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0802.1098	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 68529, "num_imgs": 0, "llama3_tokens_count": 20363 }	[]	# Model dynamics on a multigrid across multiple length and time scales A. J. Roberts Computational Engineering and Sciences Research Centre, Dept Maths and Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia. ###### Abstract Most methods for modelling dynamics posit just two time scales: a fast and a slow scale. But many applications, including many in continuum mechanics, possess a wide variety of space-time scales; often they possess a continuum of space-time scales. I discuss an approach to modelling the discretised dynamics of advection and diffusion with rigorous support for changing the resolved spatial grid scale by just a factor of two. The mapping of dynamics from a finer grid to a coarser grid is then iterated to generate a hierarchy of models across a wide range of space-time scales, all with rigorous support across the whole hierarchy. This approach empowers us with great flexibility in modelling complex dynamics over multiple scales. ###### Contents Contents * 1 Introduction * 2 Centre manifold theory supports multiscale models * 3 Coarsen linear advection-dispersion * 3.1 Computer algebra approximates the slow manifold * 3.2 The coarse scale, slow dynamics model precisely * 3.3 Coarse scale dynamics are attractive * 3.4 A normal form projects initial conditions * 3.5 Extend elements for a multigrid hierarchy * 4 Multiscale modelling iterates transformations * 4.1 Diffusion emerges from discrete dispersion * 4.2 Renormalise advection-dispersion * 4.3 Approximate the nonlinear Burgers’ dynamics * 5 Conclusion ## 1 Introduction Multiscale methods promise efficient computation and simulation of many important physical systems [3, e.g.]. Dolbow et al. [10] identify that critical applications include fuel cells, subsurface contaminant transport, protein folding, climate simulations, and general networks. Most multiscale mathematical methods for modelling disparate scales presume just two scales: small lengths and large lengths; fast times and long times; fast variations and slow variations; microscopic and macroscopic [10, 11, e.g.]. Most such methods then seek _effective_ models or properties on the large/long/slow macroscales by ‘averaging/homogenising over’ the small/fast microscales [11, 22, e.g.]. Here we explore a novel mathematical framework to modelling dynamics over many length and time scales; the framework is supported by modern dynamical systems theory. Of course most two scale modelling methods will also work over many scales. The same techniques that construct and support a slow model of rapid variations may also apply to construct and support a superslow model of the slow variations. The same techniques may then also apply to construct and support a megaslow model of the superslow variations; and so on. Crucially, in most established methods each of these constructions require a large ‘spectral gap’; that is, they require an absolutely clear separation between the micro and the macro scales; a parameter such as $\epsilon$ measures the scale separation, and the requirement for extreme scale separation is provided by theorems invoking “as $\epsilon\to 0$”. In contrast, multigrid iteration for solving linear equations transforms between length scales that are different by (usually) a factor of two [5, e.g.]; some variants of multigrid iteration use an even smaller ratio of length scales [28, e.g.]. Recently Brandt [4] proposed a method for molecular dynamics without large scale separation using so-called systematic upscaling. Analogously, here we explore modelling _dynamics_ on a hierachy of length scales that differ by a factor of two and hence the ‘spectral gap’ is finite and typically much smaller than required by popular extant methods for modelling dynamics. Section 2 rigorously supports such models with centre manifold theory [6, e.g.]. Recall that multigrid iteration, using restriction and prolongation operators, transforms between length scales differing by a factor of two [5, e.g.]. The full multigrid iteration involves iterating the restriction and prolongation transformations to cross large changes in length scales by taking many coarsening or refining transforms where each step changes the resolved length scales by a factor of two. Similarly, Section 3 starts our modelling of dynamics by exploring a transformation of dynamics from one length scale to another that is a factor of two coarser. Section 3, see (13), demonstrates that to some controlled approximation the discrete advection diffusion equation \[\frac{du_{j}}{dt}=-c\rat 12(u_{j+1}-u_{j-1})+d(u_{j+1}-2u_{j}+u_{j-1})\,,\] (1) for evolving grid values $u_{j}(t)$ on a grid of spacing $h$ (and hence with ‘advection speed’ $ch$ and ‘diffusion’ $dh^{2}$) is, on the coarser grid of spacing ${\bar{h}}=2h$ , justifiably modelled by \[\frac{d{\bar{u}}_{j}}{dt}\approx-\bar{c}\rat 12({\bar{u}}_{j+1}-{ \bar{u}}_{j-1})+\bar{d}({\bar{u}}_{j+1}-2{\bar{u}}_{j}+{\bar{u}}_{j-1})\] where \[\bar{c}=\rat 12c\qtq{and}\bar{d}=\rat 14d+\frac{c^{2}}{16d}\,,\] (2) for evolving coarse grid values ${\bar{u}}_{j}$; these coarse grid values are defined to be the fine grid values at every second point on the fine grid, ${\bar{u}}_{j}=u_{2j}$ . Intriguingly, the key to the approach is to take one step backwards in order to take two steps forward: at any level we embed the dynamics (1) in a higher dimensional problem, then analysis systematically derives the lower dimensional, macroscale model (2). The geometric approach to modelling of both centre manifold theory [6, e.g.] and normal form theory [20, e.g.] justifies the model (2) using a similar approach to that of holistic discretisation [26, e.g.]. The enhancement of the diffusion by $c^{2}/(16d)$ evident in (2) on the coarse grid comes from resolving the dynamics on the finer grid in constructing the model on the coarser grid:¹ the enhancement ensures the coarse model (2) stably models the fine grid dynamics (1); intriguingly this enhanced dispersion is _precisely_ that implicit in cyclic reduction, a multigrid method, to find an equilibrium of such advection-dispersion problems, but here derived for dynamic problems with a different theoretical base. The coarse model (2) implicitly prescribes a ‘restriction operator’ that transforms the dynamics of advection-dispersion from one grid to another with twice the spacing. [FOOTNOTE:1][ENDFOOTNOTE] [FIGURE:S1.F1][ENDFIGURE] Others also explore dynamics across space-time scales. Griebel, Oeltz & Vassilevski [15] developed space-time multigrid numerics to find optimal control of problems governed by parabolic differential equations. They base their cross-scale transformation on an algebraic multigrid. The systematic upscaling by Brandt [4] uses multigrid ideas to progressively coarsen atomic simulations of polymer folding. These approaches are largely computational whereas here we develop algebraic transformations that then are used computationally. Another major difference is that the slow manifolds constructed here provides a coarsening and interpolation, across length scales, that is specifically adapted to the dynamics of the problem rather than being imposed on the problem. Section 3.1 constructs the slow manifolds by systematically approximating exact closures provided by the fine scale dynamics. Section 4 explores iterating our transformation to model dynamics across each and every intervening length scale. For example, repeating the transformation from fine (1) to coarse (2) gives a hierarchy of models all of the form of the advection-dispersion equation (1) but with differing coefficients. At the $\ell$th level, with grid spacing $h^{(\ell)}=2^{\ell}h$ , the corresponding grid values $u_{j}^{(\ell)}$ evolve according to (1) but with coefficients $c^{(\ell)}$ and $d^{(\ell)}$ determined by the recurrence \[c^{(\ell+1)}=\rat 12c^{(\ell)}\quad\text{and}\quad d^{(\ell+1)}=\rat 14d^{( \ell)}+\frac{{c^{(\ell)}}^{2}}{16d^{(\ell)}}\,.\] (3) On successively coarser grids the coefficients thus are \[c^{(\ell)}=\frac{c}{2^{\ell}}\qtq{and}d^{(\ell)}=\frac{\|c\|}{2^{\ell+1}}\tilde{ d}^{(\ell)}\qtq{where}\tilde{d}^{(\ell+1)}=\frac{1}{2}\left(\tilde{d}^{(\ell)} +\frac{1}{\tilde{d}^{(\ell)}}\right)\,.\] (4) Observe that $\tilde{d}^{(\ell)}\to 1$ quickly as $\ell$ increases (as (4) is equivalent to Newton’s iteration to find the zeros of $\tilde{d}^{2}-1$); hence as the grid coarsens, the $\ell$th level model quickly becomes simply the upwind model \[\frac{du^{(\ell)}_{j}}{dt}\approx-c\frac{u^{(\ell)}_{j}-u^{(\ell)}_{j-1}}{2^{ \ell}}\quad\text{when }c>0\,.\] Our multigrid modelling transformation naturally recognises that advection dominates diffusion on coarse grids: the cross scale transformation, the map from fine (1) to coarse (2) as summarised by (3), not only preserves the advection speed, but also models the advection in a stable scheme that preserves non-negativity. Further, in the absence of advection, $c=0$ , the transformation (3) preserves the effective diffusion across all scales: $d^{(\ell+1)}=\rat 14d^{(\ell)}$ . These are some simple results. Section 4 explores further issues in transforming both linear and nonlinear discrete dynamics across many scales. The centre manifold and normal form [12, 8, 20, e.g.] approach established here provides a framework for dynamical modelling that links what are conventionally called multigrid [5, e.g.], wavelets [9, e.g.], multiple scales [23, e.g.], and singular perturbations [36, e.g.]. This framework applies to not only the linear dynamical systems that are the main focus of this article, but also applies to nonlinear systems [26, 18, e.g.] and to stochastic systems [7, 1, 30, 34, e.g.]. Here, because it is simplest, we focus on transforming dynamics within the same algebraic form, but in principle the methodology can support the emergence, via nonlinear interactions, of qualitatively different dynamics on macroscales (as promoted by the heterogeneous multiscale method [11, e.g.]). By rationally transforming across both space and time scales, a long term aim of this approach is to empower efficient simulation and analysis of multiscale systems at whatever level of detail is required and to a controllable error. This approach to transformation from one scale to anther may in the future illuminate complex systems simulations on both lattices and with cellular automata. ## 2 Centre manifold theory supports multiscale models [FIGURE:S2.F2][ENDFIGURE] This section establishes new theoretical support for coarsening dynamics from a fine grid to a coarse grid of twice the spacing. Suppose the fine grid has grid points $x_{j}$, spacing $h$ as shown in Figure 2, and has grid values $u_{j}(t)$ evolving in time. The figure also shows the coarse grid points ${\bar{x}}_{j}=x_{2j}$ , spacing ${\bar{h}}=2h$ , and the definition of the evolving coarse grid values² [FOOTNOTE:2][ENDFOOTNOTE] \[{\bar{u}}_{j}(t)=u_{2j}(t)\,.\] (5) Mostly, an overbar denotes variables and operators on the coarser grid, and unadorned variables are those on the finer grid. Using overdots to denote time derivatives, the question is: how do we transform the evolution $\dot{u}_{j}={\cal L}u_{j}$ , for some fine scale local operator ${\cal L}$, to a coarse evolution $\dot{\bar{u}}_{j}=\bar{{\cal L}}{\bar{u}}_{j}$ on the coarse grid? The theoretical support for multiscale modelling outlined by this section applies equally well to nonlinear dynamics: Section 4.3 briefly explores the specific nonlinear advection-dispersion of a discrete Burgers’ equation. Assume the fine spatial grid is periodic with $m$ grid points: that is, for definiteness assume the grid is periodic in space $x$ with period $mh$. For conciseness, write equations in terms of centred mean $\mu$ and difference $\delta$ operators [21, Ch. 7, e.g.] acting on the fine grid. Thus the advection-dispersion equation (1), but now including some ‘nonlinearity’ $f_{j}$ with some parameters $\vec{\epsilon}$, is \[\dot{u}_{j}=\big{\{}-c\mu\delta+d\delta^{2}\big{\}}u_{j}+f_{j}(\vec{u},\vec{ \epsilon})\,.\] (6) I give three illustrative examples of such nonlinearity: a local reaction could be prescribed by $f_{j}=\epsilon u_{j}-u_{j}^{3}$ ; a nonlinear advection by $f_{j}=u_{j}\mu\delta u_{j}/(2h)$ as in the discrete Burgers’ equation (41); whereas linear diffusion in a random medium could be encompassed by $f_{j}=\epsilon\delta(\kappa_{j}\delta u_{j})/h^{2}$ for some stochastic diffusivities $\kappa_{j}$. Centre manifold supportWe now describe how to support and construct the model on the coarse grid of the fine scale, nonlinearly modified, advection-dispersion dynamics. [FIGURE:S2.F3][ENDFIGURE] Analogous to holistic discretisation of s [26, 18, e.g.], divide the $m$-periodic fine grid into $m/2$ overlapping elements. Notionally let the $j$th coarse element stretch from $x_{2j-2}$ to $x_{2j+2}$ as shown ‘exploded’ in Figure 3. As shown, denote the evolving fine grid field in the $j$th element as the 5-tuple $\vec{v}_{j}=(v_{j,-2},v_{j,-1},v_{j,0},v_{j,1},v_{j,2})$ , so that at this stage we have just renamed the fine grid variables, $u_{2j+i}=v_{j,i}$ . Note that the elements overlap: the fine grid values $u_{2j\pm 1}=v_{j,\pm 1}=v_{j\pm 1,\mp 1}$ ; this overlap empowers us to couple the dynamics in neighbouring elements to derive consistent models as similarly derived for holistic discretisation [29].³ The interelement coupling conditions (8) determine the fine grid values $v_{j,\pm 2}$, at the extremes of each element, and so these are not extra dynamic variables. But, importantly, consider the overlapping fine grid values $v_{j,\pm 1}$ and $v_{j\pm 1,\mp 1}$ as independent dynamic variables satisfying the fine scale discrete equation (6), namely [FOOTNOTE:3][ENDFOOTNOTE] \[\dot{v}_{j,i}=\big{\{}-c\mu\delta+d\delta^{2}\big{\}}v_{j,i}+f_{j}(\vec{v}_{j} ,\vec{\epsilon})\,,\quad i=0,\pm 1\,,\] (7) where these differences and means operate over the fine grid index $i$. In essence I extend the dynamics of the $m$ fine grid variables $u_{j}(t)$ by an extra $m/2$ variables. This is the ‘one step backwards’ referred to in the Introduction: in order to rigorously support the modelling of the $m$-dimensional fine scale dynamics by $m/2$ coarse scale variables, I embed the fine scale system in the $3m/2$-dimensional dynamics of these overlapping elements. Section 3.4 shows how to choose these $m/2$ extra degrees of freedom to make forecasts from any given fine grid scale initial condition. Also analogous to holistic discretisation of s [26, 18, e.g.], couple neighbouring elements with the conditions \[v_{j,\pm 2}=\bar{\gamma}v_{j\pm 1,0}+(1-\bar{\gamma})v_{j,0}\,,\] (8) where the coupling parameter $\bar{\gamma}$ controls the interaction and information flow between elements: * when $\bar{\gamma}=1$ the elements are fully coupled and the condition (8) reduces to the statement that the extrapolation of the $j$th element field to the neighbouring coarse grid points, $v_{j,\pm 2}$, is identical to the neighbouring coarse grid values $v_{j\pm 1,0}(={\bar{u}}_{j\pm 1})$; * when $\bar{\gamma}=0$ the elements are completely isolated from each other and thus, _linearly_, the new fine grid values $v_{j,i}$ evolve quickly to be constant in each element. This equilibrium when $\bar{\gamma}=0$ , or space of equilibria depending upon the nonlinearity $f_{j}$, forms the base for the slow manifold model which when evaluated at $\bar{\gamma}=1$ gives the desired model for the fully coupled dynamics. I use the overbar in $\bar{\gamma}$ because it moderates information flow between the elements forming the coarse grid. By working to an error $\Ord{\bar{\gamma}^{n}}$ we account for interactions between the dynamics in an element and its $n-1$ neighbours on either side. Thus we transform _local_ dynamics on a fine grid to _local_ dynamics on a coarse grid as in other multiscale approaches [4, e.g.]. The size of the locality depends upon the order of error in the coupling parameter $\bar{\gamma}$. The decoupled dynamics have a useful spectral gapSet $\bar{\gamma}=0$ to decouple the elements, and neglect the nonlinearity by linearisation. Then, independently of all other elements, the linear dynamics in the $j$th element are governed by the differential-algebraic system \[\begin{bmatrix}0\\ \dot{v}_{j,-1}\\ \dot{v}_{j,0}\\ \dot{v}_{j,1}\\ 0\end{bmatrix}=\begin{bmatrix}1&0&-1&0&0\\ \rat 12c+d&-2d&-\rat 12c+d&0&0\\ 0&\rat 12c+d&-2d&-\rat 12c+d&0\\ 0&0&\rat 12c+d&-2d&-\rat 12c+d\\ 0&0&-1&0&1\end{bmatrix}\begin{bmatrix}v_{j,-2}\\ v_{j,-1}\\ v_{j,0}\\ v_{j,1}\\ v_{j,2}\end{bmatrix}\,.\] (9) Seeking solutions proportional to $e^{\lambda t}$ this set of linear s has three eigenvalues and three corresponding eigenvectors: \[\lambda=0,-2d,-4d\,;\qtq{and}\begin{bmatrix}1\\ 1\\ 1\\ 1\\ 1\end{bmatrix},\quad\begin{bmatrix}0\\ \rat 12c-d\\ 0\\ \rat 12c+d\\ 0\end{bmatrix},\quad\begin{bmatrix}1\\ -1\\ 1\\ -1\\ 1\end{bmatrix}.\] (10) From these, any zig-zag structures within an element decay exponentially quickly, and hence these decoupled dynamics results in constant solutions in each element arising on a time scale of $1/d$ . Over all the $m/2$ decoupled elements these piecewise constant solutions form an $m/2$ dimensional linear subspace of equilibria, the so-called slow subspace, in the $3m/2$ dimensional state space of the fine grid values $v_{j,0}$ and $v_{j,\pm 1}$. Centre manifold theory for deterministic systems [6, 20, e.g.] or for stochastic systems [2, 1, §8.4, e.g.] then assures us of the following three part theorem. For some domain of finite non-zero coupling parameter $\bar{\gamma}$, and if nonlinear, some neighbourhood of the origin in $(\vec{u},\vec{\epsilon})$: 1. there exists an $m/2$-dimensional, invariant _slow manifold_ ${\cal M}$ of the coupled dynamics of the discrete nonlinearly perturbed, advection-dis-persion (7) with coupling conditions (8)—with a dimension corresponding to each of the $m/2$ coarse grid elements; 2. the dynamics on the slow manifold ${\cal M}$ are approached exponentially quickly, roughly like $\exp(-2dt)$, by all initial conditions $v_{j,i}(0)$ of the fine grid values in some finite neighbourhood of ${\cal M}$—that is, the slow manifold dynamics faithfully model for long times generic solutions of the coupled dynamics; 3. we may construct the slow manifold model to some order of error in $\bar{\gamma}$, $\|\vec{\epsilon}\|$ and $\|\vec{\bar{u}}\|$ by solving the governing, nolinear, discrete advection-dispersion (6) with coupling conditions (8) to residuals of the same order. Two broad cases arise: if the nonlinearity $f_{j}=0$ whenever $v_{j,i}$ is independently constant in each element—for example the Burgers’-like nonlinearity $f_{j}=u_{j}\mu\delta u_{j}/(2h)$—then the approximation is global in the coarse grid variables $\vec{\bar{u}}$; alternatively, whenever $f_{j}\neq 0$ for $v_{j,i}$ independently constant in each element—for example the reaction $f_{j}=\epsilon u_{j}-u_{j}^{3}$—then the approximation is local to the origin in $\vec{\bar{u}}$ Like systematic upscaling [4, pp.6,9] and other multiscale methods, this approach uses equilibrium concepts. But one crucial difference is that centre manifold theory guarantees that the same separation of dynamics occurs in a _finite_ neighbourhood about equilibria and hence supports the separation of coarse scale dynamics from the fine scale occurs for nontrivial dynamics. This approach provides a systematic alternative to the heuristic Fourier or wavelet decompositions for a ‘local mode analysis’ [3, §8]: here the local modes are determined by the the dynamical system itself through the shape of the slow manifold. Finite domainAfter constructing an approximate slow manifold model, we evaluate it for coupling parameter $\bar{\gamma}=1$ to recover a coarse grid model for the fully coupled dynamics on the fine grid. Is $\bar{\gamma}=1$ in the ‘finite neighbourhood’ of theoretical support? It is for the analogous holistic discretisation of the Burgers’ [26]. Similarly, Section 3.3 demonstrates that the fully coupled case, $\bar{\gamma}=1$ , is indeed within the neighbourhood of theoretical support for the linear ($f_{j}=0$) dynamics of (7). ## 3 Coarsen linear advection-dispersion Using the theoretical support of centre manifold theory established by the previous section, this section analyses linear advection-dispersion to provide the multiscale modelling results summarised in the Introduction. ### Computer algebra approximates the slow manifold Elementary algebra readily constructs general slow manifold models [25, 18, e.g.]. We solve the fine grid, linear, discrete, advection-dispersion equation (7) with coupling conditions (8) by seeking solutions parametrised by the evolving coarse grid values ${\bar{u}}_{j}(t)$: \[v_{j,i}(t)=V_{j,i}(\vec{\bar{u}},\bar{\gamma})\qtq{suchthat}\dot{\bar{u}}_{j}= G_{j}(\vec{\bar{u}},\bar{\gamma})\,,\] (11) for some functions $V_{j,i}$ and $G_{j}$ to be determined by the iterative algorithm [32]. The base approximation is the slow subspace of equilibria: \[v_{j,i}(t)=V_{j,i}(\vec{\bar{u}},\bar{\gamma})\approx{\bar{u}}_{j}\qtq{ suchthat}\dot{\bar{u}}_{j}=G_{j}(\vec{\bar{u}}_{j},\bar{\gamma})\approx 0\,.