Spectral Statistics of Lattice Graph Percolation Models Stephen Kruzick and Jose´ M

1 Spectral Statistics of Lattice Graph Percolation Models Stephen Kruzick and Jose´ M. F. Moura, Fellow, IEEE

Abstract—In graph signal processing, the graph adja- these signals is provided by a matrix related to the net- cency matrix or the graph Laplacian commonly define work structure, such as the adjacency matrix, Laplacian the shift operator. The spectral decomposition of the shift matrix, or normalized versions thereof [1][2]. Decompo- operator plays an important role in that the eigenvalues represent frequencies and the eigenvectors provide a spec- sition of a signal according to a basis of eigenvectors tral basis. This is useful, for example, in the design of filters. of the shift operator serves a role similar to that of the However, the graph or network may be uncertain due Fourier Transform in classical signal processing [3]. In to stochastic influences in construction and maintenance, this context, multiplication by polynomial functions of and, under such conditions, the eigenvalues of the shift the chosen shift operator matrix performs shift invariant matrix become random variables. This paper examines the spectral distribution of the eigenvalues of random filtering [1]. networks formed by including each link of a D-dimensional The eigenvalues of the shift operator matrix play lattice supergraph independently with identical probability, the role of graph frequencies and are important in the a percolation model. Using the stochastic canonical equa- design and analysis of graph signals and graph filters. If tion methods developed by Girko for symmetric matrices W is a diagonalizable graph shift operator matrix with with independent upper triangular entries, a deterministic distribution is found that asymptotically approximates the eigenvalue λ for eigenvector v such that W v = λv and, empirical spectral distribution of the scaled adjacency for example, if a filter is implemented on the network matrix for a model with arbitrary parameters. The main as P (W ) where P is a polynomial, then P (W ) has results characterize the form of the solution to an impor- corresponding eigenvalue P (λ) for v by simultaneous tant system of equations that leads to this deterministic diagonalizability of powers of W [4]. The framework of distribution function and significantly reduce the number of equations that must be solved to find the solution for graph signal processing regards P (λ) as the frequency a given set of model parameters. Simulations comparing response of the filter [5]. Hence, knowledge of the eigen- the expected empirical spectral distributions and the com- values of W informs the design of the filter P when the puted deterministic distributions are provided for sample eigenvalues of P (W ) should satisfy desired properties. parameters. Furthermore, the eigenvalues relate to a notion of signal Index Terms—graph signal processing, random net- complexity known as the signal total variation, which works, random graph, random matrix, eigenvalues, spectral has several slightly different definitions depending on statistics, stochastic canonical equations context [2][3][5][6]. For purposes of motivation, taking the shift operator to be the row-normalized adjacency matrix Ab, define the lp total variation of the signal x as I.INTRODUCTION p p Much of the increasingly vast amount of data available TVG (x) = I − Ab x = Lbx (1) p p arXiv:1611.02655v1 [cs.NA] 26 Sep 2016 in the modern day exhibits nontrivial underlying structure that does not fit within classical notions of signal which sums over all network nodes the pth power of processing. The theory of graph signal processing has the absolute difference between the value of the signal been proposed for treating data with relationships and x at each node and the average of the value of x at interactions best described by complex networks. Within neighboring nodes [5][6]. Thus, if v is a normalized this framework, signals manifest as functions on the eigenvector of the row-normalized Laplacian Lb with nodes of a network. The shift operator used to analyze eigenvalue λ, v has total variation |λ|p. The eigenvectors that have higher total variation can be viewed as more The authors are with the Department of Electrical and Com- complex signal components in much the same way puter Engineering, Carnegie Mellon University, Pittsburgh, PA that classical signal processing views higher frequency 15213 USA (ph: (412)2686341; e-mail: [email protected]; [email protected]). complex exponentials. This work was supported by NSF grant # CCF1513936. As an application, consider a connected, undirected network on N nodes with normalized Laplacian Lb with the goal of achieving distributed average