
+52
+62
+72
+82
+m
+/T
+a
+music-review
+SIS
+1
+6 11 16 21 26 31 36
+k
+2
+12
+22
+32
+42
+52
+62
+72
+82
+m
+R
+b
+SIR
+1 3 5 7 9 1113151719
+k
+2
+12
+22
+32
+42
+52
+62
+72
+82
+m
+/T
+c
+house-committees
+1 3 5 7 9 1113151719
+k
+2
+12
+22
+32
+42
+52
+62
+72
+82
+m
+R
+d
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0
+500
+1000
+n
+0.5
+0.7
+0.9
+/T n
+S-rank
+R-rank
+500
+600
+700
+800
+900
+0
+1000
+n
+400
+600
+800
+R
+n
+S-rank
+R-rank
+0.675
+0.700
+0.725
+0.750
+0.775
+0.800
+0.825
+0.850
+0
+500 1000
+n
+0.7
+0.8
+0.9
+/T n
+S-rank
+R-rank
+100
+125
+150
+175
+200
+225
+0
+500 1000
+n
+100
+200
+300
+R
+n
+S-rank
+R-rank
+FIG. 3.
+Hyper-cores for seeding and localization in
+higher-order non-linear contagion processes. For the
+SIS model, panels a and c give the heatmap of the average
+fraction of time ⟨τ/T⟩ of infected nodes in the steady state as
+a function of k and m. Averages are computed over all the
+nodes of each (k, m)-hyper-core. The insets represent ⟨τ/T⟩
+averaged over the first n nodes according to the coreness rank-
+ings as a function of n. All results are obtained by averaging
+the results of 103 numerical simulations, with an observation
+window T = 103. For the SIR model, panels b and d show
+the heatmap of the average final size of the epidemic ⟨R∞⟩
+as a function of k and m, where the process is seeded in a
+single node belonging to the (k, m)-hyper-core (averaged over
+all nodes of the hyper-core). The insets represent ⟨R∞⟩ as
+a function of n averaged over the first n nodes according to
+coreness rankings. All results are obtained by averaging the
+results of 300 numerical simulations for each seed. Panels a
+and b: music-review data set with ν = 1.25, λ = 5 × 10−4
+(a) and ν = 3, λ = 5 × 10−4 (b). Panels c and d: house-
+committees data set with ν = 1.25, λ = 5 × 10−4 (c) and
+ν = 4, λ = 5 × 10−5 (d). In all panels µ = 0.1.
+in the I state during that window. This allows to iden-
+tify the nodes on which the epidemic is mainly localized
+in the steady state, i.e. the nodes that drive and sustain
+the process.
+Figure 3 reports results of simulations performed on
+two data sets (the music-review and house-committees
+data sets). Similar results are shown in the SM for the
+other considered data sets. Panels 3a and 3c show that
+nodes in (k, m)-hyper-cores with either increasing k or m
+tend to be more often infectious, as the values of τ(j)/T
+averaged over all nodes of each (k, m)-hyper-core increase
+with k and m. This implies that the process is more local-
+ized in the (k, m)-hyper-cores with large k (which favors
+connectedness, hence mutual reachability) and m (i.e.,
+large hyperedges where large values of i can be obtained
+yielding large infection rates).
+The insets of the pan-
+els moreover show the average of τ/T over the n nodes
+with highest hyper-coreness R or highest s-coreness S.
+The nodes with highest coreness tend to be more often in
+the infectious state, and this tendency is stronger for the
+
+5
+hyper-coreness than for the s-coreness: among the nodes
+with largest value of s-coreness, the hyper-coreness al-
+lows to distinguish which ones are most involved in the
+higher-order spreading processes. Moreover, in the SM
+we show that a similar phenomenology is obtained with a
+different model of contagion involving higher-order mech-
+anisms [37, 54].
+D.
+High hyper-coreness seeds increase total
+outbreak size
+Nodes belonging to large interaction groups have also
+been shown to be optimal seeds of higher-order non-linear
+contagions in terms of spreading speed at the beginning
+of an SIS outbreak [36]. Which nodes have the largest
+spreading power in the long run, i.e., in terms of final
+size reached by a SIR process [12], remains however an
+open question for higher-order spreading processes. We
+thus consider the higher-order non-linear SIR model, in
+which the dynamics, starting from a single seed, evolves
+until no individual is in the state I anymore (only nodes
+in states S or R remain).
+To quantify the “spreading
+power” of each node j considered separately as seed, we
+average the final epidemic size R∞(j), i.e., the number
+of nodes in state R at the end of the process, over 300
+stochastic runs for each seed.
+Figure 3b and 3d show
+that this average final epidemic size ⟨R∞(j)⟩, averaged
+over all nodes of each (k, m)-hyper-core, increase with k
+and m. The insets also show that the nodes with higher
+hyper-coreness lead to larger epidemics, determining a
+hierarchy even among the nodes with highest s-coreness.
+In summary, nodes with higher connectedness along
+groups of larger sizes can seed more efficiently, and the
+hyper-coreness provides a good identification of the nodes
+with highest spreading power in higher-order non-linear
+contagion processes (similar results are shown in the SM
+for another higher-order contagion model [37, 54]).
+E.
+Hypercore seeding facilitates systemic takeover
+by minority norms
+Group interactions can also play an important role in
+the formation of a consensus and the emergence of shared
+conventions in a population. According to critical mass
+theory, regular individuals might then benefit –towards
+addressing societal challenges– from the presence of a
+committed minority that aims at overturning the sta-
+tus quo [56]. Recently, it has been shown that groups
+can modulate this takeover [28]. An important issue in
+this respect concerns the best “seeding” strategy –where
+should the committed minority start from in order to best
+achieve the takeover? Here we show how hyper-coreness
+can provide an answer.
+We consider the well-known naming-game (NG) model
+[38], which describes how a shared convention can emerge
+in a population of locally interacting agents [57, 58], and
+has been shown to account for the outcome of controlled
+experiments of social coordination [59]. In the minimal
+version of this model, recently modified to take group in-
+teractions into account [28], individuals are represented
+by the N nodes of a static hypergraph, and each node is
+endowed with a dictionary that can contain at most two
+names (representing conventions or norms), A and B.
+At each time-step a hyperedge is chosen randomly and
+a speaker is randomly selected within it.
+The speaker
+randomly chooses a name from its dictionary and com-
+municates it to the other hyperedge members (the listen-
+ers), who can agree or not on the proposed name. To
+determine the possibility of an agreement within the hy-
+peredge, we consider two distinct alternatives [28]: (i)
+the union rule, for which an agreement can be reached if
+at least one of the listeners has the proposed name in its
+dictionary; (ii) the unanimity rule, for which the agree-
+ment can be reached only if all the nodes in the group
+have the proposed name in their dictionary. A parame-
+ter β ∈ [0, 1] modulates the social influence by controlling
+the propensity of the listeners to actually accept the local
+consensus: the agreement in the group becomes effective
+only with probability β. In this case, all nodes in the hy-
+peredge add the accepted name to their dictionary, if it
+was not already present, deleting all others. If instead no
+agreement is reached, the listeners simply add the name
+given by the speaker to their dictionaries.
+Crucially, we include in the population a committed
+minority of Np individuals who do not obey the aforemen-
+tioned rules whenever they are listeners, but they instead
+stick to their norm, a single name A, and their dictionary
+is never updated. We initiate the process with the rest of
+the population, i.e. the majority, having only the name
+B in their dictionaries. The system can evolve towards
+different regimes of co-existence of the two names or of
+dominance of one name over the other, depending on β,
+on the considered rule, and on the relative size of the mi-
+nority p = Np/N. In particular, the committed minority
+can overcome the majority, with the whole population
+eventually converging on A, for a range of intermediate
+values of β and for large enough p.
+When committed
+individuals are chosen at random in the population, this
+range increases when the hypergraph contains hyperedges
+of larger sizes [28]. This naturally raises the question of
+whether the committed minority might also benefit from
+belonging to specific substructures of a given hypergraph,
+such as hyper-cores with large connectedness and group
+sizes.
+Here we investigate this issue through numerical simu-
+lations of the higher-order NG process on empirical static
+hypergraphs for varying values of the parameter β and of
+the fraction p of committed individuals. In our numeri-
+cal experiments committed individuals are selected with
+different seeding strategies: (i) at random from the entire
+population (random); (ii) as the Np ones with the high-
+est hyper-coreness R (top hyper-coreness); (iii) as the Np
+ones with the highest s-coreness (top s-coreness) in the
+projected graph. In each experiment we measure the frac-
+tion nA of nodes holding only A in their dictionary (both
+committed or not), and focus on its large time limit n∗
+A.
+This limit can be either 1, if the population reaches the
+absorbing state in which all nodes agree on A, or, if the
+absorbing state is not reached before tmax time-steps, we
+average over 100 values of nA(t) sampled from the last T
+time-steps.
+Figure 4 reports the simulation results for two empir-
+ical data sets, congress-bills (a-d) and the email-EU (e-
+h). The results for the other data sets can be found in
+
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+a
+10
+2
+congress-bills
+Random
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+b
+10
+2
+s-coreness
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+c
+10
+2
+n *
+A
+hyper-coreness
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+e
+10
+2
+email-EU
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+f
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+g
+10
+2
+n *
+A
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+d
+Random
+R-rank
+s-rank
+103
+104
+105
+t
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+h
+Random
+R-rank
+s-rank
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. 4. Comparison of seeding strategies for committed minorities in a higher-order naming-game process. In
+the heatmaps a-c, e-g the stationary fraction n∗
+A of nodes supporting only the name A is shown as a function of the fraction
+of committed nodes p and the agreement probability β (steps correspond to the fact that p varies by increments of 1/N). a-c:
+congress-bills data set with unanimity rule. e-g: email-EU data set with union rule. Committed nodes are selected through
+different strategies, in particular random seeding (a,e), top s-coreness (b,f), and top hyper-coreness (c,g). Panels d,h show
+the temporal evolution of nA(t) according to the different seeding strategies and for fixed values of β and p (cross markers in
+the heatmaps), (d): (β, p) = (0.48, 2.3 × 10−2); (h): (β, p) = (0.55, 9.2 × 10−3). The minority takeover, i.e. n∗
+A = 1, takes
+place for 7.9% of the explored parameter space in panel a, 16.3% in b, 41.5% in c, 37.0% in e, 45.9% in f, and 56.4% in g. All
+simulations are run until the absorbing state n∗
+A = 1 is reached or the dynamics has evolved for tmax = 5 × 105 time steps. The
+stationary fraction n∗
+A is obtained by averaging over 100 values sampled in the last T = 5 × 104 time-steps. Results refer to the
+median values obtained over 200 simulations for each pair of parameter values.