\] Computer algebra code [32] systematically refine these slow manifold approximations. The refining iteration is based upon the residuals of the discrete equation (7) with coupling conditions (8). Centre manifold theory then assures us that the error in approximating the slow manifold model is of the same order in coupling parameter $\bar{\gamma}$ as any remaining residual. For example, to errors $\Ord{\bar{\gamma}^{2}}$, computer algebra [32] constructs the slow manifold ${\cal M}$ in the $j$th element as \[\vec{V}_{j}=\begin{bmatrix}(1-\bar{\gamma})+\bar{\gamma}\bar{E}^{-1}\hfill\\ 1-\rat 12\bar{\gamma}\bar{\mu}\bar{\delta}+\rat 18\bar{\gamma}(1+\rat{c}{d}- \rat{c^{2}}{4d^{2}})\bar{\delta}^{2}\\ 1\hfill\\ 1+\rat 12\bar{\gamma}\bar{\mu}\bar{\delta}+\rat 18\bar{\gamma}(1-\rat{c}{d}- \rat{c^{2}}{4d^{2}})\bar{\delta}^{2}\\ (1-\bar{\gamma})+\bar{\gamma}\bar{E}\hfill\end{bmatrix}{\bar{u}}_{j}+\Ord{\bar {\gamma}^{2}}\,,\] (12) in terms of the coarse grid centred difference and mean operators, $\bar{\delta}$ and $\bar{\mu}\bar{\delta}$, and shift operator $\bar{E}$ (define $\bar{E}{\bar{u}}_{j}={\bar{u}}_{j+1}$ or equivalently $\bar{E}=E^{2}$). The terms in (12) which are independent of advection $c$, for the fully coupled $\bar{\gamma}=1$ , are classic quadratic interpolation from the surrounding coarse grid values ${\bar{u}}_{j}$. The terms involving advection, flagged by $c$, arise through accounting for the dynamics of the fine grid values $u_{2j\pm 1}$ and their interaction with the surrounding grid values. Equation (12) corresponds to the multigrid prolongation operator, but here it is derived by accounting for the fine scale dynamics rather than being imposed. The evolution on the slow manifold (12) is then the coarse grid model \[\dot{\bar{u}}_{j}=\bar{\gamma}\left[-\rat 12c\bar{\mu}\bar{\delta}+\left(\rat 1 4d+\frac{c^{2}}{16d}\right)\bar{\delta}^{2}\right]{\bar{u}}_{j}+\Ord{\bar{ \gamma}^{2}}\,.\] (13) Neglecting the $\Ord{\bar{\gamma}^{2}}$ error, evaluate (13) at the physically relevant coupling $\bar{\gamma}=1$ to deduce the coarse grid model (2) discussed in the Introduction. ### The coarse scale, slow dynamics model precisely Consider further the linear advection-dispersion (7) with interelement coupling conditions (8). The previous subsection constructed an approximation to errors $\Ord{\bar{\gamma}^{2}}$; this subsection gives exact formula for all coupling $\bar{\gamma}$. Seek solutions with structure within the finite elements of the formal operator form $\vec{v}_{j}=\exp(t{\cal K}_{n})\vec{e}_{n}$ , where, generalising (10) to non-zero coupling, ${\cal K}_{n}$ is the $n$th ‘operator eigenvalue’ of the advection-dispersion (7)–(8) and $\vec{e}_{n}$ is the corresponding ‘operator eigenvector’. Elementary algebra for any coupling $\bar{\gamma}$ reveals the three operator eigenvalues are precisely \[{\cal K}_{1,3}=2d\left\{-1\pm\sqrt{1+\bar{\gamma}\left[\frac{1}{4}\left(1+ \frac{c^{2}}{4d^{2}}\right)\bar{\delta}^{2}-\frac{c}{2d}\bar{\mu}\bar{\delta} \right]}\right\}\qtq{and}{\cal K}_{2}=-2d\,.\] (14) The smallest (least negative) of these, namely ${\cal K}_{1}$, governs the longest time scales in the coupled dynamics. For example, the Taylor expansion⁴ in the coupling $\bar{\gamma}$ of operator ${\cal K}_{1}$ (the plus case above), upon using the identity $\bar{\mu}^{2}=1+\rat 14\bar{\delta}^{2}$ , agrees with the $\Ord{\bar{\gamma}^{2}}$ evolution (13), to the $\Ord{\bar{\gamma}^{3}}$ approximation (40), and to the $\Ord{\bar{\gamma}^{5}}$ approximation (31) of isotropic dynamics. That is, the coarse grid evolution operator $\bar{{\cal L}}={\cal K}_{1}$ . [FOOTNOTE:4][ENDFOOTNOTE] The coarse grid operator $\bar{{\cal L}}={\cal K}_{1}$ reproduces exactly the fine grid operator of linear advection-dispersion when the elements are fully coupled. In the fully coupled limit, $\bar{\gamma}=1$ , the three operator eigenvalues (14) reduce to \[\bar{{\cal L}}={\cal K}_{1}=d(2\bar{\mu}-2)-\rat 12c\bar{\delta}\,,\quad{\cal K }_{2}=-2d\,,\quad{\cal K}_{3}=-d(2\bar{\mu}+2)+\rat 12c\bar{\delta}\,.\] (15) Relate to the fine grid operators, via the coarse grid shift operator $\bar{E}$ and the fine grid shift operator $E(=\bar{E}^{1/2})$, by observing \[2\bar{\mu}-2=\bar{E}^{1/2}+\bar{E}^{-1/2}-2=E+E^{-1}-2=\delta^{2}\] and \[\rat 12\bar{\delta}=\rat 12\big{(}\bar{E}^{1/2}-\bar{E}^{-1/2} \big{)}=\rat 12\big{(}E-E^{-1}\big{)}=\mu\delta\,.\] Consequently, the coarse grid model \[\dot{\bar{u}}_{j}=\bar{{\cal L}}{\bar{u}}_{j}=\big{[}d(2\bar{\mu}-2)-\rat 12c \bar{\delta}\big{]}{\bar{u}}_{j}=\big{[}d\delta^{2}-c\mu\delta\big{]}{\bar{u}} _{j}\,,\] is _precisely_ the fine grid, linear, advection-dispersion equation (6), except that, having half the grid points, it does not resolve the fine scale, high wavenumber, spatial structures that the fine grid can resolve. Thus the operator $\bar{{\cal L}}$ does indeed model on the coarse grid all the coarse dynamics inherent in the fine grid advection-dispersion dynamics. It is only the approximation of $\bar{{\cal L}}$ by a truncated Taylor series, such as in the $\Ord{\bar{\gamma}^{2}}$ model (13), that induces errors in the coarse scale model of the long term dynamics of linear advection-dispersion. ### Coarse scale dynamics are attractive Consider the spectrum of the advection-dispersion dynamics implicitly described by the operator eigenvalues (14). On any regularly spaced grid, the centred mean and difference operators act on Fourier modes as \[\mu e^{ikj}=\cos(k/2)e^{ikj}\qtq{and}\delta e^{ikj}=2i\sin(k/2)e^{ikj}\] for a component of spatial wavenumber $k$ relative to the grid spacing; the wavenumber domain is $-\pi<k\leq\pi$ . Upon taking the discrete Fourier transform, the operators $\bar{\mu}$ and $\bar{\delta}$ thus transform to $\cos(\bar{k}/2)=\cos k$ and $2i\sin(\bar{k}/2)=2i\sin k$ , respectively, as the fine grid wavenumber $k=\bar{k}/2$ in terms of the coarse grid wavenumber $\bar{k}$. Thus from (14) the advection-dispersion dynamics on the fine grid elements has spectrum \[\lambda_{1,3}=2d\left\{-1\pm\sqrt{1+\bar{\gamma}\left[-\left(1+ \frac{c^{2}}{4d^{2}}\right)\sin^{2}\rat{\bar{k}}2-i\frac{c}{2d}2\sin\rat{\bar{ k}}2\cos\rat{\bar{k}}2\right]}\right\}\] \[\text{and}\quad\lambda_{2}=-2d\,,\] (16) for coarse grid wavenumbers $\|\bar{k}\|\leq\pi$ . Extensive numerical computations strongly suggest that $0\leq\Re{\sqrt{\cdot}}\leq 1$ , where $\sqrt{\cdot}$ denotes the square root in (16), for all wavenumbers $\|\bar{k}\|\leq\pi$ , for all coupling $0\leq\bar{\gamma}\leq 1$ , and for all advection relative to diffusion, $c/d$. Consequently, the numerics suggest the spectral ordering $\Re\lambda_{3}\leq\Re\lambda_{2}\leq\Re\lambda_{1}\leq 0$ is maintained across the whole relevant parameter domain. Thus, not only does the coarse grid model $\dot{u}_{j}=\bar{{\cal L}}{\bar{u}}_{j}={\cal K}_{1}{\bar{u}}_{j}$ accurately model the fine grid dynamics, the coarse grid model is the _slowest_ dynamics of the fine grid advection-dispersion. _Theorem 2 ensures an atttractive slow manifold exists in some neighbourhood of coupling $\bar{\gamma}=0$ ; the spectrum (16) demonstrates that the neighbourhood extends to include the case of fully coupled elements, $\bar{\gamma}=1$ ._ We usually cannot construct slow manifolds exactly, as done above; instead we usually approximate slow manifold by a multivariate power series. Thus the practical issue is not just whether a slow manifold exists, but how well a truncated power series approximates the slow manifold. Elementary algebra shows that a Taylor series of (16) in $\bar{\gamma}$ converges at $\bar{\gamma}=1$ provided \[\big{[}(1-{\mathfrak C}^{2})^{2}\sin^{2}k+4{\mathfrak C}^{2}\big{]}\sin^{2}k<1\,,\] where ${\mathfrak C}=c/(2d)$ measures the advection relative to the dispersion. For all parameter ${\mathfrak C}$ there is a finite range of small wavenumbers $k$ satisfying this inequality. This argument leads to the following lemma. Finite truncations of the Taylor series of the slow operator eigenvalue ${\cal K}_{1}$ provide accurate approximations of the evolution of the coarse grid variables provided the solutions vary slowly enough across the grid. ### A normal form projects initial conditions Suppose we know the fine grid values $u_{j}(0)$ at the initial time $t=0$ . This subsection addresses the question: what coarse grid values should we give to ${\bar{u}}_{j}(0)$ for the coarse grid model to make accurate long term predictions? The obvious answer is wrong [24, 8, 27, e.g.]: even though we define ${\bar{u}}_{j}(t)=u_{2j}(t)$ , we nonetheless should _not_ set the initial ${\bar{u}}_{j}(0)=u_{2j}(0)$ . The reason is that the transient dynamics of the subgrid scale dynamics modifies the appropriate initial value for ${\bar{u}}_{j}(0)$; this modification is sometimes called ‘initial slip’ in physics [14, 13, e.g.]. In this subsection, a normal form coordinate transform of the fine grid dynamics clearly displays the correct initial conditions for the coarse dynamics. In this discussion, restrict attention on initial conditions to the fully coupled case of coupling $\bar{\gamma}=1$ . This restriction simplifies by avoiding the complicating detail of having variable $\bar{\gamma}$, and it focusses on the physically relevant case of full interelement coupling. Consider the spectral decomposition of the dynamics of the fine grid of all the elements. Transform the fine grid evolution to its coarse grid ‘normal form’ of the spectral decomposition \[\vec{v}_{j}(t)=\vec{e}_{j,1}{\bar{u}}_{j}(t)+\vec{e}_{j,2}{\bar{v }}_{j}(t)+\vec{e}_{j,3}{\bar{w}}_{j}(t)\,,\] where \[\dot{\bar{u}}_{j}={\cal K}_{1}{\bar{u}}_{j}\,,\quad\dot{\bar{v}}_ {j}={\cal K}_{2}{\bar{v}}_{j}\,,\quad\dot{\bar{w}}_{j}={\cal K}_{3}{\bar{w}}_{ j}\,,\] for the operators ${\cal K}_{n}$ in (15) and for intraelement structure operators \[\vec{e}_{j,2}=\begin{bmatrix}\rat 12c-d\\ 0\\ \rat 12c+d\end{bmatrix}\qtq{and}\vec{e}_{j,n}=\begin{bmatrix}\rat 12c(\bar{E}^ {-1}-1)+d(\bar{E}^{-1}+1)\\ {\cal K}_{n}+2d\\ \rat 12c(\bar{E}-1)+d(\bar{E}+1)\end{bmatrix}\] (17) for $n=1,3$ . I do not record the two extreme components $v_{j,\pm 2}$ in these $\vec{e}_{j,n}$ as $v_{j,\pm 2}$ are identical to $v_{j\pm 1,0}$ when fully coupled, $\bar{\gamma}=1$ . Within each of the fully coupled elements, a formal expression for the complete evolution on the fine grid is thus \[\vec{v}_{j}(t)=\vec{e}_{j,1}\exp(t{\cal K}_{1}){\bar{u}}_{j}(0)+\vec{e}_{j,2} \exp(t{\cal K}_{2}){\bar{v}}_{j}(0)+\vec{e}_{j,3}\exp(t{\cal K}_{3}){\bar{w}}_ {j}(0)\,,\] (18) for some constants ${\bar{u}}_{j}(0)$, ${\bar{v}}_{j}(0)$ and ${\bar{w}}_{j}(0)$. For example, from (15), when advection $c=0$ the intraelement structure operators simplify to \[\vec{e}_{j,2}\propto\begin{bmatrix}-1\\ 0\\ 1\end{bmatrix}\qtq{and}\vec{e}_{j,n}\propto\begin{bmatrix}\rat 12(\bar{E}^{-1} +1)\\ \pm\bar{\mu}\\ \rat 12(\bar{E}+1)\end{bmatrix}\approx\begin{bmatrix}1\\ \pm 1\\ 1\end{bmatrix}\] where this last approximate equality holds for fields varying slowly enough along the grids. Thus $\vec{e}_{j,1}\approx(1,1,1)$ represents the smoothest variations within each element, whereas $\vec{e}_{j,2}\approx(-1,0,1)$ and $\vec{e}_{j,3}\approx(1,-1,1)$ represents fine grid scale fluctuations within an element.⁵ Since these fine grid scale fluctuations decay rapidly in time $t$, the long term slow dynamics on the slow manifold is just the restriction of (18) to ${\bar{v}}_{j}={\bar{w}}_{j}=0$ , namely [FOOTNOTE:5][ENDFOOTNOTE] \[\vec{v}_{j}(t)=\vec{e}_{j,1}\exp(t{\cal K}_{1}t){\bar{u}}_{j}(0)\,.\] (19) We must choose the initial condition, ${\bar{u}}_{j}(0)$, for the coarse grid values so that this evolution exponentially quickly equals the fine grid dynamics $u_{j}(t)=\exp(t{\cal L})u_{j}(0)$_from the specified initial condition_. Such a choice for the initial coarse grid value ${\bar{u}}_{j}(0)$ then realises the theoretical promise by Theorem 2-2 of long term fidelity between coarse grid model and fine grid dynamics. Elementary linear algebra determines the coarse grid values ${\bar{u}}_{j}(0)$ through evaluating the general solution (18) at time $t=0$ , \[\vec{v}_{j}(0)=\vec{e}_{j,1}{\bar{u}}_{j}(0)+\vec{e}_{j,2}{\bar{v}}_{j}(0)+ \vec{e}_{j,3}{\bar{w}}_{j}(0)\,,\] (20) and then take the inner product with the left eigenvector \[\vec{z}_{j,1}=\begin{bmatrix}\rat 12c+d\\ {\cal K}_{1}+2d\\ -\rat 12c+d\end{bmatrix},\] to deduce the following lemma. For linear advection-dispersion, the initial coarse grid values are \[{\bar{u}}_{j}(0)=\frac{\vec{z}_{j,1}\cdot\vec{v}_{j}(0)}{\vec{z}_{j,1}\cdot \vec{e}_{j,1}}\,,\] (21) in terms of specified fine element values $\vec{v}_{j}(0)$. Despite the definition that the coarse grid values ${\bar{u}}_{j}(t)=u_{2j}(t)$ , the normal form coordinate transform accounts for dynamics in fast time initial transients so that the correct initial conditions for the coarse grid model is the nonlocal and weighted projection (21). The initial condition mapping (21) relates to multigrid iteration. When advection $c=0$ \[\vec{z}_{j,1}\propto\begin{bmatrix}1\\ 2\bar{\mu}\\ 1\end{bmatrix}\approx\begin{bmatrix}1\\ 2\\ 1\end{bmatrix},\] and in the case of slowly varying grid values, this projection from the fine grid initial values $\vec{v}_{j}(0)$ to the coarse grid initial values is the classic multigrid restriction operator [5, e.g.]: namely, that the coarse grid value is the average of the nearest fine grid values with a weighting of $1:2:1$ . Uniquely prescribe fine element valuesWe have an additional complication: on the fine grid, the odd grid values $u_{2j\pm 1}$ are shared between two neighbouring elements. The grid value $u_{2j+1}$ is represented as both $v_{j,1}$ and $v_{j+1,-1}$, and both of these variables are treated as separate independent variables in the dynamics on each element. We must resolve this separation. Two independent suggestions resolve the separation with the same result. My first suggestion to avoid conflict between the values of $v_{j,1}$ and $v_{j+1,-1}$ is to require that $v_{j,1}=v_{j+1,-1}$ at the initial time. The shift operators rewrite this identity as \[Ev_{j,0}=\bar{E}E^{-1}v_{j,0}\,,\] (22) where the coarse grid shift $\bar{E}$ operates on the coarse grid, first subscript of $v_{j,i}$, whereas the fine grid shift operator $E$ operates on the fine grid, second subscript of $v_{j,i}$. For a domain with $m$ fine grid points, that is, $m/2$ coarse grid elements, the compatibility condition (22) provides an additional $m/2$ constraints to determine uniquely the $3m/2$ initial values $v_{j,i}(0)$ within the elements from the $m$ fine grid values $u_{j}(0)$. My second suggestion is to choose $v_{j,\pm 1}(0)$ so that the unphysical intermediate mode vanishes in the solution (18), that is, so that ${\bar{v}}_{j}(0)=0$ in the solution (18). Then there will be no intermediate scale dynamics $\exp(-2dt)$ and the approach to the slow manifold model will be the quickest: the only rapidly decaying mode will be the $\vec{e}_{j,3}\exp({\cal K}_{3}t){\bar{w}}_{j}(0)$ mode which, from the spectrum (16), decays more rapidly than $\exp(-2dt)$. Now relate ${\bar{v}}_{j}(0)$ directly to $\vec{v}_{j}(0)$ by multiplying (20) by the left eigenvector corresponding to ${\cal K}_{2}$ namely \[\vec{z}_{j,2}=\begin{bmatrix}\rat 12c(\bar{E}-1)-d(\bar{E}+1)\\ 0\\ \rat 12c(\bar{E}^{-1}-1)+d(\bar{E}^{-1}+1)\end{bmatrix}.\] Thus, noting $v_{j,\pm 1}=E^{\pm 1}v_{j,0}$ , \[{\bar{v}}_{j}(0) \propto \left\{\left[\rat 12c(\bar{E}-1)-d(\bar{E}+1)\right]E^{-1}+\left[ \rat 12c(\bar{E}^{-1}-1)+d(\bar{E}^{-1}+1)\right]E\right\}v_{j,0}\] \[= \left\{(\rat 12c-d)(\bar{E}E^{-1}-E)+(\rat 12c+d)(\bar{E}^{-1}E-E ^{-1})\right\}v_{j,0}\,.\] Consequently, ensure the mode $\exp(-2dt)$ does not appear at all, ${\bar{v}}_{j}(0)=0$ , by requiring $(\bar{E}E^{-1}-E)v_{j,0}=0$ which is precisely (22), and by requiring $(\bar{E}^{-1}E-E^{-1})v_{j,0}=0$ which is again (22) but just shifted to the left by the multiplication by the coarse grid shift $\bar{E}^{-1}$. Thus the condition (22) ensures that neighbouring elements agree at their common points _and_ that the slow manifold, long term model is approached quickest. Choosing the embedding to $3m/2$-dimensions to satisfy (22) at the initial time ensures that (22) is satisfied for all time in the linear, advection-dispersion dynamics on the fully coupled finite elements. ### Extend elements for a multigrid hierarchy As discussed briefly in the Introduction, we aim to transform dynamics across a wide range of space-time scales using the multigrid hierarchy illustrated in Figure 1. The Introduction used a model of $\Ord{\bar{\gamma}^{2}}$, see §3.1, to transform advection-dispersion on a fine grid to advection-dispersion _of the same form_ on a coarser grid. This transform iterates simply across all scales. However, when we seek more accuracy, say errors $\Ord{\bar{\gamma}^{n}}$ for $n>2$ , the linear advection-dispersion dynamics (6) transforms into a model of the form $d{\bar{u}}_{j}/dt=G(\bar{u}_{j-n+1},\ldots,\bar{u}_{j+n-1},\bar{\gamma})$ that involves $2(n-1)$ neighbouring coarse grid values. For example, to errors $\Ord{\bar{\gamma}^{4}}$, fine scale isotropic dispersion (equation (6) with $c=f_{j}=0$) transforms to the coarser scale dispersion [32] \[\frac{d{\bar{u}}_{j}}{dt}=d\left[\rat 14\bar{\gamma}\bar{\delta}^{2}-\rat 1{64 }\bar{\gamma}^{2}\bar{\delta}^{4}+\rat 1{512}\bar{\gamma}^{3}\bar{\delta}^{6} \right]{\bar{u}}_{j}+\Ord{\bar{\gamma}^{4}}\,,\] (23) that through $\bar{\delta}^{4}$ and $\bar{\delta}^{6}$ involves $\bar{u}_{j\pm 2}$ and $\bar{u}_{j\pm 3}$ . Consequently, to empower us to transform coarse models over a hierarchy of grids we must widen the elements defined in Figure 3 to include more fine grid points. This subsection widens the elements while maintaining the spectrum (10) ensuring the centre manifold support [6, 20, e.g.]. This subsection, as seen in equation (23), avoids the overdots for time derivatives as we invoke different time scales on each level of the hierarchy. Interestingly, it eventuates that not only do we overlap the elements, but also, in some sense, overlap the time scales. The general form of linear dynamics on a gridSuppose at some level of the multigrid hierarchy we know the discrete operator governing the evolution of grid values $u_{j}(t)$. Decompose the discrete evolution operator as the sum \[\frac{du_{j}}{dt}=\left[{\cal L}_{1}+{\cal L}_{2}+{\cal L}_{3}+\cdots+{\cal L} _{n-1}\right]u_{j}\,,\] (24) where the $k$th discrete operator ${\cal L}_{k}$ has stencil width $2k+1$ ; that is, ${\cal L}_{k}u_{j}$ only involves $u_{j-k},\ldots,u_{j+k}$ . This decomposition terminates, as written in (24), when we restrict attention, by working to errors $\Ord{\gamma^{n}}$, to operators of some maximum finite width. The decomposition is not unique as specified so far; however, as apparent in (23), a specific unique decomposition naturally arises when we generate the models to errors $\Ord{\gamma^{n}}$ in some coupling parameter $\gamma$. Thus suppose there is a natural ‘ordering’ parameter $\gamma$ such that, instead of (24), the discrete evolution equation may be written \[\frac{du_{j}}{dt}=\left[{\cal L}_{1}+\gamma{\cal L}_{2}+\gamma^{2}{\cal L}_{3} +\cdots+\gamma^{n-2}{\cal L}_{n-1}\right]u_{j}\,.\] (25) At all levels, except the very finest level $0$, this natural parameter $\gamma$ is to be the coupling parameter of the finite elements of the grid one level finer than than current level. As always, we suppose that evaluation of (25) at $\gamma=1$ gives the physically relevant model (24), whereas $\gamma=0$ provides a base for theory to support models at non-zero $\gamma$. Additionally insisting on the operator ${\cal L}_{1}$ being conservative implies ${\cal L}_{1}$ must represent advection-dispersion dynamics and implies that the $\gamma=0$ dynamics, $du_{j}/dt={\cal L}_{1}u_{j}$ , provides the same sound base for applying centre manifold theory. Note that the coupling parameter of finite elements at the current level is still $\bar{\gamma}$. That is, still couple neighbouring elements with the condition (8). Anticipating the support by centre manifold theory, derived in a couple of paragraphs, we expect to construct a coarse grid model of the dynamics (25) in the form \[\frac{d\bar{u}_{j}}{dt}=\left[\bar{\gamma}\bar{{\cal L}}_{1}+\bar{\gamma}^{2} \bar{{\cal L}}_{2}+\bar{\gamma}^{3}\bar{{\cal L}}_{3}+\cdots+\bar{\gamma}^{n-1 }\bar{{\cal L}}_{n-1}\right]\bar{u}_{j}+\Ord{\bar{\gamma}^{n},\gamma^{n-1}}\,,\] (26) for some coarse grid operators $\bar{{\cal L}}_{k}$ (implicitly a function of the artificial $\gamma$) which will be of stencil width $2k+1$ as the parameter $\bar{\gamma}$ counts the number of interelement communications. The renormalising transformation requires two extra ingredients: first remove the fine grid ordering by setting $\gamma=1$ (so operators $\bar{{\cal L}}_{k}$ are no longer a function of $\gamma$); and second introduce a coarse grid time scale $\bar{t}=t/\bar{\gamma}$ (which is the same time when $\bar{\gamma}=1$), then, upon dividing by $\bar{\gamma}$, the coarse grid dynamics become \[\frac{d\bar{u}_{j}}{d\bar{t}}=\left[\bar{{\cal L}}_{1}+\bar{\gamma}\bar{{\cal L }}_{2}+\bar{\gamma}^{2}\bar{{\cal L}}_{3}+\cdots+\bar{\gamma}^{n-2}\bar{{\cal L }}_{n-1}\right]\bar{u}_{j}+\Ord{\bar{\gamma}^{n-1}}\,.