consensus via a graph filter. For such a network, there is a simple Laplacian eigenvalue λ1(Lb) = 0 corresponding to the averaging eigenvector v1 = 1 and other eigenvalues 0 < λi(Lb) ≤ 2 for 2 ≤ i ≤ N. Any filter P such that P (0) = 1 and |P (λi(Lb))| < 1 for 2 ≤ i ≤ N will asymptotically transform an initial signal x0 with average µ to the average consensus signal ¯x = µ1 upon Fig. 1: The illustration shows an example lattice graph iterative application [7]. If the eigenvalues λi(Lb) for with three dimensions of size 4 × 3 × 3 with some node 2 ≤ i ≤ N are known, consensus can be achieved in tuples labeled. Each group of circled nodes, which differ finite time by selecting P to be the unique polynomial by exactly one symbol, represents a complete subgraph. of degree N − 1 with P (0) = 1 and P (λi(Lb)) = 0 [4]. Note that the averaging eigenvector has total variation 0 by the above definition (1) and that all other, more the empirical spectral distribution [10]. The stochastic complex eigenvectors are completely removed by the canonical equation techniques of Girko detailed in [8], filter. Thus, a finite time consensus filter represents the which allow analysis of matrices with independent but most extreme version of a non-trivial lowpass filter. With non-identically distributed elements as necessary when polynomial filters of smaller fixed degree d, knowledge studying the adjacency matrices of many random graph of the eigenvalues can be used to design filters of a given models, provide a method to obtain such deterministic length for optimal consensus convergence rate, which is equivalents. given by This paper analyzes the empirical eigenvalue distribu- 1/d 1/d tions of the adjacency matrices of random graphs formed ρ P Lb = max P λi Lb (2) 2≤i≤N by independent random inclusion or exclusion of each as studied in [7]. This can also be attempted in situations link in a certain supergraph, a bond percolation model in which the graph is a random variable, leading to [11]. Specifically, the paper examines adjacency matrices uncertainty in the eigenvalues. For example, [7] also proof percolation models of D-dimensional lattice graphs, poses two methods to accomplish this for certain random in which each D-tuple with Md possible symbols in switching network models. One relies on the eigenvalues the dth dimension has an associated graph node and in of the expected iteration matrix. The other attempts to which two nodes are connected by a link if the cor- filter over a wide range of possible eigenvalues without responding D-tuples differ by only one symbol. These taking the model into account. Neither takes into account graphs generalize other commonly encountered graphs, any information about how the eigenvalue distribution such as complete graphs and the cubic lattice graph spreads. This type of information could be relevant to [12], to an arbitrary lattice dimension. An illustration graph filter design, especially when the network does for the three dimensional case appears in Figure 1, in not switch too often relative to the filter length, rendering which a complete graph connects each group of circled static random analysis applicable. nodes. The stochastic canonical equation techniques of The empirical spectral distribution FW (x) of a Hermi- Girko provide the key mathematical tool employed here tian matrix W , which counts the fraction of eigenvalues to achieve our results. The resulting deterministic ap- of W in the interval (−∞, x], describes the set of eigen- proximations to the adjacency matrix empirical spectral values [8]. For matrices related to random graphs, the distribution can, furthermore, provide information about empirical spectral distributions are, themselves, function- the empirical spectral distributions of the normalized valued random variables. Often, it is not possible to adjacency matrices and, thus, the normalized Laplacians. know the joint distribution of the eigenvalues, but the This work is similar to that of [13], which analyzes behavior of the empirical spectral distribution can be the adjacency matrices of a different random graph described asymptotically. For instance, the empirical model known as the stochastic block model using similar spectral distributions of Wigner matrices converge to the tools and suggests applications related to the study of integral of a semicircle asymptotically in the matrix size epidemics and algorithms based on random walks. as described by Wigner’s semicircle law [9]. In other Section II discusses relevant background information cases, a limiting distribution may or may not exist, but a on random graphs and the spectral statistics of random sequence of deterministic functions can still approximate matrices. It also introduces an important mathematical