+the SM. For the random strategy, we recover the results
+of Ref. [28]: for low values of β, a co-existence state of
+A and B is observed; at low fraction of committed and
+large β values , the majority remains B. Contrarily, at
+intermediate values of β, the minority takes over and the
+whole population converges on A (in a way favored by the
+union rule w.r.t. the unanimity rule). Non-random seed-
+ing strategies yield the same phenomenology but enhance
+the range of parameters in which the minority overturns
+the majority. This is especially the case when we place
+committed individuals in the most central nodes accord-
+ing to their hyper-coreness. In this case, a tiny fraction
+of committed is able to take over on a wide range of β
+values, while for lower β values, a co-existence regime
+is always observed –due to the small propensity to ac-
+cept a local consensus [28]. For example, with the top
+hyper-coreness strategy a fraction p = 1.51 × 10−2 in
+the congress-bills data set with unanimity rule is enough
+to allow the minority takeover over a range of β values
+whose extension is ∆β ≳ 0.5. This cannot be achieved
+with the other two seeding strategies, for which below
+p = 2.8 × 10−2 only ∆β ≳ 0.2 can be reached (see
+Fig. 4a-c). Analogously, in the email-EU data set with
+the union rule a fraction p = 4.1 × 10−3 is enough to ob-
+tain the minority dominance over ∆β ≳ 0.5 when seeded
+according the top hyper-coreness strategy. In this case,
+with the top s-coreness and the random strategies the
+same result is obtained only for p = 1.33 × 10−2 and
+p = 1.74 × 10−2 respectively (see Fig. 4e-g). To further
+quantify the differences among these strategies we can
+also calculate the value of critical mass pc necessary to
+bring the system to the tipping point while keeping β con-
+stant. In the congress-bills data set with unanimity rule
+and β = 0.62 for instance, the critical mass for the top
+hyper-coreness strategy is pc = 6.4 × 10−3 as compared
+to pc = 2.68 × 10−2 and pc = 2.04 × 10−2 obtained with
+the random and the top s-coreness strategies respectively
+(see Fig. 4a-c); similarly, in the email-EU data set with
+union rule and β = 0.83, these values are respectively
+pc = 3.1 × 10−3, pc = 1.53 × 10−2, pc = 9.2 × 10−3 (see
+Fig. 4e-g).
+The hyper-coreness centrality is thus particularly effec-
+tive in identifying nodes with a crucial role in higher-order
+NG processes. Indeed, nodes belonging to (k, m)-hyper-
+cores with large values of k and m, if committed, can
+convince many others through their simultaneous pres-
+ence in several large groups, and this can be efficiently
+sustained by their large connectedness, favouring conver-
+gence on their convention even outside the committed
+minority. In addition, the temporal evolution diplayed in
+Fig. 4d,h illustrate how, even when all seeding strategies
+lead to the agreement on the convention initially sup-
+
+7
+ported by the minority, the convergence is much faster
+for the hyper-coreness seeding strategy, followed by the
+s-coreness and the random.
+III.
+DISCUSSION
+We have considered here a systematic procedure to ex-
+tract, from a given hypergraph, structures of increasing
+connectedness along increasing group sizes: the (k, m)-
+hyper-cores, in which each node is connected to the other
+by at least k hyperedges (representing higher-order in-
+teractions) of sizes at least m. Using the maximal con-
+nectedness values of each node we define a new concept of
+centrality in hypergraphs: a node hyper-coreness summa-
+rizes its relative depth in the hierarchies of hyper-cores at
+all orders (interaction sizes). Applying these concepts to
+empirical data describing a variety of higher-order sys-
+tems, we have shown how the (k, m)-hyper-cores pro-
+vide a fingerprint of empirical hypergraphs.
+Crucially,
+we have also highlighted how hyper-cores with increas-
+ing k and m play important roles in several dynamic
+processes with higher-order mechanisms unfolding upon
+hypergraphs, such as contagion processes and consensus
+formation.
+The hyper-coreness centrality in particular
+identifies nodes with high spreading power and on which
+stationary contagion processes tend to localize; moreover
+nodes with high hyper-coreness, if belonging to a commit-
+ted minority, can be particularly efficient at overturning
+a majority convention.
+Our work opens the door to several research directions
+in the expanding field of hypergraphs structure and dy-
+namics. It can provide an additional systematic charac-
+terization of both empirical and model hypergraphs, and
+thus potentially a model validation tool as well as a com-
+parison method between hypergraphs (e.g. by computing
+distances between the (k, m)-hypercore profiles of Fig. 2).
+For specific systems where additional properties of the
+nodes are known, the shell indices and hyper-coreness
+values of nodes could be compared in more detail to pro-
+vide insights into their relative positions and roles in the
+system.
+Moreover, while here we focused on static hypergraphs,
+many such systems evolve in time [60, 61]. Hyper-cores
+and hyper-coreness could be used to investigate the evo-
+lution of the higher-order interactions at multiple scales,
+from the global evolution of the structure described by
+hyper-core sizes, to the shell indices and hyper-coreness
+of individual nodes from one period to the next [8]. An
+interesting case study in this direction could be for in-
+stance the evolution of the hyper-core positions of scien-
+tists in co-authorship “networks”, which are in fact by
+construction evolving hypergraphs [60].
+IV.
+MATERIALS AND METHODS
+A.
+Data description and preprocessing
+Several data sets we considered are publicly available in
+the form of static hypergraphs, thus they do not require
+any preprocessing. These data sets describe:
+• email communications: within a European institu-
+tion (email-EU [44]), and within Enron, between
+a core-set of workers (email-Enron [45, 46]). Each
+node corresponds to an email address and a hy-
+peredge includes the sender and all receivers of an
+email.
+• interactions in legislative bills in the U.S. Congress
+(congress-bills) and in the U.S. Senate (senate-bills)
+[46, 49, 51, 52]: each node corresponds to a member
+of the U.S. Congress or Senate and a hyperedge
+involves sponsors and co-sponsors of legislative bills
+discussed in the Congress or Senate.
+• interactions in committees in the U.S. House of
+Representatives (house-committees) and in the U.S.
+Senate (senate-committees) [46, 49, 50]: each node
+corresponds to a member of the U.S. House of Rep-
+resentatives or Senate and each hyperedge involves
+nodes that share membership in a committee.
+• online interactions (3 data sets):
+exchanges be-
+tween users of MathOverflow on algebra top-
+ics
+(algebra-questions)
+or
+on
+geometry
+topics
+(geometry-questions), in which each node corre-
+sponds to a user of MathOverflow and each hyper-
+edge involves those users who have answered a spe-
+cific question belonging to the topic of algebra or ge-
+ometry [46, 48]; interactions between Amazon users
+on music (music-review [46, 47]), in which each node
+corresponds to an Amazon user and each hyperedge
+involves users who have reviewed a specific product
+belonging to the category of blues music.
+Moreover, we built static hypergraphs from several
+data sets of time-resolved face-to-face human interac-
+tions, as in [26, 28]. The data sets are provided by the
+SocioPatterns collaboration [40–42] and by the Contacts
+among Utah’s School-age Population (CUSP) project
+[43] and describe interactions between individuals in sev-
+eral contexts :
+a hospital (LH10 [62]), a workplace
+(InVS15 [41, 63]), a conference (SFHH [41]), a high-school
+(Thiers13 [64]), two primary-schools (LyonSchool [65],
+Elem1 [43]) and a middle-school (Mid1 [43]). For these
+data sets we carried out an aggregation procedure to ob-
+tain static hypergraphs: (i) we aggregate the data over
+time windows of 15 minutes; (ii) we identify the cliques
+in each time window, i.e. groups of nodes forming a fully
+connected cluster, (iii) we identify in each temporal win-
+dow the maximum cliques, i.e.
+cliques not completely
+contained in a larger clique, and promote them to a hy-
+peredge status.
+Overall, the data sets considered describe interactions
+in several different environments, mediated by different
+mechanisms. They correspond to a wide variety of sta-
+tistical properties (e.g. data set size, hyperedges size dis-
+tributions), as shown in the SM where these statistical
+properties of the data sets are reported in details.
+
+8
+B.
+Models and stochastic simulations
+1.
+Higher-order non-linear contagion
+We performed stochastic numerical simulations of the
+higher-order non-linear contagion model on each empir-
+ical static hypergraph.
+The simulations are performed
+with discrete time-steps. The S → I infection mecha-
+nism is the same for the SIR and the SIS models: for
+each time-step ∆t, given a hyperedge of size m contain-
+ing i infected nodes, each of the susceptible nodes in it
+can be infected with probability (1 − e−λiν). Therefore,
+the probability that a node j is infected in a time-step
+∆t is:
+pj = 1 −
+�
+e∈E(j)
+e−λiν
+e ,
+(2)
+where E(j) denotes the set of hyperedges in which the
+node j is involved and ie is the number of infected nodes
+in the hyperedge e.
+Each infected node heals (return-
+ing susceptible in SIS or gaining immunity in SIR) with
+probability µ in each time-step.
+In the SIS process the population is initialized with a
+single infectious seed randomly selected in the population
+and the process is iterated until the system reaches a
+steady state with a fluctuating number of infectious. An
+observation time window T is then considered and the
+time τ spent in the infectious state is estimated for all
+nodes over that time-window. The results are averaged
+over 103 simulations.
+In the SIR process the population is initialized with a
+single infectious seed j and the dynamic process is iter-
+ated until no more infectious nodes are present: the final
+epidemic size R∞(j) obtained by seeding the infection in
+j is defined as the final number of nodes in the R state.
+The results are averaged over 300 simulations for each
+infection seed j.
+2.
+Higher-order NG process
+We also performed numerical simulations of the higher-
+order NG process on the empirical hypergraphs.
+The
+system with N nodes is initialized by fixing Np nodes as
+belonging to the committed minority (equivalently, with
+a fraction p = Np/N of committed nodes), with only the
+name A in their dictionary, and setting the dictionaries
+of all the other nodes of the majority with only the name
+B. The committed nodes are selected following one of
+the three seeding strategies, i.e. randomly from the whole
+population or as the Np nodes with highest s-coreness or
+hyper-coreness. If several nodes have the same coreness
+value, the committed nodes are randomly selected within
+the coreness class.
+The simulations are performed in discrete time-steps:
+at each time-step a hyperedge is randomly selected (ac-
+tivation of the group) and within it a node is randomly
+chosen as the speaker, while the other nodes behave as
+listeners. The speaker randomly selects a name in their
+dictionary and all nodes in the group update their dictio-
+nary according to the chosen agreement rule (except for
+the committed nodes). The process is iterated until the
+system reaches the absorbing state where all nodes have
+only the name A in their dictionary, i.e. nA(t) = n∗
+A = 1,
+or until the system has evolved for tmax time-steps: in
+this last case the stationary fraction of nodes with the
+name A in their dictionary n∗
+A is obtained by averaging
+nA(t) over 100 values sampled in the last T = 50, 000
+time-steps. The results refer to the median values ob-
+tained over 200 simulations.
+ACKNOWLEDGEMENTS
+M.M. and A.B. acknowledge support from the Agence
+Nationale de la Recherche (ANR) project DATAREDUX
+(ANR-19-CE46-0008).
+I.I. acknowledges support from
+the James S. McDonnell Foundation 21st Century Sci-
+ence Initiative Understanding Dynamic and Multi-scale
+Systems - Postdoctoral Fellowship Award.