\] (27) The coarse model (27) has exactly the same form as the fine model (25). By introducing the coupling (8) across all levels of the hierarchy, and by introducing a hierarchy of times, which all collapse to the same real time when $\bar{\gamma}=1$ , and working to some order of error in coupling, models of the form (25) are transformed and renormalised across the entire multigrid hierarchy. Widen the elementsAssume we wish to construct slow manifolds to errors $\Ord{\bar{\gamma}^{n},\gamma^{n-1}}$ with the aim of using centre manifold theory to support the modelling of (25) by (27). Extend Section 2 by widening the $j$th element to extend over the interval $[x_{j-n},x_{j+n}]$ and also to possess the $(2n+1)$ fine grid variables $\vec{v}_{j}=(v_{j,-n},\ldots,v_{j,n})$ . These extra variables are not extra degrees of freedom. Let these fine grid variables evolve according to \[\frac{dv_{j,i}}{dt}={\cal L}_{1}v_{j,i}+\gamma{\cal L}_{2}v_{j,i}+\gamma^{2}{ \cal L}_{3}v_{j,i}+\cdots+\gamma^{n-2}{\cal L}_{n-1}v_{j,i}\,,\quad\|i\|<n\,,\] (28) where we adopt the unusual convention that when applied within the elements, the operator ${\cal L}_{k}v_{j,i}$ is its original definition when $\|i\|+k\leq n$ but is _zero otherwise_ (for $\|i\|+k>n$). Adopting this convention ensures that the operators on the right-hand side of (28) do not ‘poke outside’ of the $j$th element; in effect, this convention truncates the sum in (28) to remain within the $j$th element. Such truncation incurs an error $\Ord{\gamma^{n+1-\|i\|}}$ in the evolution of a variable $v_{j,i}$. However, as variable $v_{j,i}$ only affects the crucial central core variables of the element, $v_{j,i^{\prime}}$ for $\|i^{\prime}\|\leq 2$ as shown in Figure 3, via terms of $\Ord{\gamma^{\|i\|-2}}$, the net effect of this conventional truncation is an error $\Ord{\gamma^{n-1}}$ which is the same as the assumed order of error of the analysis. The coupling condition (8) closes the dynamics on these widened elements. In essence we do not have new dynamics outside of the central core of each element, instead, in effect, we simply extrapolate the dynamics to the outside of the central core. Centre manifold theory supportWhen the fine grid ‘ordering’ parameter $\gamma=0$ and interelement coupling parameter $\bar{\gamma}=0$ the dynamics on the $m/2$ elements reduces to \[\frac{dv_{j,i}}{dt}={\cal L}_{1}v_{j,i}\quad\text{for }\|i\|<n\,,\qtq{and}v_{j, \pm 2}=v_{j,0}\,.\] (29) Each element is decoupled from the others. The general conservative, linear, operator is the advection-dispersion operator, ${\cal L}_{1}=-c\bar{\mu}\bar{\delta}+d\delta^{2}$ for some constants $c$ and $d$. As for the earlier (9), for each of the extended elements there are still precisely three eigenvalues of (29), namely $\lambda=0,-2d,-4d$ . Corresponding eigenvectors are the constant $v_{j,i}\propto 1$ , the artificial $v_{j,i}\propto\sin(i\pi/2)\left[({1+{\mathfrak C}})/({1-{\mathfrak C}})\right] ^{i/2}$, and the zig-zag mode $v_{j,i}\propto(-1)^{i}$ . Consequently, centre manifold theory implies Theorem 2 also applies to the system (28) with coupling conditions (8) to ensure: firstly, that an $m/2$ dimensional slow manifold exists for the dynamics of the coupled elements; secondly, the coarse scale dynamics on the slow manifold are attractive; and thirdly, that we may construct the slow manifold to any desired error—this section assumes errors $\Ord{\gamma^{n-1},\bar{\gamma}^{n}}$. The next subsection proceeds to briefly explore the resultant models of advection and dispersion over a hierarchy of multiscale grids as supported by this theory. ## 4 Multiscale modelling iterates transformations This section explores three example applications of transforming dynamics repeatedly across the wide range of length and time scales on a multigrid hierarchy. Section 4.1 shows how continuum diffusion emerges from microscale dispersion. Section 4.2 deonstrates that the nonlinear transformation from one scale to another of linear advection-dispersion has a fixed point of a stable upwind model. Section 4.3 discusses briefly the transformation of the nonlinear Burgers’ . ### Diffusion emerges from discrete dispersion The multiscale modelling of discrete dispersion, when the advection coefficient $c=0$ , reduces to a remarkably simple linear transformation. Here we explore the exact slow manifold transformation from a fine grid to a coarser grid. Iterating this transformation proves that, in the absence of advection, the continuum diffusion equation naturally emerges very quickly on macroscales. Linear dynamics which are left-right symmetric (isotropic) can be expressed in terms of only even order central differences. Our slow manifold, multiscale modelling preserves this form. Suppose the evolution at grid level $\ell$ is governed by \[\frac{du_{j}^{(\ell)}}{dt^{(\ell)}}=\sum_{p=1}^{\infty}\gamma^{p-1}c_{p}^{( \ell)}{\delta^{(\ell)}}^{2p}u_{j}^{(\ell)}\,,\] (30) for some coefficients $c_{p}$ ; for example, the second order difference coefficient $c_{1}=d$ used earlier. In practical constructions, invoking an error $\Ord{\gamma^{n-1}}$ truncates to a finite sum this ‘in principle’ infinite sum. Computer algebra [32], supported by the theory of Section 3.5, derives the dynamics at the next coarser level of the multiscale hierarchy, namely \[\frac{du_{j}^{(\ell+1)}}{dt^{(\ell+1)}} = \big{[}\rat 14c_{1}^{(\ell)}{\delta^{(\ell+1)}}^{2}\] (31) \[{}+\bar{\gamma}\Big{(}-\rat 1{64}c_{1}^{(\ell)}+\rat 1{16}c_{2}^{ (\ell)}\Big{)}{\delta^{(\ell+1)}}^{4}\] \[{}+\bar{\gamma}^{2}\Big{(}\rat 1{512}c_{1}^{(\ell)}-\rat 1{128}c_ {2}^{(\ell)}+\rat 1{64}c_{3}^{(\ell)}\Big{)}{\delta^{(\ell+1)}}^{6}\] \[{}+\bar{\gamma}^{3}\Big{(}-\rat 5{16384}c_{1}^{(\ell)}+\rat 5{409 6}c_{2}^{(\ell)}-\rat 3{1024}c_{3}^{(\ell)}+\rat 1{256}c_{4}^{(\ell)}\Big{)}{ \delta^{(\ell+1)}}^{8}\] \[{}\big{]}u_{j}^{(\ell+1)}+\Ord{\bar{\gamma}^{4}}\,.\] That is, to model level $\ell$ dynamics at level $(\ell+1)$ the coefficients in (30) transform according to the linear transform \[\vec{c}^{(\ell+1)}={\cal T}\vec{c}^{(\ell)}\qtq{where}{\cal T}=\begin{bmatrix} \rat 14&0&0&0&\cdots\\ -\rat 1{64}&\rat 1{16}&0&0&\cdots\\ +\rat 1{512}&-\rat 1{128}&\rat 1{64}&0&\cdots\\ -\rat 5{16384}&\rat 5{4096}&-\rat 3{1034}&\rat 1{256}&\\ \vdots&\vdots&\vdots&&\ddots\end{bmatrix}.\] (32) By induction, the level $\ell$ dynamics have centred difference coefficients $\vec{c}^{(\ell)}={\cal T}^{\ell}\vec{c}^{(0)}$ . Consequently, the dynamics that emerge on macroscale grids are determined by the powers ${\cal T}^{\ell}$ for large $\ell$. Since ${\cal T}$ is triangular the powers are simple: the dominant structure for large $\ell$ corresponds to that of the leading eigenvalue $1/4$; its eigenvector gives the centred difference coefficients that emerge on the macroscale as \[\vec{c}^{(\ell)}\sim\frac{c_{1}^{(0)}}{4^{\ell}}\big{(}1,-\rat 1{12},\rat 1{90 },-\rat 1{560},\ldots)\quad\text{as }\ell\to\infty\,.\] (33) Recognise in this vector the coefficients of various powers of centred differences in a discrete representation of the continuum diffusion operator $\partial^{2}/\partial x^{2}$. That is,⁶_continuum diffusion emerges on the macroscale for all isotropic, conservative, linear, continuous time, microscale dynamics provided there is some component of $\delta^{2}$ in the microscale ($c_{1}^{(0)}\neq 0$)._ [FOOTNOTE:6][ENDFOOTNOTE] What is novel here? That continuum diffusion emerges on macroscopic scales has been well known for centuries. The novelty is the centre manifold theory framework I set up to prove this well known fact. This framework illuminates issues and empowers us to analyse much more difficult nonlinear dynamics, Section 4.3, and potentially stochastic problems. Furthermore, the framework shows that a consistent truncation in the interelement coupling parameter $\bar{\gamma}$ generates a macroscopic approximation to continuum diffusion that is of the same order of error in $\bar{\gamma}$. (For example, in the Introduction we discussed multiscale modelling with truncations to $\Ord{\bar{\gamma}^{2}}$.) This consistency follows because truncating the mapping operator ${\cal T}$, given in (32), simply truncates its spectrum _and_ truncates its eigenvectors (as ${\cal T}$ is triangular). ### Renormalise advection-dispersion Now consider advection-dispersion on a multiscale hierarchy. Although the dynamics are linear, and in contrast to the previous subsection, the transformation from one level to another in the hierarchy is nonlinear. For example, suppose the microscopic dynamics is simply the discrete $\dot{u}_{j}=[-\mu\delta+\delta^{2}]u_{j}$ . Then computer algebra [32] derives that the multigrid hierarchy of dynamic models is \[\frac{du_{j}^{(0)}}{dt^{(0)}} = [-\mu\delta+\delta^{2}]u_{j}^{(0)}\,,\] (34) \[\frac{du_{j}^{(1)}}{dt^{(1)}} = \rat 12\big{[}-\mu\delta+\rat 5{8}\delta^{2}+\gamma(-\rat 1{8} \delta^{2}+\rat 5{32}\mu\delta^{3}-\rat{41}{512}\delta^{4})\big{]}u_{j}^{(1)}+ \Ord{\gamma^{2}}\,,\] (35) \[\frac{du_{j}^{(2)}}{dt^{(2)}} =\] (36) \[{}+\Ord{\gamma^{2}}\,,\] \[\frac{du_{j}^{(3)}}{dt^{(3)}} = \rat 18\big{[}-\mu\delta+0.50015\delta^{2}\] \[{}+\gamma(-0.37515\delta^{2}+0.37527\mu\delta^{3}-0.18763\delta^{ 4})\big{]}u_{j}^{(3)}+\Ord{\gamma^{2}}\,,\] \[\vdots\] \[\frac{du_{j}^{(9)}}{dt^{(9)}} = \rat 1{2^{9}}\big{[}-\mu\delta+0.50000\delta^{2}\] (38) \[{}+\gamma(-0.49805\delta^{2}+0.49805\mu\delta^{3}-0.24902\delta^{ 4})\big{]}u_{j}^{(9)}+\Ord{\gamma^{2}}\,,\] where for simplicity I omit the level of the discrete mean and difference operators. Evidently, as the level $\ell$ increases, and upon renormalising the time scale by the factor of $2^{\ell}$ (the grid step), these models approach a fixed point corresponding to an upwind model of the advection. As is well known, advection dominates diffusion on large scales. This centre manifold supported multiscale transformation preserves the advection speed, and does so with stable upwind differencing. Now explore the general mapping of linear conservative dynamics from one level on the multigrid to another. Generalise the form (30) for isotropic dynamics to the general finite difference representation of conservative linear operators: \[\frac{du_{j}}{dt}=\sum_{p=1}^{\infty}\gamma^{p-1}\left\{\sum_{k=1}^{p}\left(c_ {p,2k-1}\mu{\delta}^{2k-1}+c_{p,2k}{\delta}^{2k}\right)\right\}u_{j}\,,\] (39) for some coefficients $c_{p,k}$ where $c_{1,1}=-c$ and $c_{1,2}=d$ as used earlier. The operator in braces $\{\,\}$, called ${\cal L}_{p}$ earlier, represents a general operator of stencil width $2p+1$. For example, working to error $\Ord{\gamma^{3},\bar{\gamma}^{2}}$, computer algebra [32] derives the model at the next level of the the multiscale hierarchy to be, in gory detail, \[\frac{d\bar{u}_{j}}{d\bar{t}} = \left\{\rat 12\big{[}c_{1,1}+c_{2,1}\big{]}\bar{\mu}\bar{\delta}+ \left[\frac{1}{4}(c_{1,2}+c_{2,2})\right.\right.\] (40) \[\left.\left.\quad{}+\frac{c_{1,1}}{16c_{1,2}}(c_{1,1}+2c_{2,1}-4c _{2,3})+\frac{c_{1,1}^{2}}{16c_{1,2}^{2}}(-c_{2,2}+4c_{2,4})\right]\bar{\delta }^{2}\right\}\bar{u}_{j}\] \[{}+\bar{\gamma}\left\{\left[\frac{c_{1,1}}{16c_{1,2}}(-c_{1,1}-2c _{2,1}+4c_{2,3})+\frac{c_{1,1}^{2}}{16c_{1,2}^{2}}(c_{2,2}-4c_{2,4})\right] \bar{\delta}^{2}\right.\] \[\left.\quad{}+\left[\frac{1}{16}(-c_{1,1}-c_{2,1}-2c_{2,3})+\frac {c_{1,1}^{2}}{64c_{1,2}^{2}}(-c_{1,1}-3c_{2,1}+3c_{2,3})\right.\right.\] \[\left.\left.\qquad{}+\frac{c_{1,1}^{3}}{32c_{1,2}^{3}}(c_{2,2}-4c _{2,4})\right]\bar{\mu}\bar{\delta}^{3}\right.\] \[\left.\quad{}+\left[\frac{1}{64}(-c_{1,2}-c_{2,2}-4c_{2,4})+\frac {3c_{1,1}}{128c_{1,2}}(-c_{1,1}-c_{2,1}+c_{2,3})\right.\right.\] \[\left.\left.\qquad{}+\frac{3c_{1,1}^{4}}{1024c_{1,2}^{4}}(c_{2,2} -4c_{2,4})\right]\bar{\delta}^{4}\right\}\bar{u}_{j}\] \[{}+\Ord{\bar{\gamma}^{2}}\,.\] This general mapping from (39) to (40) governs the particular exapmple hierarchy of models (34)–(38). Fine grid scale interactions generate the nonlinear dependence upon coefficients shown in the transformation to (40). The example hierarchy (34)–(38) shows that when we scale time by a further factor of two in each transformation, to correspond to the time scale of advection of a grid of twice the spacing, then the multiscale transformation possess a fixed point. Returning to the general transformation from (39) to (40), rescaling time by a factor of two, computer algebra finds precisely two non-trivial fixed points of the multiscale transformation: \[\frac{du_{j}^{()}}{dt^{()}} = c_{}\left\{\mp\mu\delta+\rat 12\delta^{2}+\gamma\big{[}-\rat 12 \delta^{2}\pm\rat 12\mu\delta^{3}-\rat 14\delta^{4}\big{]}\right\}u_{j}^{()}+ \Ord{\gamma^{2}}\,,\] for some speed $c_{}$ (positive) which will depend upon the precise microscopic system. These fixed points are purely upwind macroscale models of the advection and dispersion dynamics. Such stable upwind models naturally emerge from our rational transformation of dynamics based upon dynamical systems theory. ### Approximate the nonlinear Burgers’ dynamics Burgers’ partial differential equation, $\partial u/\partial t+u\,\partial u/\partial x=\partial^{2}u/\partial x^{2}$ , is frequently invoked as a benchmark problem in nonlinear spatio-temporal dynamics as it involves the important physical mechanisms of dissipative diffusion and nonlinear advection. As an example of a nonlinear application of our multiscale methodology, suppose Burgers’ is spatially discretised to \[\frac{du_{j}^{(0)}}{dt^{(0)}}=\delta^{2}u_{j}^{(0)}-\alpha u_{j}^{(0)}\mu \delta u_{j}^{(0)}\,,\] (41) where the time scale $t^{(0)}$ is chosen to make the coefficient unity for the centred difference approximation $\delta^{2}u_{j}$ of the diffusion $\partial^{2}u/\partial x^{2}$. Take equation (41) to be the microscale discrete nonlinear dynamics. The parameter $\alpha$ measures the importance of the nonlinear advection on this microscopic scale. Section 2 places the coarse grid modelling of such nonlinear discrete dynamics within the purview of centre manifold theory. For relatively small parameter $\alpha$, straightforward modifications of the computer algebra for the earlier linear dynamics [32] analyses nonlinear problems. The reason is that as long as the nonlinearity is relatively weak, small $\alpha$, the dominant mechanism in each element is the linear dissipation of $\delta^{2}$ just as for the linear dynamics. Our multiscale modelling transforms the fine grid dynamics (41) into the level one dynamics (42); applying the multiscale modelling again transforms the level one dynamics (42) into the level two dynamics (43). \[\frac{du_{j}^{(1)}}{dt^{(1)}} = \left[\frac{1}{4}+\frac{1}{16}\alpha^{2}{u_{j}^{(1)}}^{2}\right] \delta^{2}u_{j}^{(1)}-\frac{\gamma}{64}\delta^{4}u_{j}^{(1)}-\frac{\alpha}{2}u _{j}^{(1)}\delta^{2}u_{j}^{(1)}\] (42) \[{}+\frac{\alpha\gamma}{64}\left[4u_{j}^{(1)}\mu\delta^{3}u_{j}^{( 1)}+(\delta^{2}u_{j}^{(1)})\mu\delta^{3}u_{j}^{(1)}+(\delta^{4}u_{j}^{(1)})\mu \delta u_{j}^{(1)}\right]\] \[{}+\Ord{\gamma^{2}+\alpha^{3}}\,,\] \[\frac{du_{j}^{(2)}}{dt^{(2)}} = \left[\frac{1}{16}+\frac{5}{64}\alpha^{2}{u_{j}^{(2)}}^{2}\right] \delta^{2}u_{j}^{(2)}-\frac{5\gamma}{1024}\delta^{4}u_{j}^{(2)}-\frac{\alpha}{ 4}u_{j}^{(2)}\delta^{2}u_{j}^{(2)}\] (43) \[{}+\frac{5\alpha\gamma}{512}\left[4u_{j}^{(2)}\mu\delta^{3}u_{j}^ {(2)}+(\delta^{2}u_{j}^{(2)})\mu\delta^{3}u_{j}^{(2)}+(\delta^{4}u_{j}^{(2)}) \mu\delta u_{j}^{(2)}\right]\] \[{}+\Ord{\gamma^{2}+\alpha^{3}}\,.\] Evidently we could continue this transformation across many more levels of a multigrid hierarchy. The transformation from (41) to (42), to (43) is based upon small nonlinearity $\alpha$. However, as we should expect, the nonlinear advection appears to become more important at larger scales: the relative magnitude of the nonlinear enhancement to dissipation, $\alpha^{2}\big{(}u_{j}^{(\ell)}\big{)}^{2}$, increases when going from level $0$ to level $2$. After transforming over enough levels, the nonlinearity will begin to dominate the linear basis of the analysis here; at that length scale I expect the discrete dynamics to morph into a qualitatively new form, one dominated by nonlinear advection. Although these emergent dynamics cannot be captured by the transformation used here, a generalisation of the algebra to being based about a nonlinear subspace of piecewise constant solutions may be feasible. In that case, the centre manifold theory of Section 2 would still support the multiscale modelling of the strongly nonlinear dynamics. ## 5 Conclusion This article introduces a new dynamical systems approach to modelling and linking dynamics across a multigrid hierarchy. Because we recover continuum diffusion, §4.1, and upwind advection, §4.2, on macroscales we are reassured that the process of modelling from one grid to the next coarser grid is indeed sound, as claimed by centre manifold theory, §2. Further, errors do not appear to accumulate when we iterate the modelling transformation across many changes in length scales. At all lengths scales in the hierarchy of models, centre manifold theory assures us that the model on each scale is exponentially attractive, §3.3, and provides an estimate of the rate of attraction. This theoretical support applies for the finite spectral gaps on the multigrid hierarchy. The geometric picture of invariant slow manifolds also provides a rationale for providing initial conditions for the models at each length scale [24, 8, 27, e.g.]. Section 3.4 connects appropriate initial conditions with the restriction projection of multigrid solution of linear equations [5, e.g.]. In addition to providing the dynamics at all length scales in the hierarchy, this approach also provides intraelement structures realised by the dynamics of the grid values: in terms of the level $(\ell+1)$ dynamic variables, equation (19) with (17) describe the corresponding structures on the level $\ell$ grid. In some sense, equation (17) provide ‘wavelets’ for each grid scale [9, 16, e.g.]. Our modelling connects the dynamics of wavelets across a hierarchy of length scales. All the analysis herein is for dynamics in one space dimension. Just as for holistic discretisation of s [19], I expect extension to higher space dimensions will be straightforward. This article focussed on dynamics which to a first approximation could be modelled by advection-dispersion equations; extension to dynamics of necessarily higher-order, such as a discrete Kuramoto-Sivashinsky equation, could also be analogous to the approach of holsitic discretisation [17, 18]. Similarly, extension to stochastic mutliscale dynamics could be analogous to that of holistic discretisation of stochastic dynamics [31, 33, e.g.]. ## References [1] Ludwig Arnold. _Random Dynamical Systems_. Springer Monographs in Mathematics. Springer, June 2003. * [2] P. Boxler. A stochastic version of the centre manifold theorem. _Probab. Th. Rel. Fields_, 83:509–545, 1989. * [3] Achi Brandt. Multiscale scientific computation: review 2001. In T. F. Chan T. J. Barth and R. Haimes, editors, _Multiscale and Multiresolution Methods: Theory and Applications_, pages 1–96. Springer–Verlag, Heidelberg, 2001. * [4] Achi Brandt. Methods of systematic upscaling. Technical report, Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, March 2006. * [5] William L. Briggs, Van Emden Henson, and Steve F. McCormick. _A multigrid tutorial, second edition_. SIAM, 2nd edition, 2001. * [6] J. Carr. _Applications of centre manifold theory_, volume 35 of _Applied Math. Sci._ Springer–Verlag, 1981. * [7] Xu Chao and A. J. Roberts. On the low-dimensional modelling of Stratonovich stochastic differential equations. _Physica A_, 225:62–80, 1996. 10.1016/0378-4371(95)00387-8. * [8] S. M. Cox and A. J. Roberts. Initial conditions for models of dynamical systems. _Physica D_, 85:126–141, 1995. 10.1016/0167-2789(94)00201-Z. * [9] Ingrid Daubechies. _Ten lectures on wavelets_. SIAM, 1992. * [10] J. Dolbow, M. A. Khaleel, and J. Mitchell. Multiscale mathematics initiative: A roadmap. Report from the 3rd DoE workshop on multiscale mathematics. 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Computer algebra models dynamics on a multigrid across multiple length and time scales. Technical report, University of Southern Queensland, http://eprints.usq.edu.au/3373/, November 2007. * [33] A. J. Roberts. Subgrid and interelement interactions affect discretisations of stochastically forced diffusion. In Wayne Read, Jay W. Larson, and A. J. Roberts, editors, _Proceedings of the 13th Biennial Computational Techniques and Applications Conference, CTAC-2006_, volume 48 of _ANZIAM J._, pages C168–C187. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/36, June 2007. * [34] A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. _Physica A_, 387:12–38, 2008. * [35] G. Samaey, I. G. Kevrekidis, and D. Roose. The gap-tooth scheme for homogenization problems. _SIAM Multiscale Modeling and Simulation_, 4:278–306, 2005. 10.1137/030602046. * [36] Ferdinand Verhulst. _Methods and applications of singular perturbations: boundary layers and multiple timescales_, volume 50 of _Texts in Applied Maths_. Springer, 2005.