2 result from [8], presented in Therorem 1, which can be [15]. Additionally, the relationship between Erdos-R¨ enyi´ used to ﬁnd deterministic functions that approximate the random graphs and Wigner matrices leads to asymptotic empirical spectral distribution of certain large random spectral statistics that may be characterized [16].

matrices by relating it to the solution of a system of Similarly, starting from an arbitrary supergraph Gsup,N equations (16). Section III introduces lattice graphs and with N nodes, the graph-valued random variable addresses application of Theorem 1 to scaled adjacency Gperc (Gsup,N , p (N)) that results from including each matrices of percolation models based on arbitrary D- link of Gsup,N according to independent Bernoulli trials dimensional lattice graphs. The main contributions of that succeed with probability p (N) is known as a bond this paper appear in Theorem 2, which derives the form percolation model. Thus, an Erdos-R¨ enyi´ model is a of the solution to the system of equations (16) for bond percolation model with a complete supergraph. arbitrary parameters, and also in Corollary 1, which Typically, study of bond percolation models concerns describes the particular solution for given model pa- their asymptotic connectivity behavior, although that is rameters. The relationship of the empirical spectral not the purpose of this paper. Here, the terminology is distribution of the adjacency matrix to those of the simply used to refer to the type of random graph models symmetrically normalized adjacency matrix and the row- considered in this paper. normalized adjacency matrix is also noted. For some example lattice graphs, these results are used to provide deterministically calculated distribution functions that B. Spectral Statistics are compared against the expected empirical spectral distributions. Finally, Section IV provides concluding The eigenvalues of matrices, such as the graph adja- analysis. cency matrix, that respect the network structure can be important in the design of distributed algorithms. Hence, II.BACKGROUND an understanding of the eigenvalues of relevant random A. Random Graphs symmetric matrices is desired. For certain Hermitian random matrix models parameterized by matrix dimension An undirected graph (network) G consists of an N × N, results are available concerning the asymptotic ordered pair (V, E) in which V denotes a set of nodes behavior of the eigenvalues as N increases through a (vertices) and E denotes a set of undirected links function called the empirical spectral distribution. The (edges). These undirected links are unordered pairs definitions employed in this paper are provided below. {v , v } = {v , v } of nodes v , v ∈ V with v 6= v . i j j i i j i j Note that the eigenvalues of a Hermitian matrix must Note that self-loops, links formed as {v, v}, are excluded be real valued. Given a N × N Hermitian matrix-valued from this definition. The graph adjacency matrix A (G) random variable WN , order the N random eigenvalues encapsulates the graph information with A (G)ij = 1 such that λ1 (W ) ≤ ... ≤ λi (WN ) ≤ λi+1 (WN ) ≤ if {vj, vi} ∈ E and A (G)ij = 0 if {vj, vi} ∈/ E. ... ≤ λN (WN ). The empirical spectral measure of A subgraph Gsub of an undirected graph is a graph WN given by (Vsub, Esub) with Vsub ⊆ V and Esub ⊆ E, and G is supergraph G Random undirected said to be a of sub. N graphs, which are undirected graph-valued random vari- 1 X µw (X) = χ (λi (WN ) ∈ X) (3) ables, model uncertainty in the network topology. There N N i=1 are many commonly studied random graph models, which may be specified by a probability distribution on assigns to each subset X ⊆ R the fraction of eigenvalues the collection of possible link sets P (V × V) between a that appear in X using the indicator function χ. Related given number of nodes |V| = N. to this function, the empirical spectral distribution of One of the most frequently considered random graph WN given by [10] models, the Erdos-R¨ enyi´ model, considers a random graph G (N, p (N)) on N nodes formed by including ER FWN (x) = µWN ((−∞, x]) each link between two different nodes according to N 1 X (4) independent Bernoulli trials that succeed with proba- = χ (λ (W ) ≤ x) N i N bility p (N), which is often allowed to vary with N i=1 [14]. This distribution has important properties regarding asymptotic connectedness behavior as N increases that counts the number of eigenvalues in the interval condition on the relationship between N and p (N) (−∞, x], and the empirical spectral density of WN