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+
+Supplementary Material for ”Hyper-cores promote localization and efficient seeding
+in higher-order processes”
+Marco Mancastroppa,1 Iacopo Iacopini,2 Giovanni Petri,3 and Alain Barrat1
+1Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT,
+Turing Center for Living Systems, Marseille, France
+2Department of Network and Data Science, Central European University, 1100 Vienna, Austria
+3CENTAI, Corso Inghilterra 3, 10138 Turin, Italy
+In this Supplementary Material we present the same results as in the main text for all the considered data sets
+and also further results. We first present in detail some of the statistical properties of the data sets and of the
+static hypergraphs considered (Supplementary Section 1). In Supplementary Section 2 we present the results of
+the (k, m)-core decomposition, showing how the (k, m)-cores and (k, m)-shells are populated as a function of k
+and m, the functional form of the m-shell index Cm(i) for some nodes, the distributions of the hypercoreness and
+s-coreness centralities and their correlations. In Supplementary Section 3 we present the results of the higher-order
+non-linear contagion process [1], both in the SIS and SIR formulation. In Supplementary Section 4 we introduce
+in details the threshold higher-order process [2], its numerical implementation and its results in relation to the
+hyper-cores, both in the SIS and SIR formulation, as done for the higher-order non-linear contagion model. Finally
+in Supplementary Section 5, the results of the higher-order naming-game process [3] are presented for both the
+union and the unanimity rules.
+Supplementary Section 1.
+Properties of the data sets
+The considered data sets describe interactions in several environments, mediated by different mechanisms, and
+thus they differ in their fundamental statistical properties. This is summarized in Table I and Fig. 1: the number
+of nodes and hyperedges vary among the data sets considered, the distribution Ψ(m) of the hyperedge sizes m, the
+range of their sizes m ∈ [2, M] and the average hyperedge size ⟨m⟩ are different among the data sets.
+data set
+N
+E
+M
+⟨m⟩
+LH10
+76
+1 102
+7
+3.4
+Thiers13
+327
+4 795
+7
+3.1
+InVS15
+217
+3 279
+10
+2.8
+SFHH
+403
+6 398
+10
+2.7
+LyonSchool
+242
+10 848 10
+4.0
+Mid1
+591
+61 521 13
+3.9
+Elem1
+339
+20 940 16
+4.7
+email-EU
+979
+24 399 25
+3.5
+data set
+N
+E
+M
+⟨m⟩
+congress-bills
+1 718
+83 105
+25
+8.8
+senate-committees
+282
+302
+31
+17.6
+email-Enron
+143
+1 459
+37
+3.1
+house-committees
+1 290
+335
+82
+35.3
+music-review
+1 106
+686
+83
+15.3
+senate-bills
+294
+21 721
+99
+9.9
+algebra-questions
+423
+980
+107
+7.6
+geometry-questions
+580
+888
+230
+13.0
+Supplementary Table I: Some properties of the data sets. The tables give: the number of nodes N, the
+number of hyperedges E, the maximum size of the hyperedges M and the average size of the hyperedges ⟨m⟩.
+arXiv:2301.04235v1  [physics.soc-ph]  10 Jan 2023
+
+2
+2
+4
+6
+102
+(m)
+a
+2
+4
+6
+101
+102
+103
+b
+5
+10
+100
+101
+102
+103
+c
+5
+10
+100
+101
+102
+103
+d
+5
+10
+100
+101
+102
+103
+(m)
+e
+5
+10
+101
+102
+103
+104
+f
+5
+10
+15
+101
+102
+103
+g
+10
+20
+102
+103
+104
+h
+10
+20
+104
+(m)
+i
+10
+20
+30
+100
+101
+j
+0
+20
+40
+100
+101
+102
+103
+k
+0
+50
+100
+101
+l
+0
+50
+m
+100
+101
+(m)
+m
+0
+50
+100
+m
+100
+101
+102
+103
+n
+0
+50
+100
+m
+100
+101
+102
+o
+0
+100
+200
+m
+100
+101
+102
+p
+Supplementary Figure 1: Hyperedge size distribution. We show the hyperedge size distribution Ψ(m), i.e. the
+number of hyperedges of size m, for all the data sets: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c),
+SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills
+(panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel
+m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).
+
+3
+Supplementary Section 2.
+Hyper-core decomposition
+1
+9
+17
+25
+33
+41
+49
+2
+3
+4
+5
+6
+7
+m
+n(k, m)
+a
+100
+2 × 10
+1
+3 × 10
+1
+4 × 10
+1
+6 × 10
+1
+1
+8
+15
+22
+29
+36
+2
+3
+4
+5
+6
+7
+n(k, m)
+b
+10
+1
+100
+1
+9
+17
+25
+33
+41
+2
+3
+4
+5
+6
+7
+8
+9
+10
+n(k, m)
+c
+10
+1
+100
+1
+7
+13
+19
+25
+31
+2
+3
+4
+5
+6
+7
+8
+9
+10
+n(k, m)
+d
+10
+1
+100
+1
+28
+55
+82
+109
+136
+2
+3
+4
+5
+6
+7
+8
+9
+10
+m
+e
+10
+1
+100
+1
+47
+93
+139
+185
+231
+277
+2
+4
+6
+8
+10
+12
+f
+10
+1
+100
+1
+48
+95
+142
+189
+236
+2
+4
+6
+8
+10
+12
+14
+16
+g
+10
+1
+100
+1
+20
+39
+58
+77
+96
+2
+6
+10
+14
+18
+22
+h
+10
+1
+100
+1
+183
+365
+547
+729
+911
+1093
+2
+6
+10
+14
+18
+22
+m
+i
+10
+1
+100
+1
+6
+11
+16
+21
+26
+31
+2
+7
+12
+17
+22
+27
+j
+100
+2 × 10
+1
+3 × 10
+1
+4 × 10
+1
+6 × 10
+1
+1
+5
+9
+13
+17
+21
+25
+2
+8
+14
+20
+26
+32
+k
+100
+2 × 10
+1
+3 × 10
+1
+4 × 10
+1
+6 × 10
+1
+1
+4
+7
+10
+13
+16
+19
+2
+16
+30
+44
+58
+72
+l
+10
+1
+100
+1
+8
+15
+22
+29
+36
+k
+2
+16
+30
+44
+58
+72
+m
+m
+10
+1
+100
+1
+199
+397
+595
+793
+991
+k
+2
+18
+34
+50
+66
+82
+98
+n
+100
+2 × 10
+1
+3 × 10
+1
+4 × 10
+1
+6 × 10
+1
+1
+11
+21
+31
+41
+51
+k
+2
+20
+38
+56
+74
+92
+o
+10
+1
+100
+1
+15
+29
+43
+57
+71
+k
+2
+40
+78
+116
+154
+192
+230
+p
+10
+1
+100
+Supplementary Figure 2: Hyper-core decomposition I. All panels show colormaps giving the relative size
+n(k,m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a
+function of m and k (white regions correspond to n(k,m) = 0). The following data sets are considered: LH10
+(panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1
+(panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k),
+house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and
+geometry-questions (panel p).
+
+4
+0
+20
+40
+0.2
+0.4
+0.6
+0.8
+1.0
+n(k, m)
+a
+m = 2.0
+m = 3.0
+m = 4.0
+m = 5.0
+0
+20
+40
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+b
+m = 2.0
+m = 3.0
+m = 4.0
+m = 5.0
+0
+20
+40
+0.2
+0.4
+0.6
+0.8
+1.0
+c
+m = 2.0
+m = 3.0
+m = 4.0
+m = 5.0
+0
+10
+20
+30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+d
+m = 2.0
+m = 3.0
+m = 4.0
+m = 5.0
+0
+100
+200
+0.2
+0.4
+0.6
+0.8
+1.0
+n(k, m)
+e
+m = 2.0
+m = 3.0
+m = 4.0
+m = 5.0
+0
+200
+400
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+f
+m = 2.0
+m = 4.0
+m = 6.0
+m = 8.0
+0
+200
+400
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+g
+m = 2.0
+m = 4.0
+m = 6.0
+m = 8.0
+0
+50
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+h
+m = 2.0
+m = 6.0
+m = 10.0
+m = 14.0
+0
+500
+1000
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+n(k, m)
+i
+m = 2.0
+m = 6.0
+m = 10.0
+m = 14.0
+0
+10
+20
+30
+0.2
+0.4
+0.6
+0.8
+1.0
+j
+m = 2.0
+m = 12.0
+m = 17.0
+m = 22.0
+0
+10
+20
+0.2
+0.4
+0.6
+0.8
+1.0
+k
+m = 2.0
+m = 4.0
+m = 6.0
+m = 8.0
+5
+10
+15
+0.2
+0.4
+0.6
+0.8
+1.0
+l
+m = 2.0
+m = 15.0
+m = 41.0
+m = 54.0
+0
+20
+40
+k
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+n(k, m)
+m
+m = 2.0
+m = 10.0
+m = 18.0
+m = 32.0
+0
+500
+1000
+k
+0.2
+0.4
+0.6
+0.8
+1.0
+n
+m = 2.0
+m = 18.0
+m = 34.0
+m = 50.0
+0
+20
+40
+60
+k
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+o
+m = 2.0
+m = 8.0
+m = 14.0
+m = 20.0
+0
+25
+50
+75
+k
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+p
+m = 2.0
+m = 15.0
+m = 28.0
+m = 41.0
+Supplementary Figure 3: Hyper-core decomposition II. All panels show the relative size n(k,m) (number of
+nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a function of k for
+fixed values of m. The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c),
+SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills
+(panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel
+m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).
+
+5
+1
+9
+17
+25
+33
+41
+49
+2
+3
+4
+5
+6
+7
+m
+s(k, m)
+a
+10
+1
+1
+8
+15
+22
+29
+36
+2
+3
+4
+5
+6
+7
+s(k, m)
+b
+10
+2
+10
+1
+1
+9
+17
+25
+33
+41
+2
+3
+4
+5
+6
+7
+8
+9
+10
+s(k, m)
+c
+10
+2
+10
+1
+1
+7
+13
+19
+25
+31
+2
+3
+4
+5
+6
+7
+8
+9
+10
+s(k, m)
+d
+10
+2
+10
+1
+1
+28
+55
+82
+109
+136
+2
+3
+4
+5
+6
+7
+8
+9
+10
+m
+e
+10
+2
+10
+1
+1
+47
+93
+139
+185
+231
+277
+2
+4
+6
+8
+10
+12
+f
+10
+2
+10
+1
+1
+48
+95
+142
+189
+236
+2
+4
+6
+8
+10
+12
+14
+16
+g
+10
+2
+10
+1
+1
+20
+39
+58
+77
+96
+2
+6
+10
+14
+18
+22
+h
+10
+2
+10
+1
+1
+183
+365
+547
+729
+911
+1093
+2
+6
+10
+14
+18
+22
+m
+i
+10
+3
+10
+2
+10
+1
+1
+6
+11
+16
+21
+26
+31
+2
+7
+12
+17
+22
+27
+j
+10
+2
+10
+1
+1
+5
+9
+13
+17
+21
+25
+2
+8
+14
+20
+26
+32
+k
+10
+2
+10
+1
+1
+4
+7
+10
+13
+16
+19
+2
+16
+30
+44
+58
+72
+l
+10
+1
+1
+8
+15
+22
+29
+36
+k
+2
+16
+30
+44
+58
+72
+m
+m
+10
+3
+10
+2
+10
+1
+1
+199
+397
+595
+793
+991
+k
+2
+18
+34
+50
+66
+82
+98
+n
+10
+2
+10
+1
+1
+11
+21
+31
+41
+51
+k
+2
+20
+38
+56
+74
+92
+o
+10
+2
+10
+1
+1
+15
+29
+43
+57
+71
+k
+2
+40
+78
+116
+154
+192
+230
+p
+10
+2
+10
+1
+Supplementary Figure 4: (k, m)-shells. All panels show colormaps giving the relative size s(k,m) (number of
+nodes in the hyper-shell, divided by the total number of nodes N) of the (k, m)-shell as a function of m and k
+(white regions correspond to s(k,m) = 0). The following data sets are considered: LH10 (panel a), Thiers13 (panel
+b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel
+h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l),
+music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).