0911.3721	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 59632, "num_imgs": 0, "llama3_tokens_count": 20963 }	[]	# Optimal Paths on the Space–Time SINR Random Graph ###### Abstract We analyze a class of Signal-to-Interference-and-Noise-Ratio (SINR) random graphs. These random graphs arise in the modeling packet transmissions in wireless networks. In contrast to previous studies on the SINR graphs, we consider both a space and a time dimension. The spatial aspect originates from the random locations of the network nodes in the Euclidean plane. The time aspect stems from the random transmission policy followed by each network node and from the time variations of the wireless channel characteristics. The combination of these random space and time aspects leads to fluctuations of the SINR experienced by the wireless channels, which in turn determine the progression of packets in space and time in such a network. This paper studies optimal paths in such wireless networks in terms of first passage percolation on this random graph. We establish both “positive” and “negative” results on the associated time constant. The latter determines the asymptotics of the minimum delay required by a packet to progress from a source node to a destination node when the Euclidean distance between the two tends to infinity. The main negative result states that this time constant is infinite on the random graph associated with a Poisson point process under natural assumptions on the wireless channels. The main positive result states that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite. Poisson point process, random graph, first passage percolation, shot-noise process, SINR F. Baccelli, B. Blaszczyszyn, O. Mirsadeghi [INRIA/ENS]François Baccelli ENS DI TREC, 45 rue d’ulm, 75230 Paris, FRANCE. [INRIA/ENS and Math. Inst. University of Wrocław]BartŁomiej BŁaszczyszyn ENS DI TREC, 45 rue d’ulm, 75230 Paris, FRANCE. [INRIA/ENS and Sharif University of Technology]Mir Omid Haji Mirsadeghi ENS DI TREC, 45 rue d’ulm, 75230 Paris, FRANCE. The work of this author is part of a joint PhD programme with a co-advising by Prof. Amir Daneshgar in Tehran 60D05, 05C8090C27, 60G55 ## 1 Introduction There is a rich literature on random graphs generated over a random point process. These graphs are often motivated by physical, biological or social networks. Many interesting large scale properties of these networks related to connectivity have been studied in terms of the percolation of the associated graphs. An early example of such a study can be found in [13] where the connectivity of large networks was defined as the supercritical phase in what is today called the continuum (Boolean) percolation model. More recently, a random SINR graph model for wireless networks was studied with the same perspective in [10; 11]. The routing, and more precisely, the speed of delivery of information in networks is another example of problems, which motivated the study of the random graphs. The main object in this context is the evaluation of the so called _time constant_, which gives the asymptotic behavior of the number of edges (hops) in the paths (optimal or produced by some particular routing protocol) joining two given nodes in function of the (Euclidean) distance between these nodes, when this distance tends to infinity. In the case of a shortest (in terms of the number of hops) path, this problem is usually called the _first passage percolation problem_ and was originally stated by Broadbent and Hammersley in [8] to study the spread of fluid in a porous medium. More recently, in [20; 7], such time constants were studied on so called _small word graphs_, motivated by routing in certain social networks, where any two given nodes are joined by an edge independently with a probability that decays as some power function with the Euclidean distance between them. The complete graph on a Poisson p.p. with “nearest neighbor” routing policy was studied in this context in [6]. The first passage percolation problem on the Poisson-Delaunay graph was considered in [24; 21]. In the case of graphs whose edges are marked by some weights, one can extend the notion of time constant by studying the sum of the edge weights. First passage percolation on the complete Poisson p.p. graph, with weights proportional to some power of the distance between the nodes was studied in [14]. The present paper focuses on the speed of delivery of information in SINR graphs. In contrast to previous studies of this subject, in particular to [10; 11], we consider graphs with _space-time_ vertexes. This new model is motivated by multihop routing protocols used in wireless ad-hoc networks. In this framework, the random point process on the plane describes the locations of the users of an ad-hoc network and the discrete time dimension corresponds to successive time slots in which these nodes exchange information (here packets). As in [2] we assume the spatial Aloha policy to decide which node transmit at a given time slot. We also assume some space-time fading model (already used e.g. in [5]) to describe the variability of the wireless channel conditions (see e.g. [23]). In this _space-time SINR graph_, a directed edge represents the feasibility of the wireless transmission between two given network nodes at a given time. More precisely, the direct transmission of a packet is succeeds between two nodes in a given time slot if the ratio of the power of the signal between these nodes to the interference and noise at the receiver is larger than a threshold at this time slot. This definition has an information theoretic basis (see e.g. [23]). It is rigorously defined below using some power path-loss model and an associated shot-noise model representing the interference. We study various problems on this random graph including the law of its in- and out-degree, the number of paths originating from (or terminating at) a typical node or its connectedness. The most important results bear on the first passage percolation problem in this graph. In the case of Poisson p.p. for the node locations, we show that the time constant is infinite. We then show that when adding a periodic node infrastructure of arbitrarily small intensity to the Poisson point process, the time constant is positive and finite. These results lead to bounds on the delays in ad-hoc networks which hold for all routing algorithms. This subject, or more generally, the question of the speed of the delivery of information in large wireless ad-hoc networks currently receives a lot of attention in the engineering literature see e.g. [12; 15]. The paper is organized as follows. In Section 2 we introduce the space-time SINR graph model. The results are presented in Section 3, Most of the proofs are deferred to Section 4. Some implications on routing in ad-hoc networks are presented in Section 5. ## 2 The Model ### Probabilistic Assumptions Throughout the paper we consider a simple, stationary, independently marked (i.m.) point process (p.p.) $\widetilde{\Phi}=\{(X_{i},\mathbf{e}_{i},\mathbf{F}_{i},\mathbf{W}_{i})\}$ with finite, positive intensity $\lambda$ on $\mathbb{R}^{2}$. In this model, * $\Phi=\{X_{i}\}$ denotes the locations of the network nodes on the plane $\mathbb{R}^{2}$. The following three cases regarding the distribution of $\Phi$ will be considered: * $\Phi$ is a general (stationary, non-null, with finite intensity) p.p., * $\Phi$ is a Poisson p.p., * $\Phi=\Phi_{M}+\Phi_{G}$ is the superposition of two independent p.p.s; where $\Phi_{M}$ denotes a stationary Poisson p.p. with finite, non-null intensity $\lambda_{M}$ and $\Phi_{G}=s\mathbb{Z}^{2}+U_{G}$ a stationary, periodic p.p., whose nodes constitute a square grid with edge length $s$, randomly shifted by the vector $U_{G}$ that is uniformly distributed in $[0,s]^{2}$ (this makes $\Phi_{G}$ stationary). Note that the intensity of $\Phi_{G}$ is $\lambda_{G}=1/s^{2}$. * $\mathbf{e}_{i}=\{e_{i}(n)\}_{\in n\in\mathbb{Z}}$, where $\mathbb{Z}=\{\ldots,-1,0,1,\ldots\}$ denotes integers; the variables $\{e_{i}(n):i,n\}$ are i.i.d. (in $n$ and $i$) Bernoulli random variables (r.v.s) with ${\mathbf{P}}\{\,e=1\,\}=1-{\mathbf{P}}\{\,e=0\,\}=p$, where $e$ denotes the generic r.v. for this family. We always assume $0<p<1$. The variable $e_{i}(n)$ represents the _medium access indicator_ of node $X_{i}$ at time $n$; it says whether the node transmits (case $e=1$) or not at time $n$. * $\mathbf{W}_{i}=\{W_{i}(n)\}_{n}$; $\{W_{i}(n):i,n\}$ is a family of non-negative i.i.d. r.v.s with some arbitrary distribution. The variable $W_{i}(n)$ represents the power of the _thermal noise_ at node $X_{i}$ and at time $n$. Let $W$ denote the generic r.v. for this family. * $\mathbf{F}_{i}=\{F_{i,j}(n)\}_{j,n}$; $\{F_{i,j}(n):i,j,n\}$ is a family of non-negative i.i.d. r.v.s. The variable $F_{i,j}(n)$ represents the quality of the radio channel (also called _fading_) from node $X_{i}\in\Phi$ to node $X_{j}\in\Phi$ at time $n$. The following two cases regarding the distribution of $F$ (denoting the generic random variable for this family) will be considered: * when $F$ has some arbitrary distribution with finite mean. * when $F$ has _exponential_ distribution with mean $1/\mu$. (${}^{}$)¹ [FOOTNOTE:1][ENDFOOTNOTE] To complete the probabilistic description of the model we assume that, given $\Phi$, the random elements $\{\mathbf{e}_{i}\}_{i},\{\mathbf{W}_{i}\}_{i}$ and $\{\mathbf{F}_{i}\}_{i}$ are independent. For more on this framework, which is classical, see e.g. or [3; 4; 5]. Our stationary i.m. p.p. $\tilde{\Phi}$ is considered on some probability space with probability ${\mathbf{P}}$. We will denote by ${\mathbf{P}}^{0}$ the Palm probability with respect to $\Phi$; see (9, Ch.13). Recall that it can be interpreted as the conditional probability given $\Phi$ has a point at the origin $0$ of the plane. We will denote this point (considered under ${\mathbf{P}}^{0}$) by $X_{0}$ and call it the _typical node_. Under ${\mathbf{P}}^{0}$$\tilde{\Phi}$ is also an i.m. p.p. with marks distributed as in the original law. Moreover, in the case of Poisson p.p.s, the distribution of $\Phi$ under ${\mathbf{P}}^{0}$ is equal to the distribution of $\Phi\cup\{X_{0}=0\}$ under the stationary probability ${\mathbf{P}}$ (cf. the Slivnyak-Mecke Theorem (9, p.281)). ### SINR Marks Given the i.m.p.p. $\tilde{\Phi}$ described above, we construct another family of random variables $\{\text{SINR}_{ij}(n):i,j,n\}$, which will be interpreted as the SINR observed in the channel form $X_{i}\in\Phi$ to $X_{j}\in\Phi$ at time $n$. These variables, which have an information theoretic background, will be used to assess the success of transmissions. For defining these variables, we give ourselves some non-decreasing function $l:\mathbb{R}^{+}=\{t:t\geq 0\}\to\mathbb{R}^{+}$ that we call the _path-loss function_. A special example considered in this paper (and commonly accepted in the wireless communication context) is \[l(r)=(Ar)^{\beta}\qquad\text{with some $A>0$ and $\beta>2$}\,.\] (1) Denote by $\Phi^{1}(n)=\{X_{i}:e_{i}(n)=1\}$ the point process of _transmitters_ at time slot $n$ and by $\Phi^{0}(n)=\{X_{i}:e_{i}(n)=0\}$ that of (potential) _receivers_. For a given pair receiver $X_{j}\in\Phi^{0}$ and transmitter $X_{i}\in\Phi^{1}(n)$, we will assume that $X_{j}$ receives a signal from $X_{i}$ with power $F_{i,j}(n)/l(\|{X_{j}-X_{i}}\|)$ at time $n$. Node $X_{j}$ also receives signals from _other_ transmitters $X_{k}\in\Phi^{1}(n)$, $X_{k}\not=X_{i}$ at time $n$. The total received power is equal to \[I_{i,j}(n)=\sum_{X_{k}\in\Phi^{1}(n)\setminus\{X_{i}\}}F_{k,j}(n)/l(\|{X_{k}-X_ {j}}\|)\,.\] Let also \[I_{j}(n)=\sum_{X_{k}\in\Phi^{1}(n)\setminus\{X_{j}\}}F_{k,j}(n)/l(\|{X_{k}-X_{j }}\|)\,.\] Both $I_{i,j}(n)$ and $I_{j}(n)$ are _shot-noise_ r.v.s generated by $\Phi^{1}(n)$, the fading marks and the path-loss function. The are infinite sums of non-negative r.v.s. In order to check whether these r.v.s are a.s. finite, one can use the Campbell-Little-Mecke formula (Campbell for short; cf. (9, Prop. 13.3.II)), which implies that \[\mathbf{E}^{0}\Bigl{[}\sum_{X_{k}\in\Phi^{1}(n),\|{X_{k}}\|>\epsilon}F_{k,0}(n)/ l(\|{X_{k}}\|)\Bigr{]}=p\mathbf{E}[F]\int_{\mathbb{R}^{2}\setminus[0,\epsilon]^{ 2}}1/l(\|{x}\|)\breve{M}_{[2]}(\text{d}x)\,,\] (2) where $\breve{M}_{2}(\cdot)$ is the _reduced second order moment measure_ of $\Phi$ (cf (9, p. 238)). In what follows, we will always tacitly assume that $l(\cdot),\Phi$ are such that the integral in the right-hand side of (2) is finite for some $\epsilon\geq 0$, which implies that $I_{0}(n)$ is almost surely (a.s.) finite under ${\mathbf{P}}^{0}$ for all $n$ as well as all $I_{j}(n),I_{i,j}(n)$ under ${\mathbf{P}}$. If $\Phi$ is the homogeneous Poisson p.p., we have $\breve{M}_{[2]}(\text{d}x)=\lambda\,\text{d}x$ and it is easy see that the we have finiteness for $l(\cdot)$ given by (1) for all $\epsilon>0$. It is also relatively easy to see that it holds for the Poisson$+$Grid p.p. $\Phi=\Phi_{M}+\Phi_{G}$. The _SINR at the receiver $X_{j}\in\Phi^{0}(n)$ with respect to transmitter $X_{i}\in\Phi^{1}(n)$, at time $n$_ is defined as \[\text{SINR}_{i,j}(n)=\frac{F_{i,j}(n)/l(\|{X_{i}-X_{j}}\|)}{W_{j}(n)+I_{i,j}(n)}\,.\] (3) ### Space-Time SINR Graph Let \[\delta_{i,j}(n)=\begin{cases}\mathds{1}(\text{SINR}_{i,j}\geq T)&\text{if}\ e_ {i}(n)=1,e_{j}(n)=0,i\not=j,\\ 1&\text{if}\ i=j,\\ 0&\text{otherwise,}\end{cases}\] (4) where $T>0$ is a threshold assumed to be some given constant throughout the paper. We define the space-time SINR graph ${\mathbb{G}}$ as the _directed graph_ with the set of vertexes $\Phi\times\mathbb{Z}$ and a directed edge from $(X_{i},n)$ to $(X_{j},n+1)$ if $\delta_{i,j}(n)=1$. Let us stress an important convention in our terminology. By network node, or point, we understand a point of $\Phi$. A (graph) vertex is an element of $\Phi\times\mathbb{Z}$; i.e. it represents some network node at some time. The existence of a graph edge is to be interpreted as the possibility of a successful communication between two network nodes (those involved in the edge) at time $n$. This can be rephrased as follows. Suppose that at time $n$ the network node $X_{i}$ has a packet (containing some information). Then the set of graph neighbors of the vertex $(X_{i},n)$ describes all the nodes that can decode this packet at time $n+1$. Thus any path on the graph ${\mathbb{G}}$ represents some possible route of the packet in space and time. ## 3 Results In this section we present our results on ${\mathbb{G}}$. ### Existence of Paths All the results of this section are obtained under the general p.p. and fading assumptions of Section 2, under the assumption that the finiteness of the expression in (2) is granted. Note first that ${\mathbb{G}}$ has no isolated nodes in the usual sense. Indeed, we have always $(X_{i},n)$ connected to $(X_{i},n+1)$. We will consider directed paths on ${\mathbb{G}}$ and call them paths for short. Note that these paths are self-avoiding due to the fact that there are no loops in the time dimension. Denote by ${\cal H}_{i}^{out,k}(n)$ the number of paths of length $k$ (i.e. with $k$ edges) _originating_ from $(X_{i},n)$. Similarly, denote by ${\cal H}_{i}^{in,k}(n)$ the number of such path _terminating_ at $(X_{i},n)$. In particular ${\cal H}_{i}^{out}(n)={\cal H}_{i}^{out,1}(n)$ and ${\cal H}_{i}^{in}(n)={\cal H}_{i}^{in,1}(n)$ are respectively, the out- and in-degree of the node $(X_{i},n)$. For a general p.p. $\Phi$ and a general fading model, the in-degree ${\cal H}_{i}^{in}$ of any node of ${\mathbb{G}}$ is bounded from above by the constant $\xi=1/T+2$. Proof.: Assume there is an edge to node $(X_{j},n)$ from nodes $(X_{i_{1}},n-1),\ldots,(X_{i_{k}},n-1)$, for some $k>1$ and $i_{p}\not=j$ ($p=1,\ldots,k$). Then for all such $p$ \[\frac{F_{{i_{p}},j}}{l(\|X_{i_{p}}-X_{j}\|)}\geq\frac{T}{1+T}\left(\sum_{q=1}^{k }\frac{F_{{i_{q}},j}}{l(\|X_{i_{q}}-X_{j}\|)}\right)\,.\] When summing up all these inequalities, one gets that $Tk\leq 1+T$, that is $k\leq 1/T+1$. Considering the edge from $(X_{i},n-1)$ to $(X_{i},n)$, the in-degree of any node is bounded from above by $\xi=1/T+2$. ∎∎ Let \[h^{out,k}=\mathbf{E}^{0}[{\cal H}_{0}^{out,k}(n)]=\mathbf{E}^{0}[{\cal H}_{0}^ {out,k}(0)]\] and \[h^{in,k}=\mathbf{E}^{0}[{\cal H}_{0}^{in,k}(n)]=\mathbf{E}^{0}[{\cal H}_{0}^{ in,k}(0)]\] be the expected numbers of paths of length $k$ originating or terminating at the typical node, respectively. In particular $h^{out}=h^{out,1}$ and $h^{in}=h^{in,1}$ are the mean out- and in-degree of the typical node, respectively. For a general p.p. $\Phi$ and a general fading model \[h^{in,k}=h^{out,k}\,.\] (5) Proof.: We use the mass transport principle to get that $\mathbf{E}^{0}[{\cal H}_{0}^{out,k}(0)]=\mathbf{E}^{0}[{\cal H}_{0}^{in,k}(0)]$, which implies the desired result. Indeed, Campbell’s formula and stationarity give \[\lambda h^{out,k} = \lambda\int_{[0,1)^{2}}\mathbf{E}^{0}[{\cal H}_{0}^{out,k}(0)]\, \text{d}x\] \[= \mathbf{E}\Bigl{[}\sum_{{X_{i}}\in\Phi\cap[0,1)^{2}}{\cal H}_{i}^ {out,k}(0)\Bigr{]}\] \[= \sum_{v\in\mathbb{Z}}\mathbf{E}\Bigl{[}\sum_{X_{i}\in[0,1)^{2}} \sum_{X_{j}\in[0,1)^{2}+v}\#\,\text{of paths from $(X_{i},0)$ to $(X_{j},k)$} \Bigr{]}\] \[= \sum_{v\in\mathbb{Z}}\mathbf{E}\Bigl{[}\sum_{X_{i}\in[0,1)^{2}-v} \sum_{X_{j}\in[0,1)^{2}}\#\,\text{of paths from $(X_{i},0)$ to $(X_{j},k)$} \Bigr{]}\] \[= \lambda\int_{[0,1)^{2}}\mathbf{E}^{0}[{\cal H}_{0}^{in,k}(k)]\, \text{d}x=\lambda h^{in,k}\,,\] where $\#$ denotes the cardinality. This completes the proof. ∎∎ Here are immediate consequences of the two above lemmas. Under the assumptions of Lemma 3.1: ${\mathbb{G}}$ is locally finite (both on in- and out-degrees of all nodes are ${\mathbf{P}}$-a.s. finite). * ${\cal H}_{i}^{in,k}(n)\leq\xi^{k}$${\mathbf{P}}$-a.s for all $i,n,k$. * $h^{in,k}=h^{out,k}\leq\xi^{k}$ for all $k$. For all $X_{i},X_{j}\in\Phi$ and $n\in\mathbb{Z}$, we denote by we will call _local delay_ from $X_{i}$ to $X_{j}$ at time $n$ the quantity \[L_{i,j}(n)=\inf\{k\geq n:\delta_{i,j}(k)=1\}\] with the usual convention that $\inf\emptyset=\infty$. Note that $L_{i,j}(n)$ is the length (number of edges) of the shortest path (with the smallest number of edges) from $(X_{i},n)$ to $\{X_{j}\}\times\mathbb{Z}$ among the paths contained in the subgraph ${\mathbb{G}}\cap\{X_{i},X_{j}\}\times\mathbb{Z}$ of ${\mathbb{G}}$, which is of the form \[((X_{i},n),(X_{i},n+1)),\ldots,((X_{i},n+L_{i,j}(n)-1),(X_{i},n+L _{i,j}(n))),\] \[((X_{i},n+L_{i,j}(n)),(X_{j},n+L_{i,j}(n)+1)).\] Our next result gives a condition for the local delays to be a.s. finite. Assume a general p.p. $\Phi$ and a general fading model with $F$ having unbounded support (${\mathbf{P}}\{\,F>s\,\}>0$ for all $0<s<\infty$). Then, given $\Phi$, all local delays $L_{i,j}(n)$ are ${\mathbf{P}}$-a.s. finite geometric random variables. Proof.: Due to our assumption on the independence of marks in successive time slots, given $\Phi$, the variables $\{\delta_{i,j}(n):n\in\mathbb{Z}\}$ are (i.i.d.) Bernoulli r.v. and thus $L_{i,j}(n)$ is geometric r.v. It remains to show that ${\mathbf{P}}\{\,\delta_{i,j}(0)=1\,\|\,\Phi\,\}:=\pi_{i,j}(\Phi)>0$ for ${\mathbf{P}}$-almost all $\Phi$. For this, note that \[\pi_{i,j}(\Phi)=p(1-p){\mathbf{P}}\Bigl{\{}\,F_{i,j}(0)\geq l(\|{X_{j}-X_{i}}\|) \Bigl{(}W_{j}(0)+I_{i,j}(0)\Bigr{)}\,\Bigr{\}}\,.