3 is given by [10] almost surely [10]. Additionally, the sequence of deter- ◦ d ministic values gN (WN ) is called a deterministic equiv- fW (x) = FWN (x) alent of the sequence of random values g (W ) [10]. dx N N N (5) 1 X = δ (x − λ (W )) . C. Stochastic Canonical Equations N i N i=1 The most critical theorem for this work provides a indicates the locations of the eigenvalues using the Dirac method for finding deterministic equivalents of empirical delta function δ. The empirical spectral measure and spectral distribution functions for symmetric random empirical spectral distribution define each other and are matrices with independent entries, except for the rela- often referred to interchangeably. Note that each of these tion that the lower triangular entries are determined by definitions produces a function-valued random variable. symmetry from the upper triangular entries. Provided From them can be defined the expected empirical some regularity conditions (11), (12), and (13) hold, the Stieltjes transform (15) of a deterministic equivalent spectral measure µexp,WN = E [µWN ], the expected distribution function can be found by solving a system empirical spectral distribution Fexp,WN = E [FWN ], of equations (16) containing matrices [8]. Theorem 1 and the expected empirical spectral density fexp,WN = provides the formal statement below. This result and a E[fWN ]. Analysis of the empirical spectral distribution for Her- compilation of many other related results can be found mitian matrices often involves the Stieltjes transform in [8]. defined as below [10]. Theorem 1 (Girko’s K1 Equation [8]) Consider Z ∞ 1 a family of symmetric matrix valued random vari- SF (z) = dF (x) , Im {z}= 6 0 (6) ables W indexed by size N such that W is an −∞ x − z N N N × N symmetric matrix in which the entries on The Stieltjes transform can be inverted to obtain the the upper triangular region are independent. That corresponding density by computing the following ex- n o is, (W ) |1 ≤ i ≤ j ≤ N are independent with pression [10]. N ij (W ) = (W ) . Let W have expectation B = Z x N ji N ij N N 1 E[W ] and centralization H = W − E [W ] such F (x) = lim Im {SF (λ + i)} dλ (7) N N N N →0+ π −∞ that the following three conditions hold. Note that in 1 order to avoid cumbersome indexing, the index N will f (x) = lim Im {SF (x + i)} (8) →0+ π henceforth be omitted from most expressions involving WN , BN , and HN . For the empirical spectral distribution FN of an N × N N Hermitian matrix WN , the Stieltjes function computes X sup max |Bij| < ∞ (11) to i N j=1 1 −1 N SFW (z) = tr (WN −zIN ) , Im{z}= 6 0 (9) N N X 2 sup max E Hij < ∞ (12) i which is the normalized trace of the resolvent N j=1 −1 (WN − zIN ) [10]. N Given a sequence of Hermitian matrix-valued random X 2 lim max E Hijχ (|Hij| > τ) ∞ N→∞ i (13) variables {WN }N=1, the sequence has limiting spectral j=1 µ µ measure lim,WN provided WN converges weakly to = 0 for all τ > 0 µlim,W . Similarly, it has limiting spectral distribution N Then for almost all x, Flim,WN provided FWN converges weakly to Flim,WN . In some circumstances in which these limits may or may lim |FWN (x) − FN (x)| = 0 (14) not exist, the spectral statistics can still be describable by N→∞ a sequence of deterministic functions. Given a sequence almost surely, where FN is the distribution with Stieltjes of Hermitian matrix-valued random variables WN , a transform deterministic equivalent for WN with respect to the Z 1 SF (z) = dFN (x) sequence of functionals gN is a sequence of deterministic N x − z ◦ matrices W such that N (15) N 1 X ◦ = Ckk(z), Im {z}= 6 0 lim (gN (WN ) − gN (WN )) = 0 (10) N N→∞ k=1

4 and the functions Ckk (z) satisfy the canonical system complete graph. Denoting the adjacency matrix of a of equations complete graph on M vertices by KM , in terms of  Kronecker tensor products and adjacency matrices, these lattice graphs can be described by Ckk(z) =B−zI− ...