+
+6
+2
+4
+6
+0.00
+0.25
+0.50
+0.75
+1.00
+Cm(i)/km
+max
+a
+R = 0.08
+R = 0.7
+R = 3.1
+R = 6.0
+2
+4
+6
+0.00
+0.25
+0.50
+0.75
+1.00
+b
+R = 0.05
+R = 1.6
+R = 3.0
+R = 6.0
+2.5
+5.0
+7.5
+10.0
+0.00
+0.25
+0.50
+0.75
+1.00
+c
+R = 0.02
+R = 0.8
+R = 1.1
+R = 9.0
+2.5
+5.0
+7.5
+10.0
+0.00
+0.25
+0.50
+0.75
+1.00
+d
+R = 0.03
+R = 1.7
+R = 2.4
+R = 8.0
+2.5
+5.0
+7.5
+10.0
+0.00
+0.25
+0.50
+0.75
+1.00
+Cm(i)/km
+max
+e
+R = 0.4
+R = 3.1
+R = 3.8
+R = 6.2
+5
+10
+0.00
+0.25
+0.50
+0.75
+1.00
+f
+R = 0.3
+R = 3.0
+R = 3.6
+R = 9.6
+5
+10
+15
+0.00
+0.25
+0.50
+0.75
+1.00
+g
+R = 0.2
+R = 3.1
+R = 4.6
+R = 10.9
+10
+20
+0.00
+0.25
+0.50
+0.75
+1.00
+h
+R = 0.009
+R = 1.0
+R = 5.8
+R = 22.2
+10
+20
+0.00
+0.25
+0.50
+0.75
+1.00
+Cm(i)/km
+max
+i
+R = 0.003
+R = 7.6
+R = 14.4
+R = 24.0
+10
+20
+30
+0.00
+0.25
+0.50
+0.75
+1.00
+j
+R = 1.0
+R = 13.9
+R = 22.3
+R = 30.0
+10
+20
+30
+0.00
+0.25
+0.50
+0.75
+1.00
+k
+R = 0.1
+R = 6.3
+R = 27.5
+R = 36.0
+0
+25
+50
+75
+0.00
+0.25
+0.50
+0.75
+1.00
+l
+R = 0.2
+R = 27.1
+R = 50.4
+R = 81.0
+0
+25
+50
+75
+m
+0.00
+0.25
+0.50
+0.75
+1.00
+Cm(i)/km
+max
+m
+R = 0.03
+R = 37.8
+R = 53.5
+R = 82.0
+0
+50
+100
+m
+0.00
+0.25
+0.50
+0.75
+1.00
+n
+R = 0.1
+R = 48.4
+R = 89.4
+R = 98.0
+0
+50
+100
+m
+0.00
+0.25
+0.50
+0.75
+1.00
+o
+R = 0.02
+R = 3.1
+R = 78.4
+R = 106.0
+0
+100
+200
+m
+0.00
+0.25
+0.50
+0.75
+1.00
+p
+R = 0.01
+R = 11.2
+R = 196.5
+R = 229.0
+Supplementary Figure 5: m-shell index. All panels show the normalized m-shell index function Cm(i)/km
+max as a
+function of m for four nodes: one node is selected randomly among the nodes in the class with highest
+hyper-coreness R; one node is selected randomly among the nodes in the class with smallest hyper-coreness R; the
+two remaining node are selected from intermediate hyper-coreness classes, so that the positions in the
+hyper-coreness ranking of the four nodes are equispaced. The following data sets are considered: LH10 (panel a),
+Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g),
+email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k),
+house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and
+geometry-questions (panel p).
+
+7
+0
+50
+0
+2
+4
+6
+R(i)
+a
+0
+5
+R
+0
+20
+P(R)
+0
+200
+0
+2
+4
+6
+b
+0
+5
+R
+0
+50
+P(R)
+0
+100
+200
+0
+2
+4
+6
+8
+c
+0
+5
+R
+0
+50
+P(R)
+0
+200
+400
+0
+2
+4
+6
+8
+d
+0
+5
+R
+0
+100
+P(R)
+0
+100
+200
+0
+2
+4
+6
+8
+R(i)
+e
+2.5 5.0
+R
+0
+50
+P(R)
+0
+250
+500
+0.0
+2.5
+5.0
+7.5
+10.0
+f
+0
+10
+R
+0
+200
+P(R)
+0
+200
+0.0
+2.5
+5.0
+7.5
+10.0
+g
+0
+10
+R
+0
+100
+P(R)
+0
+500
+1000
+0
+5
+10
+15
+20
+h
+0
+20
+R
+0
+250
+P(R)
+0
+1000
+0
+10
+20
+30
+R(i)
+i
+0
+25
+R
+0
+250
+P(R)
+0
+200
+0
+10
+20
+30
+40
+j
+0
+25
+R
+0
+50
+P(R)
+0
+100
+0
+20
+40
+k
+0
+25
+R
+0
+20
+P(R)
+0
+500
+1000
+0
+25
+50
+75
+100
+l
+0
+50
+R
+0
+100
+P(R)
+0
+500
+1000
+Rank(i)
+0
+25
+50
+75
+100
+R(i)
+m
+0
+50
+R
+0
+200
+P(R)
+0
+200
+Rank(i)
+0
+50
+100
+n
+0
+100
+R
+0
+100
+P(R)
+0
+200
+400
+Rank(i)
+0
+50
+100
+o
+0
+100
+R
+0
+100
+P(R)
+0
+250
+500
+Rank(i)
+0
+100
+200
+300
+p
+0
+200
+R
+0
+200
+P(R)
+Supplementary Figure 6: Hyper-coreness centrality. In all panels the hyper-coreness R(i) is plotted as a
+function of the corresponding node rank: the insets show the distribution P(R) of the hyper-coreness. The
+following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d),
+LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i),
+senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m),
+senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).
+
+8
+0
+50
+0
+50
+100
+150
+200
+S(i)
+a
+0
+200
+S
+0
+10
+P(S)
+0
+200
+0
+50
+100
+b
+0
+100
+S
+0
+50
+P(S)
+0
+100
+200
+0
+50
+100
+150
+c
+0
+100
+S
+0
+100
+P(S)
+0
+200
+400
+0
+50
+100
+d
+0
+100
+S
+0
+100
+P(S)
+0
+100
+200
+0
+200
+400
+600
+800
+S(i)
+e
+250 500
+S
+0
+50
+P(S)
+0
+250
+500
+0
+500
+1000
+1500
+f
+0
+1000
+S
+0
+200
+P(S)
+0
+200
+0
+500
+1000
+1500
+g
+0
+1000
+S
+0
+100
+P(S)
+0
+500
+1000
+0
+250
+500
+750
+1000
+h
+0
+1000
+S
+0
+250
+P(S)
+0
+1000
+0
+2000
+4000
+6000
+S(i)
+i
+0
+5000
+S
+0
+200
+P(S)
+0
+200
+0
+100
+200
+300
+400
+j
+0
+150 300
+S
+0
+50
+P(S)
+0
+100
+0
+50
+100
+150
+k
+0
+100
+S
+0
+20
+P(S)
+0
+500
+1000
+0
+200
+400
+l
+0
+250
+S
+0
+250
+P(S)
+0
+500
+1000
+Rank(i)
+0
+100
+200
+300
+400
+S(i)
+m
+0
+250
+S
+0
+200
+P(S)
+0
+200
+Rank(i)
+0
+5000
+10000
+15000
+n
+0
+10000
+S
+0
+100
+P(S)
+0
+200
+400
+Rank(i)
+0
+100
+200
+300
+400
+o
+0
+250
+S
+0
+100
+P(S)
+0
+250
+500
+Rank(i)
+0
+250
+500
+750
+1000
+p
+0
+1000
+S
+0
+200
+P(S)
+Supplementary Figure 7: s-coreness centrality. In all panels the s-coreness S(i) is plotted as a function of the
+corresponding node rank: the insets give the distribution P(S) of the s-coreness. The following data sets are
+considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1
+(panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron
+(panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o)
+and geometry-questions (panel p).
+
+9
+0
+100
+200
+0
+2
+4
+6
+R(i)
+a
+= 0.98
+= 0.92
+0
+50
+100
+0
+2
+4
+6
+b
+= 0.93
+= 0.88
+0
+100
+0
+2
+4
+6
+8
+c
+= 0.94
+= 0.83
+0
+50
+100
+0
+2
+4
+6
+8
+d
+= 0.77
+= 0.75
+250
+500
+2
+4
+6
+R(i)
+e
+= 0.73
+= 0.51
+500
+1000
+0.0
+2.5
+5.0
+7.5
+10.0
+f
+= 0.74
+= 0.74
+0
+1000
+0.0
+2.5
+5.0
+7.5
+10.0
+g
+= 0.82
+= 0.69
+0
+500
+1000
+0
+5
+10
+15
+20
+h
+= 0.90
+= 0.85
+0
+5000
+0
+10
+20
+R(i)
+i
+= 0.92
+= 0.83
+100
+200
+300
+0
+10
+20
+30
+j
+= 0.93
+= 0.78
+0
+50
+100
+0
+10
+20
+30
+k
+= 0.53
+= 0.53
+0
+200
+0
+20
+40
+60
+80
+l
+= 0.92
+= 0.79
+0
+200
+S(i)
+0
+20
+40
+60
+80
+R(i)
+m
+= 0.74
+= 0.58
+0
+10000
+S(i)
+0
+25
+50
+75
+100
+n
+= 0.96
+= 0.85
+0
+200
+S(i)
+0
+25
+50
+75
+100
+o
+= 0.92
+= 0.88
+0
+500
+1000
+S(i)
+0
+100
+200
+p
+= 0.88
+= 0.88
+Supplementary Figure 8: Hyper-coreness vs. s-coreness centralities. All panels show scatterplots of the
+hyper-coreness R(i) vs. the s-coreness S(i) for all nodes: the text-box reports the Pearson correlation coefficient ρ
+of R(i) and S(i) and the Kendall’s τ coefficient of the corresponding node rankings (in all cases the p-value for
+both the coefficients is p ≪ 0.001). The following data sets are considered: LH10 (panel a), Thiers13 (panel b),
+InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h),
+congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l),
+music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).