\] Under our general assumptions (including finiteness of the expression in (2)) $I_{i,j}(0)$ is a finite random variable ${\mathbf{P}}$-a.s. The result follows from the assumption that $0<p<1$ and the fact that $F_{i,j}(0)$ is independent of $I_{i,j}(0),W_{i,j}(0)$ and has infinite support. ∎∎ The next result directly follows from Lemma 3.1. Under the assumptions of Lemma 3.1, ${\mathbb{G}}$ is ${\mathbf{P}}$-a.s. _connected_ in the following _weak sense_: for all $X_{i},X_{j}\in\Phi$ and all $n\in\mathbb{Z}$, there exists a path from $(X_{i},n)$ to the set $\{(X_{j},n+l):l\in\mathbb{N}\}$, where $\mathbb{N}=\{1,2,\ldots\}$. We denote by $L_{i}(n)=\inf_{j\not=i}L_{i,j}$ the length of a shortest directed path from $(X_{i},n)$ to $(\{\Phi\setminus X_{i}\})\times\mathbb{Z}$. We will call $L_{i}(n)$ the _exit delay_ from $X_{i}$ at time $n$. Finally, we denote by $P_{i,j}(n)$ the length of a shortest path of ${\mathbb{G}}$ from $(X_{i},n)$ to $\{X_{j}\}\times\mathbb{Z}$. We call $P_{i,j}(n)$ the _delay_ from $X_{i}$ to $X_{j}$ at time $n$. Obviously for $i\not=j$ we have \[L_{i}(n)\leq P_{i,j}(n)\leq L_{i,j}(n)\] (6) and thus it follows immediately from Lemma 3.1 that all the three collections of delays finite r.v.s ${\mathbf{P}}$-a.s. ### Optimal Paths — Poisson p.p. Case We have seen in the previous section that under very general assumptions, all the delays are ${\mathbf{P}}$-a.s. finite random variables. In this section we show that under some natural assumptions (such as Poisson p.p. and exponential fading), the averaging over $\Phi$ may lead to _infinite_ mean values. This averaging is expressed in terms of the expectation for the typical node under the Palm probability. The proofs of the results stated in what follows are given in Section 4.1. Denote $\ell=\mathbf{E}^{0}[L_{0}(n)]=\mathbf{E}^{0}[L_{0}(0)]$. Assume $\Phi$ to be a Poisson p.p., $F$ to be exponential and the noise $W$ to be bounded away from $0$: ${\mathbf{P}}\{\,W>w\,\}=1$ for some $w>0$. Let the path-loss function be given by (1). Then ${\mathbf{P}}^{0}\{\,L_{0}(0)\geq q\,\}\geq 1/q$ for $q$ large enough. Under the assumptions of Proposition 3.2, we have: * The mean exit delay from the typical node is infinite; $\ell=\infty$. * In any given subset of plane with positive Lebesgue measure, at a given time, the expected number of points of $\Phi$ which have exit delays larger than $q$ decreases not faster than $1/q$ asymptotically for large $q$. The fact that the mean exit delay from the typical point is infinite ($\ell=\infty$) seems to be a consequence of the potential existence of arbitrarily large “voids” (disks without points of $\Phi$) around this point. Indeed, when conditioning on the existence of another point in the configuration $\Phi$, one obtains finite mean local delays. This will be shown in Proposition 3.2 below. Before stating it we need to formalise the notion of existence of _two given points $X,Y\in\mathbb{R}^{2}$ of $\Phi$_. For this, we consider $\Phi$ under the two-fold Palm probability ${\mathbf{P}}^{X,Y}$. Since our results on the matter bear only on the Poisson p.p. case, we can assume (by Slivnyak’s Theorem) the following version of the Palm probability of the Poisson p.p. $\Phi$: \[{\mathbf{P}}^{X,Y}\{\,\Phi\in\cdot\,\}={\mathbf{P}}\Bigl{\{}\,\Phi\cup\{X,Y\} \in\cdot\,\Bigl{\}}\,.\] (7) Moreover, under ${\mathbf{P}}^{X,Y}$, the marked Poisson p.p. $\tilde{\Phi}$ is obtained by an independent marking of the points of $\Phi\cup\{X,Y\}$ according to the original distribution of marks. Slightly abusing the notation, we denote by $L_{X,Y}(n)$ the local delay from $X$ to $Y$ at time $n$ when considered under ${\mathbf{P}}^{X,Y}$. Similar convention will be adopted in the notation of other types of delays under the Palm probabilities ${\mathbf{P}}^{X}$ or ${\mathbf{P}}^{X,Y}$. Assume $\Phi$ to be a Poisson p.p., $F$ to be exponential and the noise $W$ to have a general distribution. Then for all $X,Y\in\mathbb{R}^{2}$, the mean local delay from $X$ to $Y$ is finite _given the existence of these two points in $\Phi$_. More precisely, \[\mathbf{E}^{X,Y}[L_{X,Y}(0)]<\infty\,.\] (8) The next result follows immediately from (6). Under the assumptions of Proposition 3.2, \[\mathbf{E}^{X,Y}[L_{X}(0)]\leq\mathbf{E}^{X,Y}[P_{X,Y}(0)]<\mathbf{E}^{X,Y}[L_ {X,Y}(0)]<\infty\,.\] (9) The following result is our main “negative” result concerning ${\mathbb{G}}$ in the Poisson p.p. case: Under the assumptions of Proposition 3.2, we have \[\lim_{\|{X-Y}\|\to\infty}\frac{\mathbf{E}^{X,Y}[P_{X,Y}(0)]}{\|{X-Y}\|}=\infty\,.\] (10) In other words, the _expected shortest delay necessary to send a packet between two given points of the Poisson p.p. grows faster than the Euclidean distance between these two points_. ### Filling in Poisson Voids In this section we show that adding an independent periodic pattern of points to the Poisson p.p. allows one to get a linear scaling of the shortest path delay with Euclidean distance. In order to prove the _existence and finiteness_ of the associated time constant, we adopt a slightly different approach to the notion of paths on ${\mathbb{G}}$, which will allow us to exploit a subadditive ergodic theorem. The proofs of the results stated in what follows are given in Section 4.2. For $x\in\mathbb{R}^{2}$, let $X(x)$ be the point of $\Phi$ which is closest to $x$. The point $X(x)\in\Phi$ is a.s. well defined for all given $x\in\mathbb{R}^{2}$ since $\Phi$ is assumed simple and stationary p.p. For all $x,y\in\mathbb{R}^{2}$, define $P(x,y,n)=P_{X(x),X(y)}(n)$ to be the length of a shortest path of ${\mathbb{G}}$ from vertex $(X(s),n)$ to the set $\{(X(y),n+l),l\in\mathbb{N}\}$. We will call $P(x,y,n)$ the _delay_ from $x$ to $y$ at time $n$. For all triples of points $x,y,z\in\mathbb{R}^{2}$, we have \[P(x,z,n)\leq P(x,y,n)+P\Bigl{(}y,z,n+P(x,y,n)\Bigr{)}\,.\] (11) Let \[p(x,y,\Phi)=\mathbf{E}[P(x,y,0)\,\|\,\Phi]\,.\] (12) Using the strong Markov property, we get that, conditionally on $\Phi$, the law of $P(y,z,n+P(x,y,n))$ is the same as that of $P(y,z,n)$. Then, the last relation and (11) give \[p(x,z,\Phi)\leq p(x,y,\Phi)+p(y,z,\Phi)\,.\] (13) We are now in a position to use the subadditive ergodic theorem to show the existence of the time constant \[\kappa_{\mathrm{d}}=\lim_{t\to\infty}\frac{p(0,t\mathrm{d},\Phi)}{t}\,,\] where $\kappa_{\mathrm{d}}$ may depend on the _unit vector_$\mathrm{d}\in\mathbb{R}^{2}$ representing the direction in which the delay is measured. Here is the main result of this section. Consider the Poisson$+$Grid p.p. defined in Section 2.1 with exponential fading $F$ and with the path-loss function be given by (1). Then, for all unit vectors $\mathrm{d}\in\mathbb{R}^{2}$, the non-negative limit $\kappa_{\mathrm{d}}$ exists and is ${\mathbf{P}}$-a.s. _finite_. The convergence also holds in $L_{1}$. Notice that $\kappa_{\mathrm{d}}$ is not a constant. Indeed, the superposition of the p.p.s $\Phi=\Phi_{M}$ and $\Phi_{G}$ is ergodic but not mixing due to the fact that $\Phi$ is a (stationary) grid. For $\mathrm{d}$ parallel to say the horizontal axis of the grid $\Phi_{G}$, the limit $\kappa_{\mathrm{d}}$ will depend on the distance from the line $\{t\mathrm{d}:t\in\mathbb{R}\}$ to the nearest parallel (horizontal) line of the grid $\Phi_{G}$, i.e. on the shift $U_{G}$ of the grid. Here is a more precise formulation of the result. Under the assumptions of Proposition 3.3, the limit $\kappa_{\mathrm{d}}=\kappa_{\mathrm{d}}(U_{G})$ is measurable w.r.t. the shift $U_{G}$ of the grid p.p. $\Phi_{G}$ and _does not_ depend on the Poisson component $\Phi_{M}$ of the p.p. $\Phi$. Moreover, the set of vectors $\mathrm{d}$ in the unit sphere for which $\kappa_{\mathrm{d}}(U_{G})$ is not ${\mathbf{P}}$-a.s. a constant is at most countable. The last result on this case is: Under the assumptions of Proposition 3.3, suppose that $W$ is constant and strictly positive. Then $\mathbf{E}[\kappa_{\mathrm{d}}]>0$. Finally let us remark that the method used in this section cannot be used in the case of the Poisson p.p. (without the addition of the grid point process). The main problem is the lack of integrability of $p(x,y,\Phi)$ as stated in the following result. Note however, that this does _not_ imply immediately that $\kappa_{\mathrm{d}}=\infty$. Under the assumptions of Proposition 3.2$\mathbf{E}[p(x,y,\Phi)]=\infty$ for all $x$ and $y$ in $\mathbb{R}^{2}$. ## 4 Proofs Consider the shortest path from $(X_{i},n)$ to $(\Phi\setminus\{X_{i}\})\times\mathbb{Z}$. Let ${\cal T}_{i}(n)$ be the number of edges $(X_{i},k),(X_{i},k+1)$ in this path such that $e_{i}(k)=1$. These variables are the _number of trials_ before the first exit form $X_{i}$ at time $n$. Obviously \[{\cal T}_{i}(n)\leq L_{i}(n)\,.\] (14) We will also consider an auxiliary graph $\widehat{\mathbb{G}}$, called the _(space-time) Signal to Noise Ratio (SNR) graph_, defined exactly in the same manner as the SINR graph ${\mathbb{G}}$ except that the variables $\text{SINR}_{i,j}(n)$ defined in (15) are replaced by the variables \[\text{SNR}_{i,j}(n)=\frac{F_{i,j}(n)/l(\|{X_{i}-X_{j}}\|)}{W_{j}(n)}\,.\] (15) Note that this modification consists in suppressing the interference term $I_{i,j}(n)$ in the SINR condition in (4). The edges of ${\mathbb{G}}$ form a subset of the edges of $\widehat{\mathbb{G}}$ (both graph share the same vertexes), which will be denoted by \[{\mathbb{G}}\subset\widehat{\mathbb{G}}\,.\] (16) In what follows we will denote the delays, local delays, exit delays and numbers of trials related to $\widehat{\mathbb{G}}$ by $\widehat{P}_{i,j}(n),\widehat{L}_{i,j}(n),\widehat{L}_{i}(n)$ and $\widehat{\cal T}_{i}(n)$, respectively. The inclusion ${\mathbb{G}}\subset\widehat{\mathbb{G}}$ implies immediately that $\widehat{P}_{i,j}(n)\leq P_{i,j}(n)$ and the same inequalities hold for the three other families of variables mentioned above. ### Proofs of Results of Section 3.2 Proof.: (_of Proposition 3.2_) The inclusion (16) and the inequality (14) yield \[\widehat{\cal T}_{i}(n)\leq{\cal T}_{i}(n)\leq L_{i}(n)\,,\] which holds for all $i,n$. The results follow from the above inequalities and the next lemma. ∎∎ Under the assumptions of Proposition 3.2, ${\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)\geq q\,\}\geq 1/q$ for $q$ large enough. Proof.: Under ${\mathbf{P}}^{0}$, denote by $\tau_{k}$ the $k\,$th time slot in $\{0,1,\ldots\}$, such that $e_{0}(k)=1$. For all $q\geq 0$ we have \[{\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)>q\,\|\,\Phi\,\} = {\mathbf{P}}^{0}\Bigl{\{}\,\forall_{0\leq k\leq q}\forall_{0\not= X_{i}\in\Phi}\;\delta_{0,i}(\tau_{k})=0\,\Bigl{\|}\,\Phi\,\Bigr{\}}\] \[= {\mathbf{P}}^{0}\Bigl{\{}\,\forall_{0\leq k\leq q}\forall_{0\not= X_{i}\in\Phi}\;e_{i}(\tau_{k})=1\text{\;or\;}\text{SNR}_{0,i}(\tau_{k})<T\, \Big{\|}\,\Phi\,\Bigr{\}}\] and by the conditional independence of marks given $\Phi$ \[{\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)>q\,\|\,\Phi\,\} = \prod_{0\not=X_{i}\in\Phi}\Bigl{(}p+(1-p){\mathbf{P}}\{F<Tl(\|{X_{ i}}\|)W\}\Bigr{)}^{q}\] \[= \exp\Bigl{\{}q\sum_{0\not=X_{i}\in\Phi}\log\Bigl{(}p+(1-p)(1-e^{- \mu Tl(\|{X_{i}}\|)W})\Bigr{)}\Bigr{\}}\,,\] where $F,W$ are independent generic random variables representing fading and thermal noise, independent of $\Phi$, $F$ is exponential with mean $1/\mu$, Using the Laplace functional formula for $\Phi$ and the assumption that $W>w$ a.s. we have \[{\mathbf{P}}^{0}\{\,\widehat{\cal T}_{0}(0)\geq q\,\} \geq \exp\left(-2\pi\lambda\int_{v>0}\left(1-\left(1-(1-p)e^{-w\mu l(v )T}\right)^{q}\right)v\,\text{d}v\right)\] (17) \[= \exp\left(-\pi\lambda\int_{v>0}\left(1-\left(1-f(v)\right)^{q} \right)\,\text{d}v\right)\,,\] where \[f(v):=(1-p)\exp(-Kv^{\beta/2})\quad\text{and}\quad K=w\mu TA^{\beta}\,.\] In what follows we will show that the expression in (17) is not smaller than $1/q$ for $q$ large enough. To this regard denote by $v_{q}$ the unique solution of $f(v)=\frac{1}{q}$. We have \[v_{q}=\frac{1}{A^{2}\left(\mu Tw\right)^{2/\beta}}(\log(q(1-p)))^{2/\beta}.\] It is clear that $f(v)$ tends to 0 when $v$ tends to infinity and that $v_{q}$ tends to infinity as $q$ tends to infinity. Therefore, there exists a constant $Q=Q(\mu,w,A,T)<\infty$ such that for all $q\geq Q$ and for all $v\geq v_{q}$, \[(1-f(v))\geq\exp(-f(v)).\] Hence, for all $q\geq Q$, \[\int_{v>0}\left(1-\left(1-f(v)\right)^{q}\right)\,\text{d}v \leq\] \[\leq v_{q}+\int_{v=v_{q}}^{\infty}\left(1-\exp(-qf(v)\right)\,\,\text {d}v\] \[\leq v_{q}+\int_{v_{q}}^{\infty}qf(v)\,\,\text{d}v\] \[= v_{q}+\int_{u=0}^{\infty}qf(u+v_{q})\,\,\text{d}u.\] The third inequality follows from the fact that $1-\exp(-x)\leq x$. Using now the fact that $(u+v_{q})^{\beta/2}\geq u+v_{q}^{\beta/2}$ (for $q$ large enough, say again $q\geq Q$) we get that \[\int_{u=0}^{\infty}qf(u+v_{q})\,\,\text{d}u = \int_{u=0}^{\infty}q(1-p)\exp(-K(u+v_{q})^{\beta/2}))\,\,\text{d}u\] \[\leq \int_{u=0}^{\infty}q(1-p)\exp(-Ku-Kv_{q}^{\beta/2})\,\,\text{d}u= \frac{1}{K}\,,\] since $(1-p)\exp(-Kv_{q}^{\beta/2})=1/q$. Hence for $q\geq Q$ \[\int_{v>0}\left(1-\left(1-f(v)\right)^{q}\right)\,\,\text{d}v\leq v_{q}+\frac{ \alpha}{K}.\] Also it is not difficult to see that $\beta>2$ implies \[v_{q}\leq\frac{\log q}{\pi\lambda}-\frac{1}{K}\] (18) for $q$ large enough. This implies for $q$ large enough, say again $q\geq Q$, \[\exp\left(-\pi\lambda\int_{v>0}\left(1-\left(1-f(v)\right)^{q}\right)\,\text{d }v\right)\geq\exp\Bigl{(}-\pi\lambda(v_{q}+1/K)\Bigr{)}\geq\frac{1}{q}\,,\] (19) which completes the proof. ∎∎ Proof.: (_of Proposition 3.2_). Assume without loss of generality $Y=0$ and $\|{X}\|=r$. Under ${\mathbf{P}}$, consider the p.p. $\Phi\cup\{X,0\}$ and its independent marking. Given $\Phi$, the r.v. $L_{X,0}(0)$ associated with the independently marked p.p. $\Phi\cup\{X,0\}$ has a geometric distribution with parameter \[\pi_{X,0}(\Phi)=p(1-p)\Pr\Bigl{\{}\,F\geq l(r)(W+I)\Bigr{)}\,\Bigr{\}}\,,\] where $F,W,I$ are independent r.v.s, $F,W$ are generic fading and noise variables and $I=\sum_{X_{i}\in\Phi}e_{i}(0)F_{i,0}(0)/l(\|{X_{i}}\|)$. Using the exponential distribution of $F$ and the independence, we obtain \[\pi_{X,0}(\Phi)=\mathbf{E}[e^{-\mu l(r)TW}]\;\mathbf{E}[e^{-\mu l(r)TI}\,\|\, \Phi]\,.\] The mean of the geometric r.v. is known to be $\mathbf{E}^{X,0}[L_{X,0}(0)\,\|\,\Phi]=1/\pi_{X,0}(\Phi)$. By unconditioning with respect to $\Phi$, one obtains The first factor in the above expression is obviously finite. In what follows we will evaluate the second one. By the conditional independence of marks and denoting by ${\cal L}_{eF}(\cdot)$ is the Laplace transform of $eF$, where $e,F$ are independent generic variables for $e_{i}(0)$ and $F_{i,0}(0)$ we have \[= \left(\mathbf{E}\left[\exp\Bigl{(}-\mu l(r)T\sum_{X_{i}\in\Phi}e_ {i}(0)F_{i,0}(n)/l(\|{X_{i}}\|)\Bigr{)}\,\Big{\|}\,\Phi\right]\right)^{-1}\] \[= \exp\left(\sum_{X_{i}\in\Phi}\log{\cal L}_{eF}\Bigl{(}\mu Tl(r)/l (\|{X_{i}}\|)\Bigr{)}\right)\,.\] Note that ${\cal L}_{eF}(\xi)=1-p+p{\cal L}_{F}(\xi)=1-p+p\mu/(\mu+\xi)$. Using this and the Laplace functional formula for $\Phi$, (cf. (9, Eq. 9.4.17)) we obtain \[\mathbf{E}\Bigl{[}\frac{1}{\mathbf{E}[e^{-\mu l(r)TI}\,\|\,\Phi]} \Bigr{]} = \exp\biggl{\{}2\pi p\lambda\int_{0}^{\infty}\frac{vTl(r)}{l(v)+(1 -p)Tl(r)}\,\,\text{d}v\biggr{\}}\,.\] (cf. (2) Using now the fact that for the Poisson p.p., $\breve{M}_{[2]}(\text{d}x)=\lambda\text{d}x$), it is now easy to see that for any path-loss function satisfying $\int_{\epsilon}^{\infty}v/l(v)\,\text{d}v<\infty$, the integral in the exponent of the last expression is finite. This completes the proof. ∎∎ Proof.: (_of Proposition 3.2_). Using the the inclusion (16), inequality (14) and the left-hand side of (6) and we have \[\widehat{\cal T}_{i}(n)\leq{\cal T}_{i}(n)\leq L_{i}(n)\leq P_{i,j}(n)\,.\] Thus, it is enough to show \[\lim_{\|{X-Y}\|\to\infty}\frac{\mathbf{E}^{X,Y}[\widehat{\cal T}_{X}(0)]}{\|{X-Y} \|}=\infty\,.\] Without loss of generality assume $X=0$ and $\|{Y}\|=r$. Using the same arguments as in the proof of Lemma 4.1 and the representation (7) of the Palm probability with respect to Poisson p.p., we obtain \[{\mathbf{P}}^{0,Y}\{\,\widehat{\cal T}_{0}(0)>q\,\|\,\Phi\,\}\] \[\geq \prod_{0,Y\not=X_{i}\in\Phi}\Bigl{(}p+(1-p){\mathbf{P}}\{F<Tl(\|{X _{i}}\|)W\}\Bigr{)}^{q}\;\Bigl{(}p+(1-p){\mathbf{P}}\{F<Tl(\|{Y}\|)W\}\Bigr{)}^{q}\] \[\geq \exp\left(-\pi\lambda\int_{v>0}\left(1-\left(1-f(v)\right)^{q} \right)\,\text{d}v\right)\;\alpha(r)^{q}\,,\] where $\alpha(r)=1-(1-p)e^{-w\mu A^{\alpha}Tr^{\beta}}$. Using (19), which holds for large $q$, more precisely $q>Q=Q(\mu,w,A,T)$, we obtain \[\frac{\mathbf{E}^{0,Y}[\widehat{\cal T}_{0}(0)]}{r}\geq\frac{1}{r}\sum_{q>Q} \frac{\alpha(r)^{q}}{q}\,.\] It is now easy to see that \[\lim_{r\to\infty}\frac{1}{r}\sum_{q>Q}\frac{\alpha(r)^{q}}{q}=\infty.\] ∎∎ ### Proofs of Results of Section 3.3 Denote by $B_{x}(R)$ the ball centered at $x\in\mathbb{R}^{2}$ of radius $R$. Similarly as for the delays, we extend the definition of the local delays to arbitrary pairs of points $x,y\in\mathbb{R}^{2}$ by taking $L(x,y,n)=L_{X(x),X(y)}(n)$. We first establish the following technical result: Under the assumptions of Proposition 3.3 let $X_{i},X_{j}\in\Phi\cap B_{0}(R)$ for some $R>0$, where $\Phi=\Phi_{M}+\Phi_{G_{s}}$. Then the conditional expectation of the local delay $L_{i,j}(0)$ given $\Phi$ satisfies \[\mathbf{E}[L_{i,j}(0)\,\|\,\Phi]\] \[= \frac{1}{p(1-p){\cal L}_{W}(T\mu A^{\beta}\|{X_{i}-X_{j}}\|^{\beta} )}\exp\Bigl{\{}-\hskip-5.690551pt\sum_{\Phi\ni X_{k},k\not=i,k}\log{\cal L}_ {eF^{\prime}}\Bigl{(}\frac{T\|{X_{i}-X_{j}}\|^{\beta}}{\|X_{j}-X_{k}\|^{\beta}} \Bigr{)}\Bigr{\}}\] \[\leq \frac{1}{p(1-p){\cal L}_{W}(T\mu(A2R)^{\beta})}\] \[\times e^{-\Phi_{M}(B_{0}(2R))\log(1-p)}\hskip 277.41437pt(b)\] where $C(s,\beta)<\infty$ is some constant (which depends on $s$ and $\beta$ but not on $\Phi$), $F^{\prime}$ is an exponential random variable of mean 1 and ${\cal L}_{eF^{\prime}}(\cdot)$ is the Laplace transform of $eF^{\prime}$. Proof.: We first prove the equality in (4.2). When using the independence assumptions, we have \[{\mathbf{P}}\left\{\,L_{i,j}(0)>m\mid\Phi\,\right\}\] \[= {\mathbf{P}}\left\{\,\forall_{n=1}^{m}\left(e_{j}(n)=1\;\text{or} \;\right.\right.\] \[\left.\left.e_{j}(n)=0\;\text{and}\;e_{i}(n)F_{i,j}(n )\leq Tl(\|{X_{i}-X_{j}}\|)(W_{j}(n)+I_{i,j}(n))\right)\,\right\}\] \[= \prod_{n=1}^{m}\Biggl{(}p+(1-p)\biggl{(}1-p+p\Bigl{(}1-{\cal L}_{ W}(T\mu A^{\beta}\|x-y\|^{\beta})\] \[\times\prod_{\Phi \ni X_{k},k\not=i,j}{\cal L}_{eF^{\prime}}\Bigl{(}\frac{T\|{X_{i}-X_{j}}\|^{ \beta}}{\|{X_{j}-X_{k}}\|^{\beta}}\Bigr{)}\Bigr{)}\biggr{)}\Biggr{)}\,.\] The result then follows from the evaluation of \[\mathbf{E}\left[L_{i,j}(0)\,\mid\,\Phi\right]=\sum_{m=0}^{\infty}{\mathbf{P}} \left\{\,L_{i,j}(0)>m\,\mid\,\Phi\right]\,.\] The bound $\|X_{i}-X_{j}\|\leq 2R$ used in the Laplace transform of $W$ leads to the first factor of the upper bound. We now factorize the exponential function in (4.2) as the product of three exponential functions \[\alpha := \exp\Bigl{\{}-\sum_{\Phi_{G_{s}}\ni X_{k},k\not=i,j}\Bigr{\}},\] \[\beta := \exp\Bigl{\{}-\sum_{\Phi_{M}\ni X_{k},k\not=i,j\|X_{k}\|\leq 2R} \Bigr{\}},\] \[\gamma := \exp\Bigl{\{}-\sum_{\Phi_{M}\ni X_{k},\|X_{k}\|>2R}\Bigr{\}}.\] Next we prove that the last three exponentials are upper-bounded by (a), (b) and (c) in (4.2), respectively. * We use $\|{X_{i}-X_{j}}\|\leq 2R$ and Jensen’s inequality to get \[\log{\cal L}_{eF^{\prime}}\Bigl{(}\frac{T\|{X_{i}-X_{j}}\|^{\beta}} {\|X_{j}-X_{k}\|^{\beta}}\Bigr{)} \geq \log{\cal L}_{eF^{\prime}}\Bigl{(}\frac{T(2R)^{\beta}}{\|X_{j}-X_{ k}\|^{\beta}}\Bigr{)}\] \[\geq \frac{-T(2R)^{\beta}\mathbf{E}[eF^{\prime}]}{\|X_{j}-X_{k}\|^{\beta}}\] \[= -pT(2R)^{\beta}\|X_{j}-X_{k}\|^{-\beta}\,.\] We now prove that \[\sum_{\Phi_{G_{s}}\ni X_{k}:\|X_{j}-X_{k}\|>3\sqrt{2}s}\|X_{j}-X_{k}\|^{-\beta} \leq C(s,\beta),\] for some constant $C(s,\beta)$. This follows from an upper-bounding of the value of $\|X_{j}-X_{k}\|^{-\beta}$ by the value of the integral $1/s^{2}\int(\|X_{j}-x\|-\sqrt{2}s)^{-\beta}\,\,\text{d}x$ over the square with corner points $X_{k}$, $X_{k}+(s,0)$, $X_{k}+(0,s)$ and $X_{k}+(s,s)$. In this way we obtain \[\sum_{\Phi_{G_{s}}\ni X_{k}:\|X_{j}-X_{k}\|>3\sqrt{2}s}\|X_{j}-X_{ k}\|^{-\beta} \leq \frac{1}{s^{2}}\int_{\|x-X_{j}\|>2\sqrt{2}s}^{\infty}(\|X_{j}-x\|- \sqrt{2}s)^{-\beta}\,\,\text{d}x\] \[= \frac{2\pi}{s^{2}}\int_{\sqrt{2}s}^{\infty}\frac{t+\sqrt{2}s}{t^{ \beta}}\,\,\text{d}t=:C(s,\beta)<\infty\,.\] Combining this and what precedes, we get that \[\exp\left\{-\sum_{X_{k}\in\Phi_{G_{s}},\|X_{j}-X_{k}\|>2\sqrt{s}}\log{\cal L}_{ eF^{\prime}}\Bigl{(}\frac{T\|{X_{j}-X_{i}}\|^{\beta}}{\|X_{j}-X_{k}\|^{\beta}} \Bigr{)}\right\}\leq\exp(T(2R)^{\beta}C(s,\beta)).\] We also have for all $X_{k}\in\Phi_{G_{s}}$ and in particular for $\|X_{j}-X_{k}\|\leq 3\sqrt{2}s$. Hence we obtain \[\exp\{-\sum_{X_{k}\in\Phi_{G_{s}}}(\dots)\}\leq e^{-49\log(1-p)+T(2R)^{\beta}C (s,\beta)}\,,\] where 49 upper-bounds the number of points $X_{k}\in\Phi_{G_{s}}$ such that $\|X_{j}-X_{k}\|\leq 3\sqrt{2}s$. * Using the bound $\|X_{j}-X_{i}\|\leq 2R$ and the inequality $\log{\cal L}_{eF^{\prime}}(\xi)\geq\log{\cal L}_{eF^{\prime}}(\infty)=\log(1-p)$, we obtain \[\exp\{-\sum_{\Phi_{M}\ni X_{k},k\not=i,j,\|X_{i}\|\leq 2R}(\dots)\}\leq e^{- \Phi_{M}(B_{0}(2R))\log(1-p)}\,.\] * Using the bounds $\|X_{j}-X_{i}\|\leq 2R$ and $\|X_{j}-X_{k}\|\geq\|X_{k}\|-R$ (the latter follows from the triangle inequality) and the expression ${\cal L}_{eF^{\prime}}(\xi)=1-p+\frac{p}{1+\xi}$, we obtain \[\exp\Bigl{\{}-\sum_{\Phi_{M}\ni X_{k},\|X_{k}\|>2R}(\dots)\Bigr{\}}\] \[\leq\] This completes the proof. ∎∎ We can now prove the following auxiliary result. Under the assumptions of Proposition 3.3 for all points $x,y$ of $\mathbb{R}^{2}$, \[\mathbf{E}\left[\sup_{x_{1},y_{1}\in[x,y]}p(x_{1},y_{1},\Phi)\right]<\infty\,,\] where the supremum is taken over $x_{1},y_{1}$ belonging to the interval $[x,y]\subset\mathbb{R}^{2}$. Proof.: Without loss of generality, we assume that $(x+y)/2=0$ is the origin of the plane. Let $B=B_{0}(R)$ be the ball centered at $0$ and of radius $R$ such that no modification of the points in the complement of $B$ modifies $X(z)$ for any $z\in[x,y]$ (recall that $X(z)$ is the point of $\Phi$ which is the closest from $z$). Since $\Phi=\Phi_{M}+\Phi_{G_{s}}$, with $\Phi_{G_{S}}$ the square lattice p.p. with intensity $1/s^{2}$, it suffices to take $R=\|u-v\|/2+\sqrt{2}s$. Let $B^{\prime}=B_{0}(2R)$. By the above choice of $B$ and the inequality (6) we have for all $x_{1},y_{1}\in[x,y]$ \[P(x_{1},y_{1},0)\leq\sum_{X_{i},X_{j}\in\Phi\cap B}L_{i,j}(0)\,\] and consequently \[\sup_{x_{1},y_{1}\in[x,y]}p(x_{1},y_{1},\Phi)\leq\sum_{X_{i},X_{j}\in\Phi\cap B }\mathbf{E}[L_{i,j}(0)\,\|\,\Phi]\,.