−1 (16) D D N !l,j=N  XO KMd j = d X 2  A(Glat) = Xdj,Xdj = (19) ... δlj Css(z)E Hjs  IM j 6= d  j=1d=1 d s=1 l,j=1 kk for k = 1,...,N. Note that the notation (·)l,j=N l,j=1 where the jth term in the summation contributes all indicates a matrix built from the parameterized contents complete graphs along the jth lattice dimension. Such of the parentheses, such that X = (X )l,j=N , and δ ij l,j=1 lj graphs are of interest because they can be grown to a is the Kronecker delta function. There exists a unique large number of nodes by increasing the lattice dimen- solution C (z) for k = 1,...,N to the canonical kk sion sizes while the number of distinct eigenvalues of system of equations among the class L = (16) the adjacency matrix does not increase. The eigenvalues {X(z) ∈ | X (z) analytic, Im {z} Im {X (z)} > 0}. C of the lattice graph adjacency matrix are Furthermore, if 2 inf N E Hij ≥ c > 0, (17) i,j D X then λ (j1, . . . , jD) = λd (jd) , d=1 lim sup |FW (x) − FN (x)| = 0 (18) (20) N→∞ N x Md − 1 jd = 0 λd(jd) = almost surely, where FN is defined as above. −1 jd = 1 In addition to the independence properties of the entries of the random matrix, Theorem 1 specifies three for j1, . . . , jD = 0, 1. Thus, the number of distinct conditions that must be verified. The first condition eigenvalues of a D-dimensional lattice graph depends (11) bounds the absolute sum of the entries of each only on the number of dimensions D and is at most 2D. row, and the second condition (12) bounds the total The adjacency matrices of random graphs distributed variance of entries of each row. The third condition (13) according to percolation models G (G , p) of D- is closely related to Lindberg’s condition of the central perc lat dimensional lattices are symmetric matrices with in- limit theorem. Under these conditions, a deterministic dependent entries, except for relations determined by equivalent F (x) for F (x) can be found by solving N WN symmetry. In more precise terms, the matrix entries the system of equations (16) and computing the Stieltjes A (G ) for i ≤ j are independent, and A (G ) = transform via (15), which can then be inverted. Under the ij perc ji perc A (G ). Thus, the scaled adjacency matrix eigenval- additional condition (17), the supremum converges [8]. ij perc ues of percolations models formed from these lattices can be analyzed by the tools provided in Theorem 1. III.MAIN RESULTS For a lattice with arbitrary number of dimensions D and This section examines, in particular, a random graph sizes Md for d = 1,...,D along with an arbitrary link model formed by percolation of a D-dimensional lattice inclusion probability p, Theorem 2 derives the form of graph. Note that there are many types of graphs known as the unique solution Ckk (z) for k = 1,...,N to the lattice graphs in other contexts, so the precise definition system of equations (16). An outline of the methods used will be introduced here. A D-dimensional lattice used can be found in the initial paragraph of the proof. graph with size Md along the dth dimension is a graph This result will subsequently be used in Corollary 1 to QD Glat = (V, E) in which the |V| = N = d=1 Md nodes compute the Stieltjes transform of a deterministically are identified with the ordered D-tuples in Z [1,M1] × equivalent distribution function for the empirical spectral ... × Z [1,MD] and in which two nodes are connected distribution. by a link if the corresponding D-tuples differ by exactly one symbol [12]. In this way, if the node indices are Theorem 2 (Solution Form for D-Lattice Percolation) fixed along D − 1 of the lattice dimensions, the node- Consider the D-dimensional lattice graph Glat with N induced subgraph along the remaining dimension is a nodes in which the dth dimension of the lattice had size

5 Md for d = 1,...,D such that the adjacency matrix is third paragraph uses symmetries in the random graph model to show that C (z), is, in fact, invariant under D D XO KMd j = d these permutations, yielding the result in the theorem. A(Glat) = Xdj,Xdj = (21) IM j 6= d j=1 d=1 d In order to simplify the calculations used to verify the D conditions of Theorem 1, a more explicit characterization Y N = Md. (22) of the entries Aij is ﬁrst introduced. Note that every d=1 integer 1 ≤ x ≤ N can be uniquely represented in a mixed-radix system as Form a random graph Gperc (Glat, p) by independently including each link of Glat with probability p, and denote the corresponding random scaled adjacency matrix W , D d−1  X Y expectation B, and centralization H x = 1 + β (x, d)  Mj (29) 1 d=1 j=1 W (G ) = A (G ) (23) perc γ perc

B = E [W (Gperc)] (24) for integers 0 ≤ β (x, d) ≤ Md −1. Hence, node indices H (G ) = W (G ) − E[W (G )] (25) (i, j) can be described by their digit sequences β (i) = perc perc perc D D {β (i, d)}d=1 and β (j) = {β (j, d)}d=1 in this system. where Let kβ (i) − β (j)k0 count the differences between these D digit sequences. Using this to describe the entries of the X γ = γ (M1,...,MD) = (Md − 1) . (26) supergraph adjacency matrix yields d=1