+
+10
+Supplementary Section 3.
+Higher-order non-linear contagion process
+data set
+ν
+λ
+LH10
+4.0
+5 × 10−4
+Thiers13
+3.0
+5 × 10−4
+InVS15
+4.0
+5 × 10−4
+SFHH
+4.0
+5 × 10−4
+LyonSchool
+2.0
+5 × 10−4
+Mid1
+3.0
+5 × 10−5
+Elem1
+1.5
+5 × 10−5
+email-EU
+2.5
+5 × 10−5
+data set
+ν
+λ
+congress-bills
+2.0
+5 × 10−6
+senate-committees
+1.25
+5 × 10−4
+email-Enron
+2.0
+5 × 10−4
+house-committees
+1.25
+5 × 10−4
+music-review
+1.25
+5 × 10−4
+senate-bills
+1.5
+5 × 10−6
+algebra-questions
+1.25
+5 × 10−4
+geometry-questions
+1.5
+5 × 10−5
+Supplementary Table II: Parameters for Figs. 9-11. The tables summarize the parameters of the higher-order
+non-linear SIS contagion process considered for each data set in Figs. 9-11.
+data set
+ν
+λ
+LH10
+1.5
+0.010
+Thiers13
+4.0
+0.001
+InVS15
+4.0
+0.001
+SFHH
+4.0
+0.010
+LyonSchool
+4.0
+0.001
+Mid1
+4.0
+5 × 10−5
+Elem1
+4.0
+10−4
+email-EU
+4.0
+5 × 10−5
+data set
+ν
+λ
+congress-bills
+1.5
+5 × 10−5
+senate-committees
+4.0
+10−4
+email-Enron
+4.0
+5 × 10−4
+house-committees
+4.0
+5 × 10−5
+music-review
+3.0
+5 × 10−4
+senate-bills
+4.0
+5 × 10−5
+algebra-questions
+4.0
+0.001
+geometry-questions
+4.0
+5 × 10−4
+Supplementary Table III: Parameters for Figs. 12-14. The tables summarize the parameters of the
+higher-order non-linear SIR contagion process considered for each data sets in Figs. 12-14.
+
+11
+1
+9
+17
+25
+33
+41
+49
+2
+3
+4
+5
+6
+7
+m
+/T
+a
+0.70
+0.75
+0.80
+0.85
+0
+50
+n
+0.7
+0.8
+0.9
+/T n
+1
+8
+15
+22
+29
+36
+2
+3
+4
+5
+6
+7
+/T
+b
+0.4
+0.5
+0.6
+0.7
+0
+250
+n
+0.4
+0.6
+/T n
+1
+9
+17
+25
+33
+41
+2
+3
+4
+5
+6
+7
+8
+9
+10
+/T
+c
+0.6
+0.7
+0.8
+0
+200
+n
+0.6
+0.8
+/T n
+1
+7
+13
+19
+25
+31
+2
+3
+4
+5
+6
+7
+8
+9
+10
+/T
+d
+0.60
+0.65
+0.70
+0.75
+0.80
+0
+250
+n
+0.6
+0.7
+0.8
+/T n
+1
+28
+55
+82
+109
+136
+2
+3
+4
+5
+6
+7
+8
+9
+10
+m
+e
+0.78
+0.80
+0.82
+0.84
+0.86
+0.88
+0
+200
+n
+0.80
+0.85
+0.90
+/T n
+1
+47
+93
+139
+185
+231
+277
+2
+4
+6
+8
+10
+12
+f
+0.74
+0.76
+0.78
+0.80
+0.82
+0
+500
+n
+0.75
+0.80
+/T n
+1
+48
+95
+142
+189
+236
+2
+4
+6
+8
+10
+12
+14
+16
+g
+0.1
+0.2
+0.3
+0
+200
+n
+0.1
+0.2
+0.3
+/T n
+1
+20
+39
+58
+77
+96
+2
+6
+10
+14
+18
+22
+h
+0.4
+0.5
+0.6
+0
+400 800
+n
+0.4
+0.6
+/T n
+1
+183
+365
+547
+729
+911
+1093
+2
+6
+10
+14
+18
+22
+m
+i
+0.14
+0.16
+0.18
+0.20
+0
+1000
+n
+0.15
+0.20
+/T n
+1
+6
+11
+16
+21
+26
+31
+2
+7
+12
+17
+22
+27
+j
+0.60
+0.65
+0.70
+0.75
+0.80
+0
+200
+n
+0.6
+0.7
+0.8
+/T n
+1
+5
+9
+13
+17
+21
+25
+2
+8
+14
+20
+26
+32
+k
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0
+100
+n
+0.6
+0.8
+/T n
+1
+4
+7
+10
+13
+16
+19
+2
+16
+30
+44
+58
+72
+l
+0.70
+0.75
+0.80
+0.85
+0
+1000
+n
+0.7
+0.8
+/T n
+1
+8
+15
+22
+29
+36
+k
+2
+16
+30
+44
+58
+72
+m
+m
+0.6
+0.7
+0.8
+0
+1000
+n
+0.6
+0.8
+/T n
+S-rank
+R-rank
+1
+199
+397
+595
+793
+991
+k
+2
+18
+34
+50
+66
+82
+98
+n
+0.60
+0.65
+0.70
+0.75
+0.80
+0
+200
+n
+0.6
+0.7
+0.8
+/T n
+S-rank
+R-rank
+1
+11
+21
+31
+41
+51
+k
+2
+20
+38
+56
+74
+92
+o
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+250
+n
+0.4
+0.6
+0.8
+/T n
+S-rank
+R-rank
+1
+15
+29
+43
+57
+71
+k
+2
+40
+78
+116
+154
+192
+230
+p
+0.4
+0.5
+0.6
+0.7
+0
+500
+n
+0.4
+0.6
+0.8
+/T n
+S-rank
+R-rank
+Supplementary Figure 9: Higher-order non-linear contagion process - SIS model - I. All panels give, as a
+heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state
+averaged over the nodes of the (k, m)-hyper-core. The insets represent τ/T averaged over the first n nodes
+according to the coreness rankings as a function of n. All results are obtained by averaging the results of 103
+numerical simulations, with a single random seed of infection and with an observation window T = 103. The
+following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),
+Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),
+music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered
+for each data set are reported in Table II.
+
+12
+0
+20
+40
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+/T
+a
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+0.4
+0.5
+0.6
+0.7 b
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85 c
+m = 2
+m = 3
+m = 4
+m = 5
+0
+10
+20
+30
+0.60
+0.65
+0.70
+0.75
+0.80 d
+m = 2
+m = 3
+m = 4
+m = 5
+0
+100
+200
+0.82
+0.84
+0.86
+0.88
+/T
+e
+m = 2
+m = 3
+m = 4
+m = 5
+0
+200
+400
+0.74
+0.76
+0.78
+0.80
+0.82
+f
+m = 2
+m = 4
+m = 6
+m = 8
+0
+200
+400
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35 g
+m = 2
+m = 4
+m = 6
+m = 8
+0
+50
+100
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60 h
+m = 2
+m = 6
+m = 10
+m = 14
+0
+500
+1000
+0.12
+0.14
+0.16
+0.18
+0.20
+/T
+i
+m = 2
+m = 6
+m = 10
+m = 14
+0
+10
+20
+30
+0.60
+0.65
+0.70
+0.75
+0.80 j
+m = 2
+m = 12
+m = 17
+m = 22
+0
+10
+20
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80 k
+m = 2
+m = 4
+m = 6
+m = 8
+5
+10
+15
+0.70
+0.75
+0.80
+0.85
+l
+m = 2
+m = 15
+m = 41
+m = 54
+0
+20
+40
+k
+0.5
+0.6
+0.7
+0.8
+/T
+m
+m = 2
+m = 10
+m = 18
+m = 32
+0
+500
+1000
+k
+0.60
+0.65
+0.70
+0.75
+0.80
+n
+m = 2
+m = 18
+m = 34
+m = 50
+0
+20
+40
+60
+k
+0.4
+0.5
+0.6
+0.7
+0.8
+o
+m = 2
+m = 8
+m = 14
+m = 20
+0
+25
+50
+75
+k
+0.4
+0.5
+0.6
+0.7
+p
+m = 2
+m = 15
+m = 28
+m = 41
+Supplementary Figure 10: Higher-order non-linear contagion process - SIS model - II. In all panels the
+average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the
+(k, m)-hyper-core is shown as a function of k at fixed values of m. All results are obtained by averaging the results
+of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.
+The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),
+Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),
+music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered
+for each data set are reported in Table II.
+
+13
+2
+4
+6
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+/T
+a
+k = 1
+k = 10
+k = 20
+k = 30
+2
+4
+6
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65 b
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85 c
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+0.60
+0.65
+0.70
+0.75
+0.80 d
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+0.80
+0.82
+0.84
+0.86
+0.88
+/T
+e
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+0.74
+0.76
+0.78
+0.80
+0.82
+f
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+15
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+g
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60 h
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+0.13
+0.14
+0.15
+0.16
+0.17
+/T
+i
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+0.60
+0.65
+0.70
+0.75
+0.80
+j
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80 k
+k = 1
+k = 7
+k = 14
+k = 21
+0
+25
+50
+75
+0.70
+0.75
+0.80
+0.85
+l
+k = 1
+k = 6
+k = 12
+k = 18
+0
+25
+50
+75
+m
+0.5
+0.6
+0.7
+0.8
+/T
+m
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+0.60
+0.65
+0.70
+0.75
+n
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+0.4
+0.5
+0.6
+0.7
+0.8
+o
+k = 1
+k = 10
+k = 20
+k = 30
+0
+100
+200
+m
+0.4
+0.5
+0.6
+0.7
+p
+k = 1
+k = 10
+k = 20
+k = 30
+Supplementary Figure 11: Higher-order non-linear contagion process - SIS model - III. In all panels the
+average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the
+(k, m)-hyper-core is shown as a function of m at fixed values of k. All results are obtained by averaging the results
+of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.
+The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),
+Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),
+music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered
+for each data set are reported in Table II.
