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Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs

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Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs Mounia Hamidouche?, Laura Cottatellucciy, Konstantin Avrachenkov ? Departement of Communication Systems, EURECOM, Campus SophiaTech, 06410 Biot, France y Department of Electrical, Electronics, and Communication Engineering, FAU, 51098 Erlangen, Germany Inria, 2004 Route des Lucioles, 06902 Valbonne, France [email protected], [email protected], [email protected]. Abstract—In this article, we analyze the limiting eigen- multivariate statistics of high-dimensional data. In this case, value distribution (LED) of random geometric graphs the coordinates of the nodes can represent the attributes of (RGGs). The RGG is constructed by uniformly distribut- the data. Then, the metric imposed by the RGG depicts the ing n nodes on the d-dimensional torus Td ≡ [0; 1]d and similarity between the data. connecting two nodes if their `p-distance, p 2 [1; 1] is at In this work, the RGG is constructed by considering a most rn. In particular, we study the LED of the adjacency finite set Xn of n nodes, x1; :::; xn; distributed uniformly and matrix of RGGs in the connectivity regime, in which independently on the d-dimensional torus Td ≡ [0; 1]d. We the average vertex degree scales as log (n) or faster, i.e., choose a torus instead of a cube in order to avoid boundary Ω (log(n)). In the connectivity regime and under some effects. Given a geographical distance, rn > 0, we form conditions on the radius rn, we show that the LED of a graph by connecting two nodes xi; xj 2 Xn if their `p- the adjacency matrix of RGGs converges to the LED of distance, p 2 [1; 1] is at most rn, i.e., kxi − xjkp ≤ rn, the adjacency matrix of a deterministic geometric graph where k:kp is the `p-metric defined as (DGG) with nodes in a grid as n goes to infinity. Then, for 8 1=p n finite, we use the structure of the DGG to approximate > Pd (k) (k) p < k=1 jxi − xj j p 2 [1; 1); the eigenvalues of the adjacency matrix of the RGG and kxi−xjkp = > (k) (k) provide an upper bound for the approximation error. : maxfjxi − xj j; k 2 [1; d]g p = 1: Index Terms—Random geometric graphs, adjacency matrix, limiting eigenvalue distribution, Levy distance. The RGG is denoted by G(Xn; rn). Note that for the case p = 2 we obtain the Euclidean metric on Rd. Typically, the function rn is chosen such that rn ! 0 when n ! 1. I. Introduction The degree of a vertex in G(Xn; rn) is the number of edges connected to it. The average vertex degree in G(Xn; rn) is In recent years, random graph theory has been applied given by [3] to model many complex real-world phenomena. A basic (d) d an = θ nrn; random graph used to model complex networks is the Erdös- Rényi (ER) graph [1], where edges between the nodes appear where θ(d) = πd=2=Γ(d=2 + 1) denotes the volume of the d- arXiv:1910.08871v1 [math.SP] 20 Oct 2019 with equal probabilities. In [2], the author introduces another dimensional unit hypersphere in Td and Γ(:) is the Gamma random graph called random geometric graph (RGG) where function. nodes have some random position in a metric space and the Different values of rn, or equivalently an, lead to different edges are determined by the position of these nodes. Since geometric structures in RGGs. In [3], different interesting then, RGG properties have been widely studied [3]. regimes are introduced: the connectivity regime in which an RGGs are very useful to model problems in which the scales as log(n) or faster, i.e., Ω(log(n))1, the thermody- geographical distance is a critical factor. For example, RGGs namic regime in which an ≡ γ, for γ > 0 and the dense have been applied to wireless communication network [4], regime, i.e., an ≡ Θ(n): sensor network [5] and to study the dynamics of a viral RGGs can be described by a variety of random matrices spreading in a specific network of interactions [6], [7]. such as adjacency matrices, transition probability matrices Another motivation for RGGs in arbitrary dimensions is and normalized Laplacian. The spectral properties of those This research was funded by the French Government through 1The notation f(n) = Ω(g(n)) indicates that f(n) is bounded the Investments for the Future Program with Reference: Labex below by g(n) asymptotically, i.e., 9K > 0 and no 2 N such that UCN@Sophia-UDCBWN. 