Let Ckk (z) for k = 1,...,N be the unique solution 1 kβ (i) − β (j)k = 1 A (G ) = 0 . (30) to the system of equations (16) among the class L ij lat 0 otherwise guaranteed to exist by Theorem 1, and write

 p Because Bij = γ Aij (Glat), the following computation C (z) = B − zIN − ... veriﬁes the ﬁrst condition (11) of Theorem 1.

−1 (27) N !l,j=N  N X 2 X X ... δlj Css (z)E Hjs  . sup max |Bij| = sup max |Bij| N i N i s=1 l,j=1 j=1 kβ(i)−β(j)k0=1 D (31) Note that Ckk (z) is the kth diagonal entry of C (z) and p X = sup (Md − 1) that uniqueness of Ckk (z) implies uniqueness of C (z). N γ For some values of α (z) for i , . . . , i = 0, 1 d=1 i1,...,iD 1 D = p < ∞ 1 D X O C (z) = αi1,...,iD (z) Ydid , 2 p(1−p) i1,...,iD =0 d=1 (28) Noting that E [( Hij xy = γ2 if (Aij (Glat))xy = 1 2 K i = 0 and that E [( Hij = 0 if (Aij (Glat)) = 0, the Y = Md d . xy xy did following computation verifies the second condition (12) IMd id = 1 of Theorem 1. Proof: The theorem is proven primarily by use of symmetry N in the random graph model and is organized in three X 2 X 2 sup max E Hij = sup max E Hij parts below. The first paragraph verifies that the three N i N i j=1 kβ(i)−β(j)k =1 conditions of Theorem 1 hold for the random matrix 0 D ! (32) model W (Gperc), guaranteeing existence of a unique p (1 − p) X = sup 2 (Md − 1) solution Ckk (z) for k = 1,...N to (16). Subsequently, N γ the second paragraph demonstrates that C (z) is of the d=1 ≤ p (1 − p) < ∞ desired form if and only if it is invariant to permutations that factor into Kronecker products of smaller permutations along each lattice dimension. Finally, the Finally, the following computation demonstrates that the

6 third condition (13) of Theorem 1 holds for any τ > 0. but that C(z)x1,y1 6= C(z)x2,y2 . Let ψ1, . . . , ψD be N permutations such that ψd (β (x1, d) + 1)−1 = β (x2, d) X 2 and ψd (β (y1, d) + 1) − 1 = β (y2, d) with correspond- lim max E Hijχ (|Hij| > τ) ≤ N→∞ i j=1 ing row permutation matrices Qd. Let Q = Q1 ⊗...QD be the row permutation matrix corresponding to permu-  2 2 X max p , (1 − p) tation ψ. Then lim max E  ... (33) N→∞ i γ2 kβ(i)−β(j)k0=1 >   QC (z) Q = C (z)ψ(x )ψ(y ) max (p, 1 − p) x1y1 1 1 (39) . . . χ  > τ = 0 = C (z) 6= C (z) . γ x2y2 x1y1