+
+14
+1
+9
+17
+25
+33
+41
+49
+2
+3
+4
+5
+6
+7
+m
+R
+a
+60
+62
+64
+66
+68
+70
+0
+50
+n
+60
+65
+70
+R
+n
+1
+8
+15
+22
+29
+36
+2
+3
+4
+5
+6
+7
+R
+b
+80
+100
+120
+140
+0
+250
+n
+100
+150
+R
+n
+1
+9
+17
+25
+33
+41
+2
+3
+4
+5
+6
+7
+8
+9
+10
+R
+c
+40
+60
+80
+0
+200
+n
+50
+100
+R
+n
+1
+7
+13
+19
+25
+31
+2
+3
+4
+5
+6
+7
+8
+9
+10
+R
+d
+340
+350
+360
+370
+380
+390
+0
+250
+n
+350
+375
+R
+n
+1
+28
+55
+82
+109
+136
+2
+3
+4
+5
+6
+7
+8
+9
+10
+m
+e
+190
+200
+210
+220
+0
+200
+n
+210
+220
+230
+R
+n
+1
+47
+93
+139
+185
+231
+277
+2
+4
+6
+8
+10
+12
+f
+100
+150
+200
+250
+0
+500
+n
+100
+200
+300
+R
+n
+1
+48
+95
+142
+189
+236
+2
+4
+6
+8
+10
+12
+14
+16
+g
+100
+125
+150
+175
+200
+225
+0
+200
+n
+150
+200
+R
+n
+1
+20
+39
+58
+77
+96
+2
+6
+10
+14
+18
+22
+h
+50
+100
+150
+200
+250
+0
+400 800
+n
+100
+200
+300
+R
+n
+1
+183
+365
+547
+729
+911
+1093
+2
+6
+10
+14
+18
+22
+m
+i
+900
+1000
+1100
+1200
+1300
+1400
+0
+1000
+n
+1000
+1250
+R
+n
+1
+6
+11
+16
+21
+26
+31
+2
+7
+12
+17
+22
+27
+j
+40
+50
+60
+70
+80
+0
+200
+n
+40
+60
+80
+R
+n
+1
+5
+9
+13
+17
+21
+25
+2
+8
+14
+20
+26
+32
+k
+40
+50
+60
+0
+100
+n
+40
+60
+R
+n
+1
+4
+7
+10
+13
+16
+19
+2
+16
+30
+44
+58
+72
+l
+100
+150
+200
+0
+1000
+n
+100
+200
+300
+R
+n
+1
+8
+15
+22
+29
+36
+k
+2
+16
+30
+44
+58
+72
+m
+m
+500
+600
+700
+800
+900
+0
+1000
+n
+400
+600
+800
+R
+n
+S-rank
+R-rank
+1
+199
+397
+595
+793
+991
+k
+2
+18
+34
+50
+66
+82
+98
+n
+240
+250
+260
+270
+280
+0
+200
+n
+240
+260
+280
+R
+n
+S-rank
+R-rank
+1
+11
+21
+31
+41
+51
+k
+2
+20
+38
+56
+74
+92
+o
+200
+250
+300
+350
+0
+250
+n
+200
+300
+R
+n
+S-rank
+R-rank
+1
+15
+29
+43
+57
+71
+k
+2
+40
+78
+116
+154
+192
+230
+p
+350
+400
+450
+500
+550
+0
+500
+n
+300
+400
+500
+R
+n
+S-rank
+R-rank
+Supplementary Figure 12: Higher-order non-linear contagion process - SIR model - I. All panels show, as
+a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR
+process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core). The insets
+represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings. All results are
+obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set
+which is the result of 10 simulations). The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c),
+SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j),
+email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and
+geometry-questions (p). The (λ, ν) values considered for each data set are reported in Table III.
+
+15
+0
+20
+40
+58
+60
+62
+64
+66
+68
+70
+R
+a
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+80
+100
+120
+140
+160 b
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+40
+60
+80
+100 c
+m = 2
+m = 3
+m = 4
+m = 5
+0
+10
+20
+30
+330
+340
+350
+360
+370
+380
+390 d
+m = 2
+m = 3
+m = 4
+m = 5
+0
+100
+200
+205
+210
+215
+220
+225
+230
+R
+e
+m = 2
+m = 3
+m = 4
+m = 5
+0
+200
+400
+100
+150
+200
+250
+f
+m = 2
+m = 4
+m = 6
+m = 8
+0
+200
+400
+140
+160
+180
+200
+220
+g
+m = 2
+m = 4
+m = 6
+m = 8
+0
+50
+100
+50
+100
+150
+200
+250
+h
+m = 2
+m = 6
+m = 10
+m = 14
+0
+500
+1000
+900
+1000
+1100
+1200
+1300
+1400
+R
+i
+m = 2
+m = 6
+m = 10
+m = 14
+0
+10
+20
+30
+40
+50
+60
+70
+80 j
+m = 2
+m = 12
+m = 17
+m = 22
+0
+10
+20
+30
+40
+50
+60
+k
+m = 2
+m = 4
+m = 6
+m = 8
+5
+10
+15
+100
+150
+200
+250 l
+m = 2
+m = 15
+m = 41
+m = 54
+0
+20
+40
+k
+400
+500
+600
+700
+800
+900
+R
+m
+m = 2
+m = 10
+m = 18
+m = 32
+0
+500
+1000
+k
+240
+250
+260
+270
+280
+290 n
+m = 2
+m = 18
+m = 34
+m = 50
+0
+20
+40
+60
+k
+200
+250
+300
+350
+o
+m = 2
+m = 8
+m = 14
+m = 20
+0
+25
+50
+75
+k
+300
+350
+400
+450
+500
+550 p
+m = 2
+m = 15
+m = 28
+m = 41
+Supplementary Figure 13: Higher-order non-linear contagion process - SIR model - II. In all panels the
+average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the
+(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m. All
+results are obtained by averaging the results of 300 numerical simulations for each seed (except for the
+congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a),
+Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),
+senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),
+algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered for each data set are reported in
+Table III.
+
+16
+2
+4
+6
+58
+60
+62
+64
+66
+68
+70
+R
+a
+k = 1
+k = 10
+k = 20
+k = 30
+2
+4
+6
+80
+90
+100
+110
+120
+130
+b
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+40
+60
+80
+100 c
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+330
+340
+350
+360
+370
+380
+390 d
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+190
+200
+210
+220
+R
+e
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+80
+100
+120
+140
+160
+180
+200
+f
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+15
+100
+120
+140
+160
+180
+200
+220
+g
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+50
+100
+150
+200
+250
+h
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+850
+900
+950
+1000
+1050
+1100
+1150
+R
+i
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+40
+50
+60
+70
+j
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+30
+40
+50
+60
+k
+k = 1
+k = 7
+k = 14
+k = 21
+0
+25
+50
+75
+100
+150
+200
+250
+l
+k = 1
+k = 6
+k = 12
+k = 18
+0
+25
+50
+75
+m
+400
+500
+600
+700
+800
+900
+R
+m
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+240
+250
+260
+270
+n
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+200
+250
+300
+350
+o
+k = 1
+k = 10
+k = 20
+k = 30
+0
+100
+200
+m
+300
+350
+400
+450
+500
+550
+p
+k = 1
+k = 10
+k = 20
+k = 30
+Supplementary Figure 14: Higher-order non-linear contagion process - SIR model - III. In all panels the
+average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the
+(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k. All
+results are obtained by averaging the results of 300 numerical simulations for each seed (except for the
+congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a),
+Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),
+senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),
+algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered for each data set are reported in
+Table III.
+
+17
+Supplementary Section 4.
+Threshold higher-order contagion process
+Here we consider another spreading process in which multi-body interactions drive the infection through a thresh-
+old effect and group contagion [2, 4]: the threshold higher-order contagion process. We consider both the SIR and
+SIS epidemic models on static hypergraphs: for each hyperedge of size m in which i individuals are in the state I,
+if the fraction of infected individuals i/m is larger or equal to a threshold θ, i.e. if i ≥ ⌈θm⌉, a group infection is
+activated at rate λ in which the susceptible nodes in the hyperedge become all infected. Note that if we consider a
+single seed of infection: for θ ≤ 1/M the group infection is activated in all the hyperedges containing the seed; for
+θ = 1/m the spreading is activated only in the hyperedges containing the seed that have size larger or equal to m;
+for θ > 1/2 the spreading is inhibited since more than one infected node is required to activate the infection in all
+hyperedges. I individuals recover independently at constant rate µ, becoming either S (SIS model) or R (SIR).
+We perform numerical simulations of this process, for both SIS and SIR models, on empirical hypergraphs: the
+simulation procedures are analogous to those described in the main text for the higher-order non-linear contagion
+process (see Methods), since the two processes only differ in the infection mechanism. In the threshold higher-order
+contagion, for each time-step ∆t, given a hyperedge of size m containing i infected nodes, if i ≥ ⌈θm⌉ a group
+infection process is activated with probability λ and all susceptible nodes in the hyperedge are infected. Thus, in
+each time-step each of the interaction groups respecting the condition i ≥ ⌈θm⌉ produces a group infection process
+with probability λ.
+Therefore, also in this case we quantify the ”spreading power” of each node considered separately as seed for the
+SIR model and the nodes on which the epidemic is mainly localized in the steady state, i.e. the nodes that drive
+and sustain the process, for the SIS model. In Figs. 15-20 are shown the results of these simulations.
+data set
+θ
+λ
+LH10
+0.03
+0.005
+Thiers13
+1/7
+0.001
+InVS15
+1/10
+0.001
+SFHH
+1/10
+0.001
+LyonSchool
+0.15
+0.001
+Mid1
+0.03
+0.001
+Elem1
+0.03
+0.001
+email-EU
+0.03
+0.001
+data set
+θ
+λ
+congress-bills
+0.03
+0.001
+senate-committees
+0.03
+0.01
+email-Enron
+1/37
+0.01
+house-committees
+0.03
+0.01
+music-review
+0.03
+0.01
+senate-bills
+1/99
+0.0001
+algebra-questions
+1/107
+0.001
+geometry-questions
+0.03
+0.001
+Supplementary Table IV: Parameters for Figs. 15-17. The tables summarize the parameters of the threshold
+higher-order SIS contagion process considered for each data set in Figs. 15-17.
+data set
+θ
+λ
+LH10
+0.03
+0.01
+Thiers13
+1/7
+0.001
+InVS15
+1/10
+0.01
+SFHH
+0.03
+0.01
+LyonSchool
+0.15
+0.01
+Mid1
+1/13
+0.001
+Elem1
+0.03
+0.001
+email-EU
+0.03
+0.001
+data set
+θ
+λ
+congress-bills
+0.03
+0.001
+senate-committees
+0.15
+0.01
+email-Enron
+0.3
+0.01
+house-committees
+1/82
+0.01
+music-review
+1/83
+0.01
+senate-bills
+0.3
+0.001
+algebra-questions
+1/107
+0.001
+geometry-questions
+1/230
+0.001
+Supplementary Table V: Parameters for Figs. 18-20. The tables summarize the parameters of the threshold
+higher-order SIR contagion process considered for each data set in Figs. 18-20.