\] Using the result of Lemma 4.2 we obtain \[\sup_{x_{1},y_{1}\in[x,y]}\overline{\|p^{}(x_{1},y_{1},\Phi)\|}\] \[\leq \frac{e^{-49\log(1-p)+(2R)^{\beta}pTC(s,\beta)}}{p(1-p){\cal L}_{ W}(T\mu A(2R)^{\beta})}\] \[\times\Bigl{(}\Phi_{M}(B)+\pi(R+\sqrt{2}s)^{2}/s^{2}\Bigr{)}e^{- \Phi_{M}(B^{\prime})\log(1-p)}\,,\] where $\pi(R+\sqrt{2}s)^{2}/s^{2}$ is an upper bound of the number of points of $\Phi_{G_{s}}$ in $B$. The first factor in the above upper bound is deterministic. The two other factors are random and independent due to the independence property of the Poisson p.p. The finiteness of the expectation of the last expression follows from the finiteness of the exponential moments (of any order) of the Poisson random variable $\Phi_{M}(B^{\prime})$. For the expectation of the second (exponential) factor, we use the known form of the Laplace transform of the Poisson SN to obtain the following expression \[\mathbf{E}\Bigl{[}\exp\Bigl{\{}-\sum(\dots)\Bigr{\}}\Bigr{]}=\exp\biggl{\{}2 \pi p\lambda_{M}\int_{R}^{\infty}\frac{T(2R)^{\beta}}{v^{\beta}+(1-p)T(2R)^{ \beta}}(v+R)\,\,\text{d}v\Bigr{\}}<\infty\,.\] ∎∎ Proof.: (_of Proposition 3.3_ The existence and finiteness of the limit $\kappa_{\mathrm{d}}$ follows from the subadditivity (13) and Lemma 4.2 by the continuous-parameter sub-additive ergodic theorem (see (19, Theorem 4)). ∎∎ Proof.: (_of Proposition 3.3_) First, we prove the second statement; i.e., that $\kappa_{\mathrm{d}}$ is constant for all $\mathrm{d}$ in the unit sphere off some countable subset. Note that the point process $\Phi$ is ergodic as the independent superposition of mixing Poisson p.p. $\Phi_{M}$ and ergodic grid process $\Phi_{G}$. This can be easily proved using e.g. the respective characterisations of above properties by means of Laplace transforms of p.p. (see (9, Prop. 12.3.VI)). From the ergodicity of $\Phi$ we _cannot_ conclude the desired property for any vector $\mathrm{d}$ since the limit $\kappa_{\mathrm{d}}=\kappa_{\mathrm{d}(\Phi)}$ is not necessarily invariant with respect to translations of $\Phi$ by _any_ vector $x\in\mathbb{R}^{2}$ but only $x=\alpha\mathrm{d}$ for any scalar $\alpha\in\mathbb{R}$. The announced result follows from (22, Th. 1). For the first statement, consider a product space with on which two independent p.p.s $(\Phi_{M},\Phi_{G})$ are defined. Fix some vector $\mathrm{d}$ and define the operator $T=T_{1}\times T_{2}$ on this product space as the product of two operators, which correspond to the shift in the direction $\mathrm{d}$, of $\Phi_{M}$ and $\Phi_{G}$ respectively. The $\sigma$-field invariant with respect to $T$ is the product of the respective $\sigma$-fields invariant with respect to $T_{1}$ and $T_{2}$. The latter is trivial since $\Phi_{M}$ is mixing (as a Poisson p.p.). Consequently every function of $(\Phi_{M},\Phi_{G})$ that is invariant with respect to the shift in the direction $\mathrm{d}$ of its first argument ($\Phi_{M}$) is a.s. constant. This concludes the proof that $\kappa_{\mathrm{d}}$ is constant in $\Phi_{M}$ and thus depends only on $U_{G}$. ∎ Proof.: (_of Proposition 3.3_) For a given path $\sigma=\{(X_{0},n_{0}),(X_{1},n_{0}+1),\ldots,\allowbreak(X_{k},n_{0}+k)\}$ on ${\mathbb{G}}$ denote by $\|{\sigma}\|=\sum_{i=1}^{k}\|{X_{i}-X_{i-1}}\|$ the Euclidean length of the projection of $\sigma$ on $\mathbb{R}^{2}$; let us call it Euclidean length of $\sigma$ for short and recall that the (graph) length of $\sigma$ is equal to $k$. For fixed $\epsilon>0$ and all $n\geq 1$ denote by $\Pi(n)=\Pi_{\epsilon}(n)$ _the event that there exists a path on ${\mathbb{G}}$ starting at $(X(0),0)$ that has (graph) length $n$ and Euclidean length larger than $n/\epsilon$_. Assume $\mathbf{E}[\kappa_{\mathrm{d}}]=0$. We show first that this implies that for any $\epsilon>0$, ${\mathbf{P}}^{0}$-a.s. the event $\Pi_{\epsilon}(n)$ holds for infinitely many $n$ \[{\mathbf{P}}^{0}\Bigl{\{}\,\bigcap_{n\geq 1}\bigcup_{k\geq n}\Pi_{\epsilon}(k) \,\Bigl{\}}=1\,.\] (21) Indeed, $\mathbf{E}[\kappa_{\mathrm{d}}]=0$ implies $\kappa_{\mathrm{d}}=0$${\mathbf{P}}$-a.s. and by Palm-Matthes definition of the Palm probability ${\mathbf{P}}^{0}$-a.s. as well. This means that $\mathbf{E}^{0}[P(0,t\mathrm{d},0)\,\|\,\Phi]/t\to 0$, when $t\to\infty$, which implies that \[\lim_{k}P(0,t_{k}\mathrm{d},0)/t_{k}\to 0\] (22) ${\mathbf{P}}^{0}$-a.s. for some subsequence $\{t_{k}:k\geq 1\}$, with $\lim_{k}t_{k}=\infty$. Recall that $P(0,t_{k}\mathrm{d},0)$ is the length of a shortest path from $(X(0),0)$ (with $X(0)=0$ under ${\mathbf{P}}^{0}$) to $\{(X(t_{k}\mathrm{d}),n):n\geq 0\}$. Denote one of such shortest paths by $\sigma_{k}$. By the triangle inequality its Euclidean length satisfies \[\|{\sigma_{k}}\|\geq\|{0-X(t_{k}\mathrm{d})}\|\geq t_{k}-\sqrt{2}s\,.\] (23) From (22) and (23) one concludes that for any $\epsilon>0$ and $k$ large enough the length of the path $\sigma_{k}$ is smaller than $\epsilon$ time its Euclidean length $\|{\sigma_{k}}\|$. Now, (21) follows from the fact that the length of the path $\sigma_{k}$ tends to infinity with $k$, which is a consequence of $t_{k}\to\infty$ and the local finiteness of the graph ${\mathbb{G}}$ (cf Corollary 3.1). We conclude the proof by showing that for $\epsilon$ small enough, \[\sum_{n}{\mathbf{P}}^{0}\{\Pi_{\epsilon}(n)\}<\infty\,,\] (24) which by the Borel–Cantelli lemma implies that $\Pi(n)$ holds ${\mathbf{P}}^{0}$-a.s. only for a finite number of integers $n$ and thus contradicts to (21). To this regard assume constant $W=w>0$ and let ${\cal P}_{w}^{n}$ denote the set of paths $\sigma$ of ${\mathbb{G}}$ of length $n$, originating from $(X(0)=0,0)$. Denote also by ${\cal P}_{0}^{n}$ the analogous set of paths on the graph constructed under assumption $W=0$. Note that by monotonicity, \[{\cal P}^{n}_{w}\subset{\cal P}^{n}_{0}.\] (25) By the definition \[{\mathbf{P}}^{0}\{\Pi_{\epsilon}(n)\,\|\,\Phi\,\}={\mathbf{P}}^{0}\Bigl{(} \bigcup_{\sigma}\Bigl{\{}\,\sigma\in{\cal P}_{w}^{n}\;\text{and}\;\|{\sigma}\| \geq n/\epsilon\,\Bigr{\}}\,\Big{\|}\,\Phi\Bigr{)}\,,\] (26) where the sum bears on all possible $n$-tuples $\sigma=((X_{j_{1}},1),\ldots,(X_{j_{n}},n))$, with $X_{j_{i}}\in\Phi$. From this we have \[{\mathbf{P}}^{0}(\Pi_{\epsilon}(n)\,\|\,\Phi)\] \[\leq \sum_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{W} ,\|\sigma\|\geq n/\epsilon\,\Big{\|}\,\Phi\,\Bigr{\}}\] \[= \sum_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{W} ,\|\sigma\|\geq n/\epsilon\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}{ \mathbf{P}}^{0}\{\,\sigma\in{\cal P}^{n}_{0}\,\|\,\Phi\}\] \[\leq \mathbf{E}^{0}[{\cal H}_{0}^{out,n;W=0}(0)\,\|\,\Phi]\sup_{\sigma} {\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{W},\|\sigma\|\geq n/\epsilon\, \Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}\,,\] (28) where ${\cal H}_{0}^{out,n;W=0}(0)$ denotes the number of paths of length $n$ originating from $(X_{0}=0,0)$ under the assumption $W=0$. But \[\sup_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{w} ,\|\sigma\|\geq n/\epsilon\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}\] \[\leq \sup_{\sigma=((X_{j_{1}},1),\ldots,(X_{j_{n}},n))\atop{\sum_{i=1} ^{n}\|X_{j_{i}}-X_{j_{i-1}}\|\geq n/\epsilon}}\mathbf{E}^{0}\Bigl{[}\prod_{i=1}^ {n}\delta_{j_{i-1},{j_{i}}}(i-1,w)\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0} \Bigr{]},\] where $X_{j_{0}}=0$ and $\delta_{j_{i-1},{j_{i}}}(i-1,w)=\delta_{j_{i-1},{j_{i}}}(i-1)$ is the indicator of the existence of the edge from $(X_{j_{i-1}},i-1)$ to $(X_{j_{i}},i)$ defined by (4), and where we add in the notation the dependence on the noise $W=w$. Using the conditional independence of marks, (4), (15) and lack of memory of the exponential distribution of $F$ of parameter $\mu$ we have for the path-loss function (1) \[\mathbf{E}^{0}\Bigl{[}\prod_{i=1}^{n}\delta_{j_{i-1},{j_{i}}}(i-1 ,w)\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\Bigr{]} = \prod_{i=1}^{n}\mathbf{E}^{0}\bigl{[}\delta_{j_{i-1},{j_{i}}}(i-1 ,w)\,\big{\|}\,\Phi,\delta_{j_{i-1},{j_{i}}}(i-1,0)=1\bigr{]}\] \[= \prod_{i=1}^{n}\exp\left(-\mu(A\|X_{j_{i-1}}-X_{j_{i}}\|)^{\beta}{ Tw}\right)\,.\] Hence \[\sup_{\sigma}{\mathbf{P}}^{0}\Bigl{\{}\,\sigma\in{\cal P}^{n}_{w},\|\sigma\|\geq n /\epsilon\,\Big{\|}\,\Phi,\sigma\in{\cal P}^{n}_{0}\,\Bigr{\}}\leq\exp\left(- \mu A^{\beta}Twn\epsilon^{-\beta}\right)\,,\] where the last inequality follows from a convexity argument. Using this and (4.2), we get \[\mathbf{E}^{0}(\Pi_{\epsilon}(n)) \leq \mathbf{E}^{0}[{\cal H}_{0}^{out,n;W=0}(0)]\exp\left(-\mu A^{ \beta}Twn\epsilon^{-\beta}\right)\] \[\leq \xi^{n}\exp\left(-\mu A^{\beta}Twn\epsilon^{-\beta}\right)\] \[\leq \exp\left(n(\log(\xi)-K/\epsilon^{\beta})\right)\,,\] where in the second inequality we used the following result of Corollary 3.1 \[\mathbf{E}^{0}[{\cal H}_{0}^{out,n;W=0}(0)]=h^{out,n;W=0}=h^{in,k,W=0}\leq\xi^ {k}\,\] and where $K$ is a positive constant. This shows (24) for $\epsilon$ small enough, and thus concludes the proof. ∎∎ Proof.: (_of Corollary 3.3_) Without loss of generality assume $x=0$. We use the left inequality in (6), (14) and the inclusion (16) to obtain \[P_{X(0),X(y)}(0)\geq L_{X(0)}(0)\geq{\cal T}_{X(0)}(0)\geq\widehat{\cal T}_{X( 0)}(0)\,\] and in consequence \[p(0,y,\Phi)\geq\mathbf{E}[\widehat{\cal T}_{X(0)}(0)\,\|\,\Phi]\,.\] Using the isotropy and the strong Markov property of the Poisson p.p. \[\mathbf{E}[\widehat{\cal T}_{X(0)}(0)\,\|\,\Phi]=\mathbf{E}^{0}[\widehat{\cal T }_{0}(0)\,\|\,\Phi\|_{\overline{B}})]\,,\] $\Phi\|_{\overline{B}})$ is the restriction of $\Phi$ to the complement of the open ball $B=B_{(0,R)}(R)$, centered at $(0,R)$ of radius $R\geq 0$, where $R$ is r.v. independent of $\Phi$ and having for density \[\frac{\,\text{d}\theta}{2\pi}2\pi\lambda r\exp(-\lambda\pi r^{2})\,.\] But since we consider here the SNR graph $\widehat{\mathbb{G}}$ \[\mathbf{E}^{0}[\widehat{\cal T}_{0}(0)\,\|\,\Phi\|_{\overline{B}})]\geq\mathbf{E }^{0}[\widehat{\cal T}_{0}(0)\,\|\,\Phi]\,.\] The result follows now from Lemma 4.1. ∎∎ ## 5 SINR space-time graph and routing Let us now translate our results regarding the SINR graph into properties of routing in ad-hoc networks. Firstly, it makes sense to assume that any routing algorithm builds paths on ${\mathbb{G}}$. This takes two key phenomena into account: contention for channel (nodes have to wait for some particular time slots to transmit a packet) and collisions (lack of capture due to insufficient SINR). Our time constant gives bounds on the delays that can be attained in the ad-hoc network by any routing algorithms. Of course, realistic routing policies cannot use information about future channel conditions. In the case of Poisson p.p. there is hence no routing algorithm with a finite time constant. The existence of such an algorithm in the case of the Poisson$+$Grid p.p. remains an open question. In the Poisson p.p. case; one can ask about the exact asymptotics of the optimal delay (we know it is not linear) and of the delay realizable by some non-anticipating algorithm. Let us discuss now the relation of our results to those obtained in [12; 15]. In these papers the so called delay-tolerant networks are considered and modeled by a spatial SINR or signal-to-noise ratio (SNR) graph with no time dimension. In these models, the time constant (defined there as the asymptotic ratio of the graph distance to the Euclidean distance) is announced to be finite, even in the pure Poisson case. The reason for the different performance of these models lays in the fact that they do not take the time required for a successful transmission from a given node in the evaluation of the end-to-end delay. The heavy-tailness of this time (which follows from that of the exit time (cf. Proposition 3.2) makes the time constant infinite in the space-time Poisson scenario. The reason for the heavy-tailness of the successful transmission time is linked to the so called “RESTART” algorithm (see e.g. [17; 18; 1; 16]). 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1304.2618	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 18829, "num_imgs": 0, "llama3_tokens_count": 5818 }	[]	# Lexicographic identifying codes Maximilien Gadouleau School of Engineering and Computing Sciences Durham University m.r.gadouleau@durham.ac.uk ###### Abstract An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this paper, we design an algorithm that finds an identifying code in a twin-free graph on $n$ vertices in $O(n^{3})$ binary operations, and returns a failure if the graph is not twin-free. We also determine an alternative for sparse graphs with a running time of $O(n^{2}d\log n)$ binary operations, where $d$ is the maximum degree. We also prove that these algorithms can return any identifying code with minimum cardinality, provided the vertices are correctly sorted. ## 1 Introduction Identifying codes were introduced in [1] for fault diagnosis in multiprocessor systems, and have since then found applications in location and detection problems. In general, an identifying code in a graph $G$ can be defined as follows. First, we denote the (closed) neighborhood of any vertex $v$ as $N(v)=\{v\}\cup\{w:vw\in E(G)\}$. An _identifying code_ is a subset of vertices which satisfies the following property: for any two vertices $v$ and $w$, we have $N(v)\cap C\neq N(w)\cap C\neq\emptyset$. Equivalently, it is any subset of vertices $C$ such that for all $v_{1},v_{2}\in V(G)$, $(N(v_{1})\Delta N(v_{2}))\cap C\neq\emptyset$, where $\Delta$ is the symmetric difference between two sets. A graph admits an identifying code if and only if it is _twin-free_[2], where twins are two vertices with the same neighborhood. We remark that the definitions above are commonly used for a so-called $1$-identifying code, where an $r$-identifying code is defined in terms of balls of radius $r$ around a vertex. Since any $r$-identifying code can be seen as a $1$-identifying code for a related graph, we do not lose any generality in considering $1$-identifying codes only. For a thorough survey of identifying codes, the reader is invited to [3], and an exhaustive literature bibliography on identifying codes and related topics is maintained in [4]. Since any superset of an identifying code is itself an identifying code, it is natural to search for the minimum cardinality $i(G)$ of an identifying code of a given graph $G$. Let us refer to an identifying code as _minimal_ if it has no proper subset which itself is an identifying code and as _minimum_ if it has the smallest cardinality amongst all codes. The problem of finding the minimum cardinality of an identifying code was shown to be NP-hard in [3]. Viewing this problem as an instance of the subset cover problem [5], a greedy heuristic was also designed and analyzed in [3]. Its running time is on the order of $O(n^{4})$ binary operations and has the following performance guarantees. It always finds an identifying code whose cardinality is less than $c_{1}i(G)\ln n$ for some nonnegative constant $c_{1}$; however, there are graphs for which the algorithm always returns a code with cardinality greater than $c_{2}i(G)\ln n$ for another nonnegative constant $c_{2}$. _Lexicographic codes_ were introduced in [6] and independently rediscovered in [7] to design large constant-weight codes, which are sets of binary vectors of equal Hamming weight with a prescribed minimum Hamming distance (see [8] for a detailed review of constant-weight codes and lexicographic codes). The principle is to first sort all the vectors with the same Hamming weight, and then construct the code as we run through them. Adding a codeword is done according to a simple criterion: it must be at distance at least $d$ from the code constructed so far. The performance of the algorithm depends on the order in which the vectors have been sorted; moreover, some modifications can be added, such as starting with a predetermined set of vectors. Many record-holding constant-weight codes have been designed using lexicographic codes. However, this idea is not limited to constant-weight codes, and their application to nonrestricted binary codes has led to many interesting results [9]. They have also been recently applied to the construction of codes on subspaces in [10], also yielding record-holding codes. In this paper, we investigate adapting the idea of lexicographic codes to identifying codes. The main contribution is an algorithm running in $O(n^{3})$ binary operations which returns an identifying code for a twin-free graph, and returns a failure if the graph is not twin-free. This algorithm is then adapted to sparse graphs to run in $O(n^{2}d\log n)$ binary operations. Both algorithms have the same guarantees in terms of cardinality of the output. Although we are unable to give an upper bound which does not depend on the ordering of the vertices, we show that provided the vertices are properly sorted, the algorithm returns a minimum identifying code. This is fundamentally different to the greedy approach in $O(n^{4})$. ## 2 Algorithm for general graphs ### Description and correctness Let $G$ be a graph on $n$ vertices with adjacency matrix ${\bf A}$, and let ${\bf B}={\bf I}_{n}+{\bf A}$. We denote the vertices as $v_{1},v_{2},\ldots,v_{n}$, thus $b_{i,j}=1$ if and only if $v_{i}\in N(v_{j})$; yet we shall abuse notation and identify a vertex with its index. For instance, we refer to the vertex with minimum index in the neighborhood of $v_{i}$ as $\mathtt{min1}(i)$. Also, the output of our algorithm is actually the set of indices of the vertices in the code. Before giving the pseudocode of Algorithm 1, we describe it schematically below. Its input is the matrix ${\bf B}$ of the graph. It then runs along all vertices $v_{j}$, adding a new codeword to the code $C$ if $N(v_{j})\cap C=\emptyset$ or $N(v_{j})\cap C=N(v_{k})\cap C$ for some $k<j$. While searching for a new codeword to add, the algorithm may return a failure if the graph is not twin-free, which we identify as $n+1\in C$. After the $j$-th step, the code $C$ then ‘identifies’ the first $j$ vertices, i.e. they are all covered in a distinct fashion. We keep track of the intersections $N(v_{i})\cap C$ in a matrix ${\bf X}$. After going through all vertices, the algorithm then returns an identifying code $C$ or a failure (if $n+1\in C$) if the graph is not twin-free. ``` $C\leftarrow\emptyset$, $X\leftarrow{\bf 0}_{n}$, $j\gets 1$ while $j\leq n$ and $n+1\notin C$ do $l\gets 0$ if ${\bf X}(j)={\bf 0}$ then {$v_{j}$ is not covered} $l\leftarrow\mathtt{min1}(j)$ else $k\gets 1$ while ${\bf X}(j)\neq{\bf X}(k)$ and $k<j$ do {$v_{j}$ is covered, so we search if it is identified} $k\gets k+1$ end while if $k<j$ then {$v_{j}$ is not identified} $l\leftarrow\mathtt{min2}(j,k)$ end if end if if $1\leq l\leq n$ then {A new codeword has been found} $C\gets C\cup\{l\}$ ${\bf X}^{T}(l)\leftarrow{\bf B}^{T}(l)$ end if $j\gets j+1$ end while return $C$ ``` Algorithm 1 Main algorithm for general graphs The subroutine $\mathtt{min2}(j,k)$ returns the first vertex which identifies $v_{j}$ if it exists and a failure otherwise, i.e. it determines the first vertex in lexicographic order in $N(v_{j})\Delta N(v_{k})$. If $N(v_{j})=N(v_{k})$, then it returns $n+1$. It is given in Algorithm 2. ``` $l\gets 1$ while $l\leq n$ and ${\bf B}(j,l)={\bf B}(k,l)$ do $l\gets l+1$ end while return $l$ ``` Algorithm 2$\mathtt{min2}(j,k)$ subroutine We now justify this claim in Lemma 1 below. Lemma 1: _The subroutine $\mathtt{min2}(j,k)$ returns the minimum element in $N(v_{j})\Delta N(v_{k})$ if this symmetric difference is non-empty, and a failure ($l=n+1$) otherwise._ Proof First, if $N(v_{j})=N(v_{k})$, then ${\bf B}(j,l)={\bf B}(k,l)$ for all $1\leq l\leq n$. Therefore, the while loop will only stop once $l=n+1$, and hence the subroutine returns a failure. Second, if $N(v_{j})\neq N(v_{k})$, then the minimum element in $N(v_{j})\Delta N(v_{k})$ is the smallest $l$ such that ${\bf B}(j,l)\neq{\bf B}(k,l)$. It is clear that the subroutine returns this value. $\Box$ Proposition 1: _Algorithm 1 returns an identifying code if the input graph is twin-free, and a failure ($n+1\in C$) otherwise._ Proof First of all, we prove that the algorithm returns a failure if and only if the graph is not twin-free. In the latter case, let $k$ be the smallest integer such that the set $\{i\neq k:N(v_{k})=N(v_{i})\}$ is not empty, and let $j$ be the minimum element of this set (hence $k<j$, $N(v_{k})=N(v_{j})$). It is easily shown that after the $k$-th step, $v_{k}$ is covered. On the $j$-th step, Algorithm 1 first checks if $v_{j}$ is covered. Since $v_{k}$ is covered and $N(v_{k})=N(v_{j})$, then $v_{j}$ is also covered. Algorithm 1 then finds that $k$ is the smallest integer satisfying ${\bf X}(k)={\bf X}(j)$, and hence calls the subroutine $\mathtt{min2}(j,k)$. By Lemma 1 this returns a failure, and hence the whole algorithm returns a failure. Conversely, the only case where the subroutine (and hence the algorithm) returns a failure is when there exist $k<j$ such that $N(v_{j})=N(v_{k})$, i.e. the graph is not twin-free. We now assume that the graph is twin-free, and hence we have $l\leq n$ at any step. We need to show that the output $C$ of Algorithm 1 is an identifying code. Let us denote the matrix ${\bf X}$ and the code $C$ obtained after $j$ steps as ${\bf X}^{j}$ as $C^{j}$, respectively. Note that for all $a$, ${\bf X}^{j}(a)$ reflects how the vertex $v_{a}$ is covered by $C^{j}$: $N(v_{a})\cap C^{j}=\mathrm{supp}({\bf X}(a))=\{b:{\bf X}^{j}(a,b)=1\}$. The following claim is the cornerstone of the proof. Claim: After step $j$, all ${\bf X}^{j}(i)$’s are nonzero and distinct for $1\leq i\leq j$. The proof goes by induction on $j$, and is trivial for $j=1$. Suppose it is true for $j-1$, then \[\mathrm{supp}({\bf X}^{j-1}(a))=N(v_{a})\cap C^{j-1}\subseteq N(v_{a})\cap C^{ j}=\mathrm{supp}({\bf X}^{j}(a)).