8n > n0 f(n) ≥ Kg(n). random matrices are powerful tools to predict and analyze where the term χ[xi s xj] takes the value 1 when there is a complex networks behavior. In this work, we give a special connection between nodes xi and xj in G(Xn; rn) and zero attention to the limiting eigenvalue distribution (LED) of the otherwise, represented as adjacency matrix of RGGs in the connectivity regime. 8 Some works analyzed the spectral properties of RGGs in <> 1; kxi − xjkp ≤ rn; i 6= j; p 2 [1; 1] different regimes. In particular, in the thermodynamic regime, χ[xi s xj] = > the authors in [8], [9] show that the spectral measure of the : 0; otherwise: adjacency matrix of RGGs has a limit as n ! 1. However, A similar definition holds for A(Dn) defined over due to the difficulty to compute exactly this spectral measure, G(Dn; rn). The matrices A(Xn) and A(Dn) are symmetric Bordenave in [8] proposes an approximation for it as γ ! 1. and their spectrum consists of real eigenvalues. We denote In the connectivity regime, the work in [6] provides a by fλi; i = 1; ::; ng and fµi; i = 1; ::; ng the sets of all real closed form expression for the asymptotic spectral moments eigenvalues of the real symmetric square matrices A(Dn) of the adjacency matrix of G(Xn; rn). Additionnaly, Bor- and A(Xn) of order n, respectively. The empirical spectral denave in [8] characterizes the spectral measure of the adja- 0 distribution functions vn(x) and vn(x) of the adjacency cency matrix normalized by n in the dense regime. However, matrices of an RGG and a DGG, respectively are defined in the connectivity regime and as n ! 1, the normalization as factor n puts to zero all the eigenvalues of the adjacency n n 1 X 1 X matrix that are finite and only the infinite eigenvalues in the v (x) = Ifµ ≤ xg and v0 (x) = Ifλ ≤ xg; n n i n n i adjacency matrix are nonzero in the normalized adjacency i=1 i=1 matrix. Motivated by this results, in this work we analyze the where IfBg denotes the indicator of an event B. behavior of the eigenvalues of the adjacency matrix without Let a0 be the degree of the nodes in G(D ; r ). In the normalization in the connectivity regime and in a wider range n n n following Lemma 1 we provide an upper bound for a0 under of the connectivity regime. n any ` -metric. First, we propose an approximation for the actual LED of p the RGG. Then, we provide a bound on the Levy distance Lemma 1. For any chosen `p-metric with p 2 [1; 1] and between this approximation and the actual distribution. More d ≥ 1, we have precisely, for > 0 we show that the LEDs of the adjacency d 0 1 d 1 matrices of the RGG and the deterministic geometric graph a ≤ d p 2 a 1 + : n n 1=d (DGG) with nodes in a grid converge to the same limit when 2an p 2 an scales as Ω(log (n) n) for d = 1 and as Ω(log (n)) for Proof. See Appendix A d ≥ 2. Then, under the `1-metric we provide an analytical approximation for the eigenvalues of the adjacency matrix of To prove our result on the concentration of the LED of RGGs by taking the d-dimensional discrete Fourier transform RGGs and investigate its relationship with the LED of DGGs (DFT) of an n = Nd tensor of rank d obtained from the first under any `p-metric, we use the Levy distance between two block row of the adjacency matrix of the DGG. distribution functions defined as follows. A B The rest of this paper is organized as follows. In Section Definition 1. ([10], page 257) Let vn and vn be two II we describe the model, then we present our main results A B distribution functions on R. The Levy distance L(vn ; vn ) on the concentration of the LED of large RGGs in the between them is the infimum of all positive such that, for connectivity regime. Numerical results are given in Section all x 2 R III to validate the theoretical results. Finally, conclusions are A B A given in Section IV. vn (x − ) − ≤ vn (x) ≤ vn (x + ) + . Lemma 2. ([11], page 614) Let A and B be two n × n Her- II. Spectral Analysis of RGGs mitian matrices with eigenvalues λ1; :::; λn and µ1; :::; µn, To study the spectrum of G(Xn; rn) we introduce an respectively. Then auxiliary graph called the DGG. The DGG denoted by 1 L3(vA; vB) tr(A − B)2; G(Dn; rn) is formed by letting Dn be the set of n grid points n n 6 n that are at the intersections of axes parallel hyperplanes with A B −1=d 0 0 where L(vn ; vn ) denotes the Levy distance between the separation n , and connecting two points xi, xj 2 Dn A B 0 0 empirical distribution functions vn and vn of the eigenvalues if kxi − xjkp ≤ rn with p 2 [1; 1].
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