+
+18
+1
+9
+17
+25
+33
+41
+49
+2
+3
+4
+5
+6
+7
+m
+/T
+a
+0.75
+0.80
+0.85
+0
+50
+n
+0.8
+0.9
+/T n
+1
+8
+15
+22
+29
+36
+2
+3
+4
+5
+6
+7
+/T
+b
+0.04
+0.06
+0.08
+0
+250
+n
+0.05
+0.10
+/T n
+1
+9
+17
+25
+33
+41
+2
+3
+4
+5
+6
+7
+8
+9
+10
+/T
+c
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0
+200
+n
+0.05
+0.10
+0.15
+/T n
+1
+7
+13
+19
+25
+31
+2
+3
+4
+5
+6
+7
+8
+9
+10
+/T
+d
+0.10
+0.15
+0.20
+0
+250
+n
+0.1
+0.2
+/T n
+1
+28
+55
+82
+109
+136
+2
+3
+4
+5
+6
+7
+8
+9
+10
+m
+e
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0
+200
+n
+0.6
+0.7
+/T n
+1
+47
+93
+139
+185
+231
+277
+2
+4
+6
+8
+10
+12
+f
+0.74
+0.76
+0.78
+0.80
+0.82
+0
+500
+n
+0.75
+0.80
+/T n
+1
+48
+95
+142
+189
+236
+2
+4
+6
+8
+10
+12
+14
+16
+g
+0.65
+0.70
+0.75
+0.80
+0
+200
+n
+0.7
+0.8
+/T n
+1
+20
+39
+58
+77
+96
+2
+6
+10
+14
+18
+22
+h
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+400 800
+n
+0.4
+0.6
+/T n
+1
+183
+365
+547
+729
+911
+1093
+2
+6
+10
+14
+18
+22
+m
+i
+0.65
+0.70
+0.75
+0.80
+0.85
+0
+1000
+n
+0.7
+0.8
+0.9
+/T n
+1
+6
+11
+16
+21
+26
+31
+2
+7
+12
+17
+22
+27
+j
+0.55
+0.60
+0.65
+0.70
+0.75
+0
+200
+n
+0.6
+0.7
+/T n
+1
+5
+9
+13
+17
+21
+25
+2
+8
+14
+20
+26
+32
+k
+0.65
+0.70
+0.75
+0.80
+0
+100
+n
+0.6
+0.7
+0.8
+/T n
+1
+4
+7
+10
+13
+16
+19
+2
+16
+30
+44
+58
+72
+l
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0
+1000
+n
+0.4
+0.6
+/T n
+1
+8
+15
+22
+29
+36
+k
+2
+16
+30
+44
+58
+72
+m
+m
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+1000
+n
+0.4
+0.6
+0.8
+/T n
+S-rank
+R-rank
+1
+199
+397
+595
+793
+991
+k
+2
+18
+34
+50
+66
+82
+98
+n
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0
+200
+n
+0.4
+0.6
+/T n
+S-rank
+R-rank
+1
+11
+21
+31
+41
+51
+k
+2
+20
+38
+56
+74
+92
+o
+0.1
+0.2
+0.3
+0.4
+0
+250
+n
+0.2
+0.4
+/T n
+S-rank
+R-rank
+1
+15
+29
+43
+57
+71
+k
+2
+40
+78
+116
+154
+192
+230
+p
+0.2
+0.3
+0.4
+0.5
+0
+500
+n
+0.2
+0.4
+/T n
+S-rank
+R-rank
+Supplementary Figure 15: Threshold higher-order contagion process - SIS model - I. All panels give, as a
+heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state
+averaged over the nodes of the (k, m)-hyper-core. The insets represent τ/T averaged over the first n nodes
+according to the coreness rankings as a function of n. All results are obtained by averaging the results of 103
+numerical simulations, with a single random seed of infection and with an observation window T = 103. The
+following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f),
+Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l),
+music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The values (λ,θ) values
+considered for each data set are summarized in Table IV.
+
+19
+0
+20
+40
+0.75
+0.80
+0.85
+0.90
+/T
+a
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10 b
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175 c
+m = 2
+m = 3
+m = 4
+m = 5
+0
+10
+20
+30
+0.10
+0.15
+0.20
+d
+m = 2
+m = 3
+m = 4
+m = 5
+0
+100
+200
+0.575
+0.600
+0.625
+0.650
+0.675
+0.700
+/T
+e
+m = 2
+m = 3
+m = 4
+m = 5
+0
+200
+400
+0.74
+0.76
+0.78
+0.80
+0.82
+f
+m = 2
+m = 4
+m = 6
+m = 8
+0
+200
+400
+0.650
+0.675
+0.700
+0.725
+0.750
+0.775
+0.800 g
+m = 2
+m = 4
+m = 6
+m = 8
+0
+50
+100
+0.3
+0.4
+0.5
+0.6
+0.7 h
+m = 2
+m = 6
+m = 10
+m = 14
+0
+500
+1000
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+/T
+i
+m = 2
+m = 6
+m = 10
+m = 14
+0
+10
+20
+30
+0.55
+0.60
+0.65
+0.70
+0.75
+j
+m = 2
+m = 12
+m = 17
+m = 22
+0
+10
+20
+0.65
+0.70
+0.75
+0.80 k
+m = 2
+m = 4
+m = 6
+m = 8
+5
+10
+15
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65 l
+m = 2
+m = 15
+m = 41
+m = 54
+0
+20
+40
+k
+0.4
+0.5
+0.6
+0.7
+0.8
+/T
+m
+m = 2
+m = 10
+m = 18
+m = 32
+0
+500
+1000
+k
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60 n
+m = 2
+m = 18
+m = 34
+m = 50
+0
+20
+40
+60
+k
+0.1
+0.2
+0.3
+0.4
+0.5 o
+m = 2
+m = 8
+m = 14
+m = 20
+0
+25
+50
+75
+k
+0.1
+0.2
+0.3
+0.4
+p
+m = 2
+m = 15
+m = 28
+m = 41
+Supplementary Figure 16: Threshold higher-order contagion process - SIS model - II. In all panels the
+average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core
+is shown as a function of k at fixed values of m. All results are obtained by averaging the results of 103 numerical
+simulations, with a single random seed of infection and with an observation window T = 103. The following data
+sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g),
+email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m),
+senate-bills (n), algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each
+data set are summarized in Table IV.
+
+20
+2
+4
+6
+0.75
+0.80
+0.85
+/T
+a
+k = 1
+k = 10
+k = 20
+k = 30
+2
+4
+6
+0.04
+0.05
+0.06
+b
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175 c
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+0.05
+0.10
+0.15
+0.20
+d
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+0.45
+0.50
+0.55
+0.60
+0.65
+/T
+e
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+0.74
+0.76
+0.78
+0.80 f
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+15
+0.65
+0.70
+0.75
+g
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+0.3
+0.4
+0.5
+0.6
+h
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+0.64
+0.66
+0.68
+0.70
+0.72
+0.74
+0.76
+/T
+i
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+0.55
+0.60
+0.65
+0.70
+0.75
+j
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+0.60
+0.65
+0.70
+0.75
+0.80 k
+k = 1
+k = 7
+k = 14
+k = 21
+0
+25
+50
+75
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+l
+k = 1
+k = 6
+k = 12
+k = 18
+0
+25
+50
+75
+m
+0.4
+0.5
+0.6
+0.7
+0.8
+/T
+m
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+0.34
+0.36
+0.38
+0.40
+0.42
+0.44
+0.46
+n
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+0.1
+0.2
+0.3
+0.4
+o
+k = 1
+k = 10
+k = 20
+k = 30
+0
+100
+200
+m
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+p
+k = 1
+k = 10
+k = 20
+k = 30
+Supplementary Figure 17: Threshold higher-order contagion process - SIS model - III. In all panels the
+average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core
+is shown as a function of m at fixed values of k. All results are obtained by averaging the results of 103 numerical
+simulations, with a single random seed of infection and with an observation window T = 103. The following data
+sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g),
+email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m),
+senate-bills (n), algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each
+data set are summarized in Table IV.
+
+21
+1
+9
+17
+25
+33
+41
+49
+2
+3
+4
+5
+6
+7
+m
+R
+a
+50
+55
+60
+0
+50
+n
+50
+60
+R
+n
+1
+8
+15
+22
+29
+36
+2
+3
+4
+5
+6
+7
+R
+b
+8
+10
+12
+0
+250
+n
+7.5
+10.0
+12.5
+R
+n
+1
+9
+17
+25
+33
+41
+2
+3
+4
+5
+6
+7
+8
+9
+10
+R
+c
+160
+170
+180
+190
+0
+200
+n
+160
+180
+R
+n
+1
+7
+13
+19
+25
+31
+2
+3
+4
+5
+6
+7
+8
+9
+10
+R
+d
+280
+300
+320
+340
+0
+250
+n
+300
+350
+R
+n
+1
+28
+55
+82
+109
+136
+2
+3
+4
+5
+6
+7
+8
+9
+10
+m
+e
+220
+225
+230
+235
+240
+0
+200
+n
+235
+240
+R
+n
+1
+47
+93
+139
+185
+231
+277
+2
+4
+6
+8
+10
+12
+f
+480
+500
+520
+0
+500
+n
+475
+500
+525
+R
+n
+1
+48
+95
+142
+189
+236
+2
+4
+6
+8
+10
+12
+14
+16
+g
+220
+240
+260
+0
+200
+n
+225
+250
+275
+R
+n
+1
+20
+39
+58
+77
+96
+2
+6
+10
+14
+18
+22
+h
+200
+250
+300
+350
+0
+400 800
+n
+200
+400
+R
+n
+1
+183
+365
+547
+729
+911
+1093
+2
+6
+10
+14
+18
+22
+m
+i
+1200
+1300
+1400
+1500
+1600
+0
+1000
+n
+1200
+1400
+1600
+R
+n
+1
+6
+11
+16
+21
+26
+31
+2
+7
+12
+17
+22
+27
+j
+10
+15
+20
+25
+0
+200
+n
+20
+40
+R
+n
+1
+5
+9
+13
+17
+21
+25
+2
+8
+14
+20
+26
+32
+k
+50
+55
+60
+65
+70
+0
+100
+n
+60
+80
+R
+n
+1
+4
+7
+10
+13
+16
+19
+2
+16
+30
+44
+58
+72
+l
+500
+600
+700
+800
+0
+1000
+n
+600
+800
+R
+n
+1
+8
+15
+22
+29
+36
+k
+2
+16
+30
+44
+58
+72
+m
+m
+400
+500
+600
+700
+800
+0
+1000
+n
+400
+600
+800
+R
+n
+S-rank
+R-rank
+1
+199
+397
+595
+793
+991
+k
+2
+18
+34
+50
+66
+82
+98
+n
+40
+50
+60
+70
+80
+0
+200
+n
+40
+60
+80
+R
+n
+S-rank
+R-rank
+1
+11
+21
+31
+41
+51
+k
+2
+20
+38
+56
+74
+92
+o
+20
+30
+40
+50
+60
+70
+0
+250
+n
+25
+50
+75
+R
+n
+S-rank
+R-rank
+1
+15
+29
+43
+57
+71
+k
+2
+40
+78
+116
+154
+192
+230
+p
+50
+75
+100
+125
+150
+0
+500
+n
+50
+100
+150
+R
+n
+S-rank
+R-rank
+Supplementary Figure 18: Threshold higher-order contagion process - SIR model - I. All panels show, as
+a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR
+process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core). The insets
+represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings. All results are
+obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set
+which is the result of 10 simulations). The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c),
+SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j),
+email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and
+geometry-questions (p). The values (λ,θ) values considered for each data set are summarized in Table V.