\] It is hence easy to show that if ${\bf X}^{j-1}(a)\neq{\bf 0}$, then ${\bf X}^{j}(a)\neq{\bf 0}$ and if ${\bf X}^{j-1}(a)\neq{\bf X}^{j-1}(b)$, then ${\bf X}^{j}(a)\neq{\bf X}^{j}(b)$ for all $a$ and $b$. It immediately follows that the vectors ${\bf X}^{j}(i)$’s are all nonzero and distinct for $1\leq i\leq j-1$, and we only have to consider ${\bf X}^{j}(j)$. Three cases occur when the algorithm reaches step $j$. * Case I: ${\bf X}^{j-1}(j)$ is nonzero and distinct to any ${\bf X}^{j-1}(i)$ for $1\leq i\leq j-1$. Then as shown above, ${\bf X}^{j}(j)$ is nonzero and distinct to all ${\bf X}^{j}(i)$’s. * Case II: ${\bf X}^{j-1}(j)$ is nonzero and equal to ${\bf X}^{j-1}(k)$ for some $k<j$. First, we remark that $k$ is unique, as ${\bf X}^{j-1}(k)\neq{\bf X}^{j-1}(i)$ for all other $i$. The $\mathtt{min2}(k,j)$ subroutine then returns an element $v_{l}\in N(v_{j})\Delta N(v_{k})$, and hence ${\bf X}^{j}(j,l)\neq{\bf X}^{j}(k,l)$. * Case III: ${\bf X}^{j-1}(j)={\bf 0}$. Then by hypothesis ${\bf X}^{j-1}(j)\neq{\bf X}^{j-1}(i)$ for all $1\leq i\leq j-1$, and hence ${\bf X}^{j}(j)\neq{\bf X}^{j}(i)$. Also, ${\bf X}^{j}(j)$ is the unit vector ${\bf e}_{\mathtt{min1}(j)}$, which is nonzero. Therefore, for the code $C^{n}=C$ obtained after $n$ steps, $N(v_{a})\cap C$ are all nonzero and distinct for all $1\leq a\leq n$. It is hence an identifying code. $\Box$ ### Performance We now investigate the performance of Algorithm 1. We are first interested in the cardinality of its output. Clearly, this depends on the order in which the vertices are sorted. We show below that provided the order is suitable, the algorithm can find any minimal identifying code, and hence can return a minimum one. Proposition 2: _Suppose that the graph is twin-free and that $M=\{v_{1},v_{2},\ldots,v_{m}\}$ forms an identifying code. Then Algorithm 1 returns an identifying code that is a subset of $M$._ Proof We know by Proposition 1 that the algorithm returns an identifying code; we only have to prove that all codewords are in $M$. At step $j$, three cases need to be distinguished. * Case I: $v_{j}$ is covered and identified, then no codeword is added. * Case II: $v_{j}$ is covered but not identified, i.e. $(N(v_{j})\Delta N(v_{k}))\cap C^{j-1}=\emptyset$ for some $k<j$. The subroutine returns the smallest element $v_{l}$ in $N(v_{j})\Delta N(v_{k})$. Since $M$ is an identifying code, the set $(N(v_{j})\Delta N(v_{k}))\cap M$ is not empty, hence $v_{l}\in M$. * Case III: $v_{j}$ is not covered. The algorithm then selects the next codeword to be $\mathtt{min1}(j)$, which is necessarily in $M$ as $N(v_{j})\cap M\neq\emptyset$. Therefore, the algorithm only adds codewords of $M$, and hence returns a subcode of $M$. $\Box$ We remark that Algorithm 1 does not necessarily return a minimal code, as seen in Figure 1. Algorithm 1 would return the code $\{1,2,3,4,5,6\}$ while $\{2,3,4,5,6\}$ is a minimal identifying code. [FIGURE:S2.F1][ENDFIGURE] On the other hand, if $M$ is minimal, then it has no proper subset that itself is an identifying code; Algorithm 1 thus returns it. We obtain the following corollary. Corollary 1: _Provided that the vertices are sorted such that $v_{1},v_{2},\ldots,v_{m}$ form a minimal identifying code for some $1\leq m\leq n$, Algorithm 1 will return this identifying code._ Proposition 2 also implies that the probability that the output has cardinality no more than $K$ is at least the probability that the first $K$ vertices form an identifying code. Hence our algorithm returns a minimum identifying code with probability at least $\frac{1}{{n\choose i(G)}}$. Proposition 3: _The running time of Algorithm 1 is $O(n^{3})$ binary operations._ Proof Clearly, we have to run the iteration for $j$ exactly $n$ times. For each iteration, the step demanding the highest number of operations is the search for $k$. We consider at most $j-1$ values of $k$, comparing at most $n$ bits to verify whether ${\bf X}(j)\neq{\bf X}(k)$. Therefore, the running time is $O(n^{3})$. $\Box$ ## 3 Algorithm for sparse graphs For sparse graphs, it is more efficient not to work with the whole adjacency matrix, but with the neighborhood array $A\in\mathcal{P}(E)^{n}$, defined as $A(v_{i})=N(v_{i})$, where the neighborhood is sorted in increasing lexicographic order. Then, instead of adding the column of the adjacency matrix corresponding to a new codeword, we only update the code array $X(v)$ for all vertices adjacent to the new codeword. THe algorithm for sparse graphs is given in Algorithm 3; its input is the neighorhood array, and it returns an identifying code $C$ or a failure ($n+1\in C$) if the graph is not twin-free. ``` $C\leftarrow\emptyset$, $X\leftarrow\emptyset^{n}$, $j\gets 1$, $f\gets 0$ while $j\leq n$ and $n+1\notin C$ do $l\gets 0$ if $X(j)=\emptyset$ then {$v_{j}$ not covered} $l\gets A(j,1)$ else $m\gets X(j,1)$, $k\gets 1$ while $X(j)\neq X(k)$ and $k<j$ do $k\gets k+1$ end while if $k<j$ then {$v_{j}$ not identified} $l\leftarrow\mathtt{min3}(j,k)$ end if end if if $1\leq l\leq n$ then $C\gets C\cup\{l\}$ for $i$ from $1$ to $d_{l}$ do $X(A(l,i))\gets X(A(l,i))\cup\{l\}$ end for end if $j\gets j+1$ end while return $C$ ``` Algorithm 3 Main algorithm for sparse graphs Similar to the general case, the $\mathtt{min3}(j,k)$ subroutine produces the first vertex $v_{l}$ which identifies $v_{j}$ if it exists and a failure otherwise, i.e. it determines the first vertex in lexicographic order which covers either $j$ or $k$, but not both. It is given in Algorithm 4. ``` $l\gets n+1$ while $a\leq\min\{d_{j},d_{k}\}$ do if $A(j,a)\neq A(k,a)$ then $l\leftarrow\min\{A(j,a),A(k,a)\}$ end if $a\gets a+1$ end while if $l=n+1$ then if $d_{j}<d_{k}$ then $l\gets A(k,d_{j}+1)$ else if $d_{k}<d_{j}$ then $l\gets A(j,d_{k}+1)$ end if end if return $l$ ``` Algorithm 4$\mathtt{min3}(j,k)$ subroutine The same results on correctness and the possibility of returning a minimum code also hold for Algorithm 3; they are summarized below. Proposition 4: _If the graph is not twin-free, then Algorithm 3 returns a failure. Otherwise, the algorithm returns an identifying code contained in $\{v_{1},v_{2},\ldots,v_{m}\}$, where $m$ is the minimum integer such that this forms an identifying code._ _The running time of Algorithm 3 is $O(n^{2}d\log n)$ binary operations._ Proof The proof of correctness of Algorithm 3 is similar to that of Algorithm 1, and is hence omitted. We hence determine the running time of the algorithm. $\Box$ ## References * [1] M. G. Karpovsky, K. Chakrabarty, and L. B. Levitin, “A new class of codes for identification of vertices in graphs,” _IEEE Trans. Info. Theory_, vol. 44, no. 2, pp. 599–611, March 1998. * [2] I. Charon, I. Honkala, O. Hudry, and A. Lobstein, “Structural properties of twin-free graphs,” _The Electronic Journal of Combinatorics_, vol. 14, no. 1, p. R16, January 2007. * [3] M. Laifenfeld and A. Trachtenberg, “Identifying codes and covering problems,” _IEEE Trans. Info. Theory_, vol. 54, no. 9, pp. 3929–3950, September 2008. * [4] A. Lobstein. Watching systems, identifying, locating-dominating and discriminating codes in graphs. [Online]. Available: http://www.infres.enst.fr/{̃}lobstein/bibLOCDOMetID.html * [5] T. Cormen, C. Leiserson, and R. Rivest, _Introduction to Algorithms_. MIT Press, 2001. * [6] V. Levenshtein, “A class of systematic codes,” _Soviet Math. Dokl. 1_, pp. 368–371, 1960. * [7] J. H. Conway and N. J. A. Sloane, “Lexicographic codes: error-correcting codes from game theory,” _IEEE Trans. Info. Theory_, vol. 32, pp. 337–348, May 1986. * [8] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, and W. D. Smith, “A new table of constant weight codes,” _IEEE Trans. Info. Theory_, vol. 36, no. 6, pp. 1334–1380, November 1990. * [9] A. Trachtenberg, “Error-correcting codes on graphs: Lexicodes, trellises, and factor graphs,” Ph.D. dissertation, University of Illinois at Urbana-Champaign, 2000. * [10] N. Silberstein and T. Etzion, “Large constant-dimension codes and lexicodes,” in _Proc. Algebraic Combinatorics and Applications_, Thurnau, Germany, April 2010.
1311.6600	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 32960, "num_imgs": 1, "llama3_tokens_count": 10209 }	[ "content_image/1311.6600/x1.png" ]	# Optimal condition for measurement observable via error-propagation Wei Zhong${}^{1}$, Xiao-Ming Lu${}^{2}$, Xiao-Xing Jing${}^{1}$ and Xiaoguang Wang${}^{1}$ ${}^{1}$ Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China. ${}^{2}$ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore. xgwang@zimp.zju.edu.cn ###### Abstract Propagation of error is a widely used estimation tool in experiments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramér-Rao theorem in parameter estimation. By invoking the Schrödinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for single parameter estimate. By applying this condition to Greenberg-Horne-Zeilinger states, we obtain the general expression of the optimal observable for separable measurements to achieve the Heisenberg-limit precision and show that it is closely related to the parity measurement. However, Jose _et al_ [Phys. Rev. A 87, 022330 (2013)] have claimed that the Heisenberg limit may not be obtained via separable measurements. We show this claim is incorrect. pacs: 06.20.Dk, 42.50.St, 03.65.Ta, 03.67.-a, † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction An essential task in quantum parameter estimation is to suppress the fundamental bound on measurement precision imposed by quantum mechanics. Various quantum strategies have been developed to enhance the accuracy of the parameter estimation, which are closely related to some practical applications, such as the Ramsey spectroscopies, atomic clocks, and the gravitational wave detection [1, 2, 3, 4, 5, 6, 7, 8]. Two approaches in common use for high-precision measurements are the parallel protocol with correlated multi-probes [9] and multi-round protocol with a single probe [10, 11]. Most recently, some novel methods, like environment-assisted metrology [12] and enhanced metrology by quantum error correction [13, 14, 15, 16], were raised to achieve high precision in realistic experiments. Rather than engineering the sensitivity-enhanced strategies, we concentrate on the problem of how to attain the maximal sensitivity in realistic experiments. In general, a noiseless procedure of the quantum single parameter estimation can be abstractly modeled by four steps (see figure 1): (i) preparing the input state $\rho_{\rm in}$, (ii) parameterizing it under the evolution of the parameter-dependent Hamiltonian, for instance, a unitary evolution $U_{\varphi}$ with $\varphi$ the parameter to be estimated, (iii) performing measurements of the observable $\hat{\mathcal{O}}$ on the output state $\rho_{\varphi}$, (iv) finally estimating the value of the parameter from the estimator $\varphi_{\rm est}$ as a function of the outcomes of the measurements. <figure><img src="content_image/1311.6600/x1.png"><figcaption>Figure 1: The schematic representation of a general scheme of (noiseless)quantum parameter estimation is composed of four components: input state ρin,parametrization process Uφ, measurements ^O, and estimator φest. Here, weconcentrate on the part in shadow to find the optimal ^O attaining the highestsensitivity to the parameter φ in ρφ.</figcaption></figure> From estimation theory, the estimation precision is statistically measured by the units-corrected mean-square deviation of the estimator $\varphi_{\rm est}$ from the true value $\varphi$[17, 18], (1) where the brackets $\langle\,\rangle_{\rm av}$ denote statistical average and the derivative $\partial_{\varphi}\langle\varphi_{\rm{est}}\rangle\equiv\partial\langle\varphi _{\rm{est}}\rangle/\partial\varphi$ removes the local difference in the “units” of $\varphi_{\rm{est}}$ and $\varphi$. Whatever is the measurement scheme employed, the ultimate limit to the precision of the unbiased estimate is given by the quantum Cramér-Rao bound (QCRB) from below as \[(\delta\varphi)^{2}_{\rm est}\geq(\upsilon\mathcal{F}_{\varphi})^{-1},\] (2) where $\upsilon$ is the repetitions of the experiment and $\mathcal{F}_{\varphi}$ is the quantum Fisher information (QFI) (see equation (7) for definition), which measures the statistical distinguishability of the parameter in quantum states. This bound is asymptotically achieved for large $\upsilon$ under optimal measurements followed by the maximum likelihood estimator [17, 18, 19, 20]. On the other hand, it is well-known that error-propagation is a widely acceptable theory in experiments [1, 3, 9, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. With this theory, to estimate the parameter $\varphi$ is reduced to measuring the average value of a physical observable $\hat{\mathcal{O}}$. After repeating the experiment $\upsilon$ times, the real accessible precision on $\varphi$ is given by the error-propagation formula as follows [1, 3, 9, 21, 22, 23, 24], \[(\delta\varphi)^{2}_{\rm ep}:=\frac{1}{\upsilon}\frac{\langle(\Delta\hat{ \mathcal{O}})^{2}\rangle}{\,\,\|\partial_{\varphi}\langle\hat{\mathcal{O}} \rangle\|^{2}},\] (3) where $\Delta\hat{\mathcal{O}}=\hat{\mathcal{O}}-\langle\hat{\mathcal{O}}\rangle$ and $\langle\hat{\mathcal{O}}\rangle={\rm Tr}(\rho_{\varphi}\hat{\mathcal{O}})$. Note that the two estimation errors defined in equations (1) and (3) are closely related. To show the relationship between the two kinds of the estimation errors, $(\delta\varphi)^{2}_{\mathrm{est}}$ and $(\delta\varphi)^{2}_{\mathrm{ep}}$, we introduce $\Delta\varphi_{\mathrm{est}}:=\varphi_{\mathrm{est}}-\langle\varphi_{\mathrm{ est}}\rangle_{\mathrm{av}}$. Then, it is easy to show that [17] \[(\delta\varphi)^{2}_{\mathrm{est}}=\frac{\langle(\Delta\varphi_{\rm est})^{2} \rangle_{\rm av}}{\|\partial_{\varphi}\langle\varphi_{\rm{est}}\rangle_{\rm av} \|^{2}}+\bigg{\langle}\frac{\varphi_{\rm{est}}}{\|\partial_{\varphi}\langle \varphi_{\rm{est}}\rangle_{\rm av}\|}-\varphi\bigg{\rangle}_{\rm av}^{2}.\] (4) When viewing the arithmetic mean of the measurement outcomes of $\hat{\mathcal{O}}$ over repetitions of the experiment as the estimator in the quantum setting, one has in general $(\delta\varphi)^{2}_{\rm est}\geq(\delta\varphi)^{2}_{\rm ep}\geq(\upsilon \mathcal{F}_{\varphi})^{-1}$ by noting that $\langle(\Delta\varphi_{\rm est})^{2}\rangle=\langle(\Delta\hat{\mathcal{O}})^{ 2}\rangle/\upsilon$ for sufficiently large $\upsilon$ according to the central limit theorem [34] and comparison of the two definitions of the errors given by equations (1) and (3). In such situation, $(\delta\varphi)^{2}_{\rm est}$ and $(\delta\varphi)^{2}_{\rm ep}$ have the same QCRB, and the saturation of the former implies that of the latter. The formula equation (3) indicates that the fluctuation of the observable $\hat{\mathcal{O}}$ propagates to the estimated values of the parameter $\varphi$. This means that what kinds observable $\hat{\mathcal{O}}$ employed directly affects the estimating precision of the parameter $\varphi$. The purpose of this paper is to address the question of with which kind of observable does the estimation error given by equation (3) achieve the QCRB given by equation (2). In this paper, we derive the necessary and sufficient (N&S) condition for the optimal observable saturating the QCRB for the single parameter estimation by using the Schrödinger-Robertson uncertainty relation (SRUR). We then apply this condition to GHZ states and find the general form of the optimal observable for separable measurements to achieve the Heisenberg-limit sensitivity (i.e., $1/N$). Moreover, we discuss the relation between the optimal separable observable and parity measurements. However, Jose _et al._, in a recent work [35], made a contradictory conclusion with respect to the above result. They claimed that separable measurements are impossible to go beyond the shot-noise limit (i.e., $1/{\sqrt{N}}$) for any entangled states. To clarify this issue, we revisit the method in [35] and show the causes for this inconsistency. This paper is structured as follows. In section 2, we first briefly review the single parameter estimation and obtain the N&S condition for the optimal observable. In section 3, we give an application of this condition to obtain the optimal separable observables for GHZ states to saturate the Heisenberg-limit precision. In section 4, we further elucidate the reasons for contradiction between the result given in [35] and ours. At last, a conclusion is given in section 5. ## 2 N&S condition for optimal observable in single parameter estimation We start by a brief review of quantum single parameter estimation via the general estimator. Consider a parametric family of density matrices $\rho_{\varphi}$ containing an unknown parameter $\varphi$ to be estimated. Suppose that the general quantum measurement performed on $\rho_{\varphi}$ is characterized by a positive-operator-valued measure $\hat{M}:=\{\hat{M}_{x}\}$ with $x$ the results of measurement. The value of the parameter is inferred via an estimator $\varphi_{\rm est}$, which maps the measurement outcomes to the estimated value. After repeating the experiment $\upsilon$ times, the standard estimation error $(\delta\varphi)^{2}_{\rm est}$ in equation (1) is bounded from below as \[(\delta\varphi)^{2}_{\rm est}\geq(\upsilon F_{\varphi})^{-1},\] (5) where \[F_{\varphi}:=\sum_{x}p_{\varphi}(x)[\partial_{\varphi}\ln p_{\varphi}(x)]^{2}\] (6) is the (classical) Fisher information of the measurement-induced probability distribution $p_{\varphi}(x)=\mathrm{Tr}(\rho_{\varphi}\hat{M}_{x})$. The maximization over all POVMs gives rise to the so-called QFI, which is defined by \[\mathcal{F}_{\varphi}:={\rm Tr}(\rho_{\varphi}\hat{L}_{\varphi}^{2}).\] (7) Hence, a more tighter bound of equation (5) is given by equation (2). Here $\hat{L}_{\varphi}$ is the symmetric logarithmic derivative (SLD) operator, which is a Hermitian operator determined by \[\partial_{\varphi}\rho_{\varphi}=\frac{1}{2}[\rho_{\varphi},\hat{L}_{\varphi}] _{+}\] (8) with $[\cdot\,,\,\cdot]_{+}$ denoting the anti-commutator, see reference [17]. It is remarkable that $\hat{L}_{\varphi}$ may not be uniquely determined by equation (8) when $\rho_{\varphi}$ is not of full rank [36]. However, in general the value of the parameter $\varphi$ may not be directly measured. The most general method of estimating the value of $\varphi$ in practice involves measurements corresponding to a physical observable $\hat{\mathcal{O}}$ which is generally $\varphi$-independent. In such cases, the estimation error is given by the error-propagation formula equation (3), in which the fluctuations on the observable $\hat{\mathcal{O}}$ propagate to the uncertainty in the estimation of $\varphi$. In the following, we follow Hotta and Ozawa [24] to derive the achievable lower bound of the estimation error $(\delta\varphi)^{2}_{\rm ep}$ by using the SRUR. Let us first recall the SRUR [37, 38], which states that the uncertainty of two non-commuting observables $\hat{X},\,\hat{Y}$ must obey the following inequality \[\langle(\Delta\hat{X})^{2}\rangle\langle(\Delta\hat{Y})^{2}\rangle\geq\frac{1} {4}\|\langle[\hat{X},\hat{Y}]\rangle\|^{2}+\frac{1}{4}\langle[\Delta\hat{X}, \Delta\hat{Y}]_{+}\rangle^{2},\] (9) where $[\cdot\,,\,\cdot]$ denotes the commutator. The SRUR follows from the Schwarz inequality for the Hilbert-Schmidt inner product, and naturally reduces to the Heisenberg uncertainty relation under the condition $\langle[\Delta\hat{X},\Delta\hat{Y}]_{+}\rangle=0$. By substituting $\hat{X}\,(\hat{Y})$ with $\hat{\mathcal{O}}\,(\hat{L}_{\varphi}$) and utilizing \[\mathcal{F}_{\varphi}=\langle\hat{L}_{\varphi}^{2}\rangle=\langle(\Delta\hat{L }_{\varphi})^{2}\rangle,\] (10) as a result of $\langle\hat{L}_{\varphi}\rangle=2\,\partial_{\theta}\mathrm{Tr}(\rho_{\varphi} )=0$, equation (9) becomes \[\langle(\Delta\hat{\mathcal{O}})^{2}\rangle\,\mathcal{F}_{\varphi}\geq\frac{1} {4}\|\langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]\rangle\|^{2}+\frac{1}{4} \langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rangle^{2}.\] (11) Moreover, since the observable operator $\hat{\mathcal{O}}$ is independent of $\varphi$, we have \[\langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rangle = \mathrm{Tr}([\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rho_{ \varphi})\] (12) \[= \mathrm{Tr}(\hat{\mathcal{O}}[\hat{L}_{\varphi},\rho_{\varphi}]_{ +})\] \[= 2\,\partial_{\varphi}\langle\hat{\mathcal{O}}\rangle,\] where the second equality is obtained by employing the cyclic property of the trace operation, and the third equality is due to the SLD equation (8). Provided that $\langle\hat{\mathcal{O}}\rangle$ is nonzero, combining equations (3), (11) and (12) yields \[(\delta\varphi)^{2}_{\rm ep} \geq \frac{1}{\upsilon\mathcal{F}_{\varphi}}\bigg{(}1+\frac{\|\langle[ \hat{\mathcal{O}},\hat{L}_{\varphi}]\rangle\|^{2}}{\langle[\hat{\mathcal{O}}, \hat{L}_{\varphi}]_{+}\rangle^{2}}\bigg{)}\] (13) \[= \frac{1}{\upsilon\mathcal{F}_{\varphi}}\bigg{[}1+\bigg{(}\frac{{ \rm Im}\langle\hat{\mathcal{O}}\hat{L}_{\varphi}\rangle}{{\rm Re}\langle\hat{ \mathcal{O}}\hat{L}_{\varphi}\rangle}\bigg{)}^{2}\bigg{]}\] (14) \[\geq (\upsilon\mathcal{F}_{\varphi})^{-1}.\] (15) The bound in equation (13) describes the achievable sensitivity of $\varphi$ when employing an observable $\hat{\mathcal{O}}$. The bound in equation (15) gives the highest precision for $\varphi$ for the optimal observable $\hat{\mathcal{O}}_{\rm opt}$, which coincides with the QCRB in equation (2). It is shown that the estimation error $(\delta\varphi)^{2}_{\rm ep}$ achieves the QCRB only when the two equalities in equations (13) and (15) hold simultaneously. Below, we consider the attainability of the above bounds and give the N&S condition for optimal observables. From the N&S condition for equality in the SRUR, the equality in equation (13) holds if and only if \[\Delta\hat{\mathcal{O}}\sqrt{\rho_{\varphi}}=\alpha\hat{L}_{\varphi}\sqrt{\rho _{\varphi}}\] (16) is satisfied with a nonzero complex number $\alpha$. Note that we restrict here $\alpha\neq 0$ at the request of $\langle[\hat{\mathcal{O}},\hat{L}_{\varphi}]_{+}\rangle\neq 0$ in the denominator of equation (13). Furthermore, the equality in equation (15) holds if and only if \[\mathrm{Im}\langle\hat{\mathcal{O}}\hat{L}_{\varphi}\rangle=0.\] (17) This condition can be combined into the condition (16) by restricting $\alpha$ to be a nonzero real number, i.e., \[\Delta\hat{\mathcal{O}}\sqrt{\rho_{\varphi}}=\alpha\hat{L}_{\varphi}\sqrt{\rho _{\varphi}}\quad\mbox{with $\alpha\in\mathbb{R}\!\setminus\!\!\{0\}$}.\] (18) This is the of the optimal observable for density matrix $\rho_{\varphi}$. It implies that the estimation error achieves the QCRB given by the QFI for $\rho_{\varphi}$_only_ when the observable that we choose satisfies equation (18). This is the main result of the paper. For pure states $\rho_{\varphi}=\|\psi_{\varphi}\rangle\langle\psi_{\varphi}\|$, the condition (18) is equivalent to \[\Delta\hat{\mathcal{O}}\|\psi_{\varphi}\rangle=\alpha\hat{L}_{\varphi}\|\psi_{ \varphi}\rangle\quad\mbox{with $\alpha\in\mathbb{R}\!\setminus\!\!\{0\}$}.\] (19) If we assume that the parameter $\varphi$ here is imprinted via a unitary operation [9], i.e., $\rho_{\varphi}=\exp({-i\hat{G}\varphi})\,\rho_{\rm in}\exp({i\hat{G}\varphi})$ with $\hat{G}$ the generator, associating with the equality $\partial_{\varphi}\rho_{\varphi}=-i[\hat{G},\rho_{\varphi}]$, then condition (19) further reduces to \[\Delta\hat{\mathcal{O}}\|\psi_{\varphi}\rangle=-2i\alpha\Delta\hat{G}\|\psi_{ \varphi}\rangle\quad\mbox{with $\alpha\in\mathbb{R}\!\setminus\!\!\{0\}$}.\] (20) This condition was alternatively obtained in Ref. [31]. It is deserved to note that their proof is only valid in the case of unitary parametrization for pure states, and cannot be generalized to obtain the condition (18). Here, we discuss the relations between the saturation of the QCRB with respect to $(\delta\varphi)^{2}_{\rm est}$ and that with respect to $(\delta\varphi)^{2}_{\rm ep}$. Following Braunstein and Caves [17], the saturation of the QCRB with respect to the error $(\delta\varphi)^{2}_{\rm est}$ can be separated as the saturation of a classical Cramér-Rao bound (CCRB) equation (5) and finding an optimal measurement attaining the QFI. The CCRB can always be asymptotically achieved by the maximum likelihood estimator, so whether the QCRB can be asymptotically saturated is determined by whether the measurement attains the QFI. The N&S condition for the optimal measurement attaining the QFI reads [17] \[\sqrt{\hat{M}_{x}}\sqrt{\rho_{\varphi}}=u_{x}\sqrt{\hat{M}_{x}}\hat{L}_{ \varphi}\sqrt{\rho_{\varphi}},\] (21) where $\{\hat{M}_{x}\}$ denotes the POVM of the measurement and $u_{x}$ are real numbers. In the following, we show that the N&S condition (18) for the saturation of the QCRB with respect to $(\delta\varphi)^{2}_{\rm ep}$ identifies an optimal measurement attaining QFI. Let $\hat{\mathcal{O}}_{\rm opt}$ be the optimal observable satisfying Eq. (18) and $P_{x}$ the eigenprojectors of $\hat{\mathcal{O}}_{\rm opt}$ with the eigenvalues $x$. Left multiplying $P_{x}$ on both sides of Eq. (18), it is easy to see that $\{P_{x}\}$ is the optimal measurement attaining the QFI. That is to say, the projective measurement $\{P_{x}\}$ followed by the maximum likelihood estimator of the measurement outcomes saturate the QCRB with respect to the standard estimation error $(\delta\varphi)^{2}_{\rm est}$. ## 3 Optimal separable observable for GHZ states Below, we apply the N&S condition to show the general optimal observable for GHZ states. Let us specifically consider an experimentally realizable Ramsey interferometry to estimate the transition frequency $\omega$ of the two-level atoms loaded in the ion trap [1, 2]. The Hamiltonian of the system with $N$ atoms is $\hat{H}=(\omega/2)\sum_{i=1}^{N}\hat{\sigma}_{z}^{i}$ where $\hat{\sigma}_{z}^{i}$ is the Pauli matrix acting on the $i$th particle. In this setup, the measurements are limited to be performed separately on each atom. The observable operator may be described as a tensor product of Hermitian matrices $\hat{\mathcal{O}}=\hat{\mathcal{O}}_{\rm q}^{\otimes N}$ with $\hat{\mathcal{O}}_{\rm q}=a_{0}\mathbb{I}+\bm{a}\cdot\hat{\bm{\sigma}}$ dependent of four real coefficients $\{a_{0},a_{1},a_{2},a_{3}\}$, where $\mathbb{I}$ is the identity matrix of dimension $2$. Suppose that the input state is the maximally entangled states, i.e., GHZ states, which provides the Heisenberg-limit-scaling sensitivity of frequency estimation in the absence of noise [1, 9, 39]. Under the time evolution $\hat{U}=\exp{(-i\hat{H}t)}$, the output state can be represented as \[\|\psi_{\rm GHZ}(\varphi)\rangle=\frac{1}{\sqrt{2}}\big{(}\|0\rangle^{\otimes N} +e^{iN\varphi}\|1\rangle^{\otimes N}\big{)},\] (22) up to an irrelevant global phase with $\varphi=\omega t$. Here, we adopt the standard notation where $\|0\rangle$ and $\|1\rangle$ are the eigenvectors of $\sigma_{z}$ corresponding to eigenvalues $+1$ and $-1$, respectively. To determine the optimal separable observable $\hat{\mathcal{O}}$, we need to find the solutions of the coefficients $\{a_{0},a_{1},a_{2},a_{3}\}$ to satisfy equation (19). With $\hat{L}_{\varphi}=2\partial_{\varphi}(\|\psi_{\varphi}\rangle\langle\psi_{ \varphi}\|)$ for pure states, the SLD operator for the state of equation (22) is given by \[\hat{L}_{\varphi}=-iNe^{-iN\varphi}\left(\|0\rangle\langle 1\|\right)^{\otimes N }+iNe^{iN\varphi}\left(\|1\rangle\langle 0\|\right)^{\otimes N}.\] (23) We find that equation (19) is always satisfied for $a_{0}=a_{3}=0$ and arbitrary real number $a_{1},\,a_{2}$ that do not vanish simultaneously. Therefore, the general expression of the optimal separable observable is given by \[\hat{\mathcal{O}}_{\rm opt}=(a_{1}\hat{\sigma}_{x}+a_{2}\hat{\sigma}_{y})^{ \otimes N},\] (24) which is independent of the parameter $\varphi$, i.e., globally optimal in the whole range of the parameter. It is easy to check that such observables saturate the Heisenberg-limit sensitivity. Actually, according to the error-propagation formula equation (3), we have \[\delta\varphi_{\rm GHZ}=\frac{1}{\sqrt{\upsilon}}\frac{\sqrt{\langle\hat{ \mathcal{O}}_{\rm opt}^{2}\rangle-\langle\hat{\mathcal{O}}_{\rm opt}\rangle^{2 }}}{\|\partial_{\varphi}\langle\hat{\mathcal{O}}_{\rm opt}\rangle\|}=\frac{1}{ \sqrt{\upsilon}N},\] (25) as a result of \[\langle\hat{\mathcal{O}}_{\rm opt}\rangle = {\rm Re}[e^{-iN\varphi}(a_{1}+ia_{2})^{N}],\] (26) \[\langle\hat{\mathcal{O}}_{\rm opt}^{2}\rangle = (a_{1}^{2}+a_{2}^{2})^{N}.\] (27) When setting $a_{1}=1,a_{0}=a_{2}=a_{3}=0$, the optimal observable in equation (24) reduces to $\hat{\sigma}_{x}^{\otimes N}$, as given in [9]. Note that here measuring the observable $\hat{\sigma}_{x}^{\otimes N}$ fails to attain the Heisenberg limit for the cases of $\varphi=k\pi/N,\,(k\in\mathbb{Z})$ in which equation (25) becomes singular. Besides, we note that measuring the spin observable $\hat{\sigma}_{y}^{\otimes N}$ also fail in these cases when $N$ is even, and it is useful except for the cases of $\varphi=(2k+1)\pi/2N,\,(k\in\mathbb{Z})$ when $N$ is odd. We next show that the optimal observable in the form of equation (24) is closely related to the parity measurement proposed originally by Bollinger _et al_[3]. As is well known, in the standard Ramsey interferometry, there are generally two Ramsey pulses applying before and after the free evolution (with an accumulated phase $\varphi$), and measurements often take place after the second pulse [1, 3]. Here the action of the pulse is modeled by a $\pi/2$-rotation operation about the $y$ axis, i.e., $R_{y}\big{[}\frac{\pi}{2}\big{]}=\exp[-i(\frac{\pi}{2})\hat{J}_{y}]$, and the measurement observable is denoted as the operator $\hat{\mathcal{O}}_{f}$. With equation (24), one has \[\hat{\mathcal{O}}_{f}=R_{y}^{\dagger}\bigg{[}\frac{\pi}{2}\bigg{]}\,\hat{ \mathcal{O}}\,R_{y}\bigg{[}\frac{\pi}{2}\bigg{]}=(a_{1}\hat{\sigma}_{z}+a_{2} \hat{\sigma}_{y})^{\otimes N}.\] (28) When setting $a_{1}=1,a_{2}=0$, equation (28) reduces to \[\hat{\mathcal{O}}_{f}=\hat{\sigma}_{z}^{\otimes N}\equiv(-1)^{j-\hat{J}_{z}}\] (29) with $j=N/2$, which is the so-called parity measurement [3]. It is shown that only a parity measurement is necessary for the optimal estimate of the phase parameter $\varphi$ for GHZ states, and it is more experimentally feasible than the detection strategy, as discussed in [9], that applies local operations and classical communication. ## 4 Further discussions However, in a recent work [35], it was pointed out that the separable measurement (the restricted readout procedure) might not be possible to go beyond the shot-noise limit even for arbitrary entangled states. It seems that this conclusion is inconsistent with ours in the above discussion. In what follows, we clarify this issue by revisiting the method in [35] and showing the causes for this inconsistency. For simplicity, let us consider the two-qubit parametric GHZ state \[\|\psi^{(2)}_{\rm GHZ}(\varphi)\rangle=\frac{1}{\sqrt{2}}\big{(}\|0 0\rangle+e^{2i\varphi}\|11\rangle\big{)}.\] (30) Following Ref. [35], we restrict the separable measurement to be the projective measurements $\{\|+\rangle\langle+\|,\|-\rangle\langle-\|\}$ for each qubit with \[\|\pm\rangle=\frac{1}{\sqrt{2}}(\|0\rangle\pm\|1\rangle).\] (31) According to the condition of equation (21), whether the above restricted measurement presented by equation (31) is the optimal measurement saturating the QCRB can be tested by asking whether or not the operators of the form \[\hat{K} = \lambda_{++}\|++\rangle\langle++\|+\lambda_{+-}\|+-\rangle\langle+-\|\] (32) \[+\,\lambda_{-+}\|-+\rangle\langle-+\|+\lambda_{--}\|--\rangle\langle --\|\] can be the SLD operator for the state of equation (30). By domenstrating that for the state in equation (30) with $\varphi=0$, there is no solution of the SLD equation (8) for the coefficients $\{\lambda_{++},\lambda_{+-},\lambda_{-+},\lambda_{--}\}$ in equation (32), the authors in Ref. [35] claimed that the projective measurement about $\{\|++\rangle,\|+-\rangle,\|-+\rangle,\|--\rangle\}$ is not the optimal measurement for the state of equation (30). However, as we showed in the Sec. 3, $\sigma_{x}\otimes\sigma_{x}$ is an optimal observable saturating the QCRB with respect to $(\delta\varphi)^{2}_{\rm ep}$ for the states (30). Although the estimation error considered in the Ref. [35] is $(\delta\varphi)^{2}_{\rm est}$, a contradiction still arises, as the projective measurement of $\sigma_{x}\otimes\sigma_{x}$ attains the QFI of states (30) (see the end in Sec. 2) so that $\{\|++\rangle,\|+-\rangle,\|-+\rangle,\|--\rangle\}$ is the optimal measurement regarding the estimation error $(\delta\varphi)^{2}_{\rm est}$. Below, we shall show that actually for any other point except for $\varphi=k\pi/2,\,(k\in\mathbb{Z})$ in the range of the parameter, there do exist the SLD operator in form of equations (32). First, note that the SLD operator for the non-full-rank density matrices is not uniquely determined, but $\hat{L}_{\varphi}\rho_{\varphi}$ (or $\hat{L}_{\varphi}\|\psi_{\varphi}\rangle$ for pure state) is uniquely determined. Second, from equation (23), we see \[\hat{L}_{\varphi}=-2ie^{-2i\varphi}\|00\rangle\langle 11\|+2ie^{2i\varphi}\|11 \rangle\langle 00\|.\] (33) is a SLD operator for the state of equation (30). Third, since $\hat{L}_{\varphi}\|\psi_{\varphi}\rangle$ is uniquely determined, then if $\hat{K}$ is the SLD operator for $\|\psi_{\varphi}\rangle$ if and only if \[\hat{L}_{\varphi}\|\psi_{\varphi}\rangle=\hat{K}\|\psi_{\varphi}\rangle\] (34) is satisfied. Thus, substituting equations (30), (32) and (33) into equation (34), we obtain the solutions for the coefficients as \[\lambda_{++}=\lambda_{--}=-2\tan\varphi,\quad\lambda_{+-}=\lambda_{-+}=2\cot\varphi.\] (35) The above solutions are singular for $\varphi=k\pi/2,\,(k\in\mathbb{Z})$, which coincide with the results discussed below equation (27). Note that here the $\varphi=0,\,(k=0)$ case is just considered in Ref. [35]. Whilst, for a general value of the parameter except those singular points, the restricted separable measurement considered here indeed saturate the Heisenberg-limit-scaling sensitivity for the parametric state of equation (30). Moreover, it is easy to check that the same results of Eq. (35) can be obtained when restricting the separable measurement to be the projective measurements $\{\|+\rangle_{y}\langle+\|,\|-\rangle_{y}\langle-\|\}$ for each qubit with \[\|\pm\rangle_{y}=\frac{1}{\sqrt{2}}(\|0\rangle\pm i\|1\rangle)\] (36) the eigenvectors of $\sigma_{y}$. This is coincided with the result shown below equation (27) that measuring the observable $\sigma_{y}^{\otimes N}$ fails to attain Heisenberg limit for the $\varphi=k\pi/N,\,(k\in\mathbb{Z})$ cases when $N$ is even. ## 5 Conclusion We have addressed the optimization problem of measurements for achieving the ultimate sensitivity determined by the QCRB. From the propagation of error, we derive the N&S condition of the optimal observables for single parameter estimate by using the SRUR. As an application of this condition, we examine the optimal observables for GHZ states to achieve the ultimate sensitivity at the Heisenberg limit. We consider an experimentally feasible case that the observable operators are restricted to separably acting on the subsystem. We then find the general expression of the optimal separable observable by applying the N&S condition, and show that it is exactly equivalent to the parity measurement when applying a $\pi/2$ pulse operation. 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1510.07423	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 86158, "num_imgs": 1, "llama3_tokens_count": 35723 }	[ "content_image/1510.07423/x1.png" ]	"# Anisotropic scaling of random grain model\n\nwith application to network traffic\n\nVytautė Pili(...TRUNCATED)
1812.06383	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 46015, "num_imgs": 0, "llama3_tokens_count": 19408 }	[]	"# Exact normalized eigenfunctions for general deformed Hulthén potentials\n\nRichard L. Hall\n\nDe(...TRUNCATED)
1902.06526	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23519, "num_imgs": 2, "llama3_tokens_count": 6742 }	[ "content_image/1902.06526/x1.png", "content_image/1902.06526/x2.png" ]	"# Exact results for the entanglement in 1D Hubbard models with spatial constraints\n\nIoannis Kleft(...TRUNCATED)
1602.00825	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 54956, "num_imgs": 21, "llama3_tokens_count": 20136 }	["content_image/1602.00825/x1.png","content_image/1602.00825/x2.png","content_image/1602.00825/x8.pn(...TRUNCATED)	"# Nonclassical Particle Transport in 1-D Random Periodic Media\n\n**R. Vasques \${}^{\\dagger,}\$(...TRUNCATED)
1009.0666	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 17134, "num_imgs": 9, "llama3_tokens_count": 3996 }	["content_image/1009.0666/x1.png","content_image/1009.0666/x2.png","content_image/1009.0666/x3.png",(...TRUNCATED)	"# The First Station of the Long Wavelength Array\n\nUniversity of New Mexico\n\nE-mail:\n\nSteven W(...TRUNCATED)
1912.09110	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 21698, "num_imgs": 1, "llama3_tokens_count": 7023 }	[ "content_image/1912.09110/x1.png" ]	"# Mixed QCD-EW two-loop corrections to Drell-Yan production\n\nPRISMA\${}^{+}\$ Cluster of Excell(...TRUNCATED)

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This project is a subproject of the 📌PIN project, focusing on the development of the largest scientific document multimodal dataset, which integrates both text and images.

📑: https://arxiv.org/abs/2406.13923

🤗: https://huggingface.co/datasets/m-a-p/PIN-14M

Dataset statistics

Source	Content Images (#)	Content Images (Size GB)	Documents (#)	Documents (Size GB)	Llama3 Tokens (#)
IN-Arxiv	3.947 M	507.99	0.715 M	37.38	11.752 B
IN-PMC	19.711 M	2154.95	5.677 M	219.00	57.500 B
Total	23.658 M	2662.94	6.392 M	256.38	69.252 B

Examples

IN-Arxiv

{
    "id": "1407.4558",
    "meta": {
        "language": "en",
        "source": "Arxiv",
        "date_download": "2024-12-03"
    },
    "quality_signals": {
        "doc_length": 55245,
        "num_imgs": 1,
        "llama3_tokens_count": 22007
    },
    "content_image": [
        "content_image/1407.4558/x1.png"
    ],
    "md": "# Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations \u2020\n[FOOTNOTE:\u2020][ENDFOOTNOTE]\n\nZhiqiang Cai\n\n Department of Mathematics ..."
}

IN-PMC

{
    "id": "PMC3231073",
    "meta": {
        "date_download": "2024-12-14",
        "language": "en",
        "source": "PMC"
    },
    "quality_signals": {
        "doc_length": 20785,
        "llama3_tokens_count": 4364,
        "num_imgs": 7
    },
    "content_image": [
        "content_image/PMC3231073/sensors-10-10663-v2f1.jpg",
        "content_image/PMC3231073/sensors-10-10663-v2f2.jpg",
        "content_image/PMC3231073/sensors-10-10663-v2f3.jpg",
        "content_image/PMC3231073/sensors-10-10663-v2f4.jpg",
        "content_image/PMC3231073/sensors-10-10663-v2f5.jpg",
        "content_image/PMC3231073/sensors-10-10663-v2f6.jpg",
        "content_image/PMC3231073/sensors-10-10663-v2f7.jpg"
    ],
    "md": "# The Use of Helmholtz Resonance for Measuring the Volume of Liquids and Solids\n\n## Abstract\n\nAn experimental investigation was undertaken to ascertain the potential of using Helmholtz resonance for volume determination and the factors that may influence accuracy. The uses for a rapid non-interference volume measurement system range from agricultural produce and mineral sampling through to liquid fill measurements. By weighing the sample the density can also measured indirectly..."
}

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