This is true because Hence, C (z) is of the form in (28) if and only if C (z) = max (p, 1 − p) PC (z) P > for all P = P ⊗ ... ⊗ P where P is a lim = 0, (34) 1 D d N→∞ γ row permutation matrix on Md symbols. so when γ becomes sufﬁciently large Let φd be a permutation function on Md symbols with corresponding row permutation matrix P for d = max (p, 1 − p) d lim χ > τ = 0. (35) 1,...,D. Also, let P = P1 ⊗ ... ⊗ PD, thus forming N→∞ γ a permutation matrix that operates on each lattice dimension. The following equations result from permuting Note that because Hij = 0 when kβ (i) − β (j)k0 6= 1 the additional condition (17) for the stronger conclusion the rows and columns of C (z) and by manipulating the of Theorem 1 does not hold. matrix inverse, expectations, and transposes. Given that C(z) is of the form in (28), let φd be a permutation function on Md symbols with corresponding  row permutation matrix Pd for d = 1,...,D and P = > PC(z)P = P B−zIN −... P1 ⊗ ... ⊗ PD with corresponding permutation function φ. It is clear that l,j=N −1   N ! 1 D X 2 > > X O > ... δlj Css(z)E Hjs  P PC(z)P = P αi ,...,i (z) Ydi P  1 D d  s=1 l,j=1 i1,...,iD =0 d=1  (40) 1 D > > X O > = PBP −zP IN P −... = αi1,...,iD (z) PdYdid Pd (36) i1,...,iD =0 d=1 −1 1 D N !l,j=N  X O X h 2 i = αi1,...,iD (z) Ydid = C(z). ... δlj Css(z)E Hφ(j)s  i1,...,iD =0 d=1 s=1 l,j=1 Note, again, that every integer 1 ≤ x ≤ N can be uniquely represented as Note that for the underlying lattice graph, > D d−1  PA (Glat) P = A (Glat). Hence W (Gperc) has X Y the same distribution as PW (G ) P > as all edge x = 1 + β (x, d)  Mj (37) perc d=1 j=1 inclusions are independent and identically distributed. Therefore, for integers 0 ≤ β (x, d) < Md and that > D d−1  B = PBP . (41) X Y φ(x) = 1+ (φd (β (x,d)+1)−1) Mj. (38) d=1 j=1 Furthermore, in terms of matrix entries, If C(z) is of the form in (28) then C(z)xy = Wφ(x)φ(y) (Gperc) has the same distribution as

αδβ(x,1)β(y,1),...,δβ(x,D)β(y,D) (z) . If C(z) is not of the Wxy (Gperc). Hence, Hφ(j)s has the same distribution as form in (28) then there are some x1, y1 and x2, y2 such Hjφ−1(s). By applying these two identities and changing that δβ(x1,d)β(y1,d) = δβ(x2,d)β(y2,d) for d = 1,...,D the order of summation by φ, the following equations

7 D result. where the 2 complex variables αi1,...,iD (z) for  i1, . . . , iD = 0, 1 solves the system PC(z)P > = B−zI −...  N 1 D X Y αi1,...,iD (z) λdid (jd) = l,j=N −1 N ! i1,...,iD =0 d=1 X h 2 i ... δ C (z)E H −1 D ! lj ss jφ (s)  p X s=1 λ (j ) −z−... (45) l,j=1 γ d0 d  (42) d=1 D ! !−1 = B−zIN −... p(1−p) X ... (M −1) α (z) γ2 d 1,...,1 −1 d=1 N !l,j=N  X 2 ... δlj Cφ(s)φ(s)(z)E Hjs  D of 2 rational equations for j1, . . . , jD = 0, 1 where s=1 l,j=1  M − 1 i = 0, j = 0 The entries Cφ(k)φ(k) (z) are the kth diagonal entries of  d d d > PC (z) P for k = 1,...,N and, thus, a solution to the λdid (jd) = −1 id = 0, jd = 1 . (46) system of equations (16). However, because the solution  1 id = 1 is unique, this implies that Proof: C (z) = PC (z) P >. (43) Using the Stieltjes transform expression (15) from Theorem 1 and the derived form for C(z) in (28) from Therefore, C (z) is of the form shown in (28). Theorem 2, the Stieltjes transform of the deterministic equivalent distribution function F specified in Theorem While Theorm 2 specifies the form of the unique N 1 for the empirical spectral distribution of W (G ) is solution to the system of equations (16), it does not perc D computed as describe how to find the 2 parameters αi1,...,iD (z) that correspond to a given set of lattice dimensions N and link probability parameter that are necessary to find 1 X S (z) = C (z) the Stieltjes transform of the deterministically equivalent FN N kk distribution function F . Corollary 1 shows that the k=1 (47) N N 1 X Stieltjes transform of FN is equal to the diagonal element = α (z) = α (z) . of C(z) described in Theorem 2. It also derives the N 1,...,1 1,...,1 k=1 system of 2D nonlinear equations to which the parameters are a solution, accomplishing this by computing In order to derive the system of equations describing the terms of (27) and noting that all computed matrices the coefficients α for i , . . . , i = 0