+
+22
+0
+20
+40
+50.0
+52.5
+55.0
+57.5
+60.0
+62.5
+65.0
+R
+a
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+6
+8
+10
+12
+b
+m = 2
+m = 3
+m = 4
+m = 5
+0
+20
+40
+160
+170
+180
+190
+c
+m = 2
+m = 3
+m = 4
+m = 5
+0
+10
+20
+30
+280
+300
+320
+340
+d
+m = 2
+m = 3
+m = 4
+m = 5
+0
+100
+200
+232
+234
+236
+238
+240
+R
+e
+m = 2
+m = 3
+m = 4
+m = 5
+0
+200
+400
+480
+500
+520
+f
+m = 2
+m = 4
+m = 6
+m = 8
+0
+200
+400
+220
+230
+240
+250
+260
+270
+280 g
+m = 2
+m = 4
+m = 6
+m = 8
+0
+50
+100
+150
+200
+250
+300
+350
+400 h
+m = 2
+m = 6
+m = 10
+m = 14
+0
+500
+1000
+1200
+1300
+1400
+1500
+1600
+R
+i
+m = 2
+m = 6
+m = 10
+m = 14
+0
+10
+20
+30
+10
+15
+20
+25
+j
+m = 2
+m = 12
+m = 17
+m = 22
+0
+10
+20
+50
+55
+60
+65
+70
+75 k
+m = 2
+m = 4
+m = 6
+m = 8
+5
+10
+15
+500
+600
+700
+800
+l
+m = 2
+m = 15
+m = 41
+m = 54
+0
+20
+40
+k
+400
+500
+600
+700
+800
+R
+m
+m = 2
+m = 10
+m = 18
+m = 32
+0
+500
+1000
+k
+40
+50
+60
+70
+80 n
+m = 2
+m = 18
+m = 34
+m = 50
+0
+20
+40
+60
+k
+20
+30
+40
+50
+60
+70 o
+m = 2
+m = 8
+m = 14
+m = 20
+0
+25
+50
+75
+k
+40
+60
+80
+100
+120
+140 p
+m = 2
+m = 15
+m = 28
+m = 41
+Supplementary Figure 19: Threshold higher-order contagion process - SIR model - II. In all panels the
+average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the
+(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m. All
+results are obtained by averaging the results of 300 numerical simulations for each seed (except for the
+congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a),
+Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),
+senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),
+algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each data set are
+summarized in Table V.
+
+23
+2
+4
+6
+50.0
+52.5
+55.0
+57.5
+60.0
+62.5
+R
+a
+k = 1
+k = 10
+k = 20
+k = 30
+2
+4
+6
+6
+7
+8
+9
+10
+b
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+160
+170
+180
+190
+c
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+280
+300
+320
+340
+d
+k = 1
+k = 10
+k = 20
+k = 30
+2.5
+5.0
+7.5
+10.0
+222.5
+225.0
+227.5
+230.0
+232.5
+235.0
+237.5
+R
+e
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+470
+480
+490
+500
+510
+f
+k = 1
+k = 10
+k = 20
+k = 30
+5
+10
+15
+210
+220
+230
+240
+250
+260
+270
+g
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+150
+200
+250
+300
+350
+h
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+1150
+1200
+1250
+1300
+1350
+R
+i
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+10
+15
+20
+25
+j
+k = 1
+k = 10
+k = 20
+k = 30
+10
+20
+30
+50
+55
+60
+65
+70
+k
+k = 1
+k = 7
+k = 14
+k = 21
+0
+25
+50
+75
+500
+600
+700
+800
+l
+k = 1
+k = 6
+k = 12
+k = 18
+0
+25
+50
+75
+m
+400
+500
+600
+700
+800
+R
+m
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+37.5
+40.0
+42.5
+45.0
+47.5
+50.0
+n
+k = 1
+k = 10
+k = 20
+k = 30
+0
+50
+100
+m
+20
+30
+40
+50
+60
+o
+k = 1
+k = 10
+k = 20
+k = 30
+0
+100
+200
+m
+40
+60
+80
+100
+p
+k = 1
+k = 10
+k = 20
+k = 30
+Supplementary Figure 20: Threshold higher-order contagion process - SIR model - III. In all panels the
+average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the
+(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k. All
+results are obtained by averaging the results of 300 numerical simulations for each seed (except for the
+congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a),
+Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i),
+senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n),
+algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each data set are
+summarized in Table V.
+
+24
+Supplementary Section 5.
+Higher-order naming-game (NG) process
+data set
+β
+p
+tmax
+T
+InVS15
+0.41
+1.8 × 10−2
+105
+104
+Mid1
+0.59
+1.5 × 10−2
+105
+104
+email-EU
+0.52
+9.2 × 10−3
+5 × 105
+5 × 104
+congress-bills
+0.59
+2.4 × 10−2
+5 × 105
+5 × 104
+house-committees
+0.55
+2.5 × 10−2
+5 × 105
+5 × 104
+music-review
+0.52
+9.0 × 10−3
+105
+104
+Supplementary Table VI: Parameters for Fig. 21 - Union rule. The table summarizes the main parameters of
+the higher-order naming-game process considered for the temporal dynamics of Fig. 21 in the various data sets
+(union rule).
+data set
+Arandom
+As−core
+Ahyper−core
+InVS15
+24.3%
+54.8%
+54.8%
+Mid1
+14.5%
+25.5%
+23.7%
+email-EU
+37.0%
+45.9%
+56.4%
+congress-bills
+40.8%
+47.1%
+49.7%
+house-committees
+63.0%
+64.0%
+64.6%
+music-review
+51.6%
+55.8%
+59.5%
+Supplementary Table VII: Minority takeover areas for Fig. 21 - Union rule. The table reports the area Ax
+of the explored parameter space in which the minority take-over, i.e. n∗
+A = 1, takes place for the different data sets
+of Fig. 21 (union rule) and for the different strategies of committed seeding x ∈ {random, s − core, hyper − core}.
+data set
+β
+p
+tmax
+T
+InVS15
+0.38
+1.4 × 10−2
+105
+104
+Mid1
+0.38
+2.7 × 10−2
+105
+104
+email-EU
+0.41
+1.7 × 10−2
+5 × 105
+5 × 104
+congress-bills
+0.48
+2.3 × 10−2
+5 × 105
+5 × 104
+house-committees
+0.41
+3.0 × 10−3
+5 × 105
+5 × 104
+music-review
+0.52
+1.0 × 10−2
+105
+104
+Supplementary Table VIII: Parameters for Fig. 22 - Unanimity rule. The table summarizes the main
+parameters of the higher-order naming-game process considered for the temporal dynamics of Fig. 22 in the
+various data sets (unanimity rule).
+
+25
+data set
+Arandom
+As−core
+Ahyper−core
+InVS15
+8.6%
+35.2%
+32.9%
+Mid1
+1.0%
+2.5%
+2.5%
+email-EU
+7.3%
+8.6%
+14.7%
+congress-bills
+7.9%
+16.3%
+41.5%
+house-committees
+54.6%
+63.3%
+64.4%
+music-review
+33.9%
+56.6%
+62.6%
+Supplementary Table IX: Minority takeover areas for Fig. 22 - Unanimity rule. The table reports the area
+Ax of the explored parameter space in which the minority take-over, i.e. n∗
+A = 1, takes place for the different data
+sets of Fig. 22 (unanimity rule) and for the different strategies of committed seeding
+x ∈ {random, s − core, hyper − core}.
+
+26
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+p
+a
+10
+2
+InVS15
+Random
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+b
+10
+2
+s-core
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+c
+10
+2
+n *
+A
+hyper-core
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+d
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+p
+e
+10
+2
+Mid1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+f
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+g
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+h
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+i
+10
+2
+email-EU
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+j
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+k
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+l
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+m
+10
+2
+congress-bills
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+n
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+o
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+p
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+q
+10
+2
+house-committees
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+r
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+s
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+t
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+u
+10
+2
+music-review
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+v
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+w
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+x
+Random
+R-rank
+s-rank
+Supplementary Figure 21:
+Higher-order NG process - Union rule. In the first three panels of each row the
+stationary fraction n∗
+A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p
+and the agreement probability β through a heat-map. We consider the union rule and the following data sets: InVS15 (first
+row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews
+(sixth row). For each row, the committed nodes are selected through: the random seeding strategy in the first panel; the
+top s-coreness seeding strategy in the second panel; the top hyper-coreness seeding strategy in the third panel. In the
+fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VI (cross
+markers in the heatmaps). The minority take-over, i.e. n∗
+A = 1, takes place over an area A of the explored parameter space:
+in Table VII we report its value for the different strategies and data sets considered. All simulations are run until the
+absorbing state with n∗
+A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗
+A is
+obtained by averaging over 100 values sampled in the last T time-steps (see Table VI for the tmax and T values for each
+data set). The results refer to the median values obtained over 200 simulations.
+
+27
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+p
+a
+10
+2
+InVS15
+Random
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+b
+10
+2
+s-core
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+3.2
+c
+10
+2
+n *
+A
+hyper-core
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+d
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+p
+e
+10
+2
+Mid1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+f
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.5
+0.8
+1.1
+1.4
+1.7
+2.0
+2.3
+2.6
+2.9
+g
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+h
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+i
+10
+2
+email-EU
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+j
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+k
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+l
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+m
+10
+2
+congress-bills
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+n
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+o
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+p
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+q
+10
+2
+house-committees
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+r
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+s
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+t
+Random
+R-rank
+s-rank
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+p
+u
+10
+2
+music-review
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+v
+10
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.4
+0.7
+1.0
+1.3
+1.6
+1.9
+2.2
+2.5
+2.8
+w
+10
+2
+n *
+A
+1/N
+0.2
+0.4
+0.6
+0.8
+1.0
+102
+103
+104
+105
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nA(t)
+x
+Random
+R-rank
+s-rank
+Supplementary Figure 22:
+Higher-order NG process - Unanimity rule. In the first three panels of each row the
+fraction n∗
+A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the
+agreement probability β through a heat-map. We consider the unanimity rule and the following data sets: InVS15 (first
+row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews
+(sixth row). For each row, the committed nodes are selected through: the random seeding strategy in the first panel; the
+top s-coreness seeding strategy in the second panel; the top hyper-coreness seeding strategy in the third panel. In the
+fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VIII (cross
+markers in the heatmaps). The minority take-over, i.e. n∗
+A = 1, takes place over an area A of the explored parameter space:
+in Table IX we report its value for the different strategies and data sets considered. All simulations are run until the
+absorbing state with n∗
+A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗
+A is
+obtained by averaging over 100 values sampled in the last T time-steps (see Table VIII for the tmax and T values for each
+data set). The results refer to the median values obtained over 200 simulations.
+
+28
+[1] St-Onge, G. et al.
+Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs.
+Communications Physics 5, 25 (2022).
+[2] de Arruda, G. F., Petri, G., Rodriguez, P. M. & Moreno, Y. Multistability, intermittency and hybrid transitions in social
+contagion models on hypergraphs. arXiv preprint - arXiv:2112.04273 (2021).
+[3] Iacopini, I., Petri, G., Baronchelli, A. & Barrat, A.
+Group interactions modulate critical mass dynamics in social
+convention. Communications Physics 5, 64 (2022).
+[4] de Arruda, G. F., Petri, G. & Moreno, Y. Social contagion models on hypergraphs. Phys. Rev. Research 2, 023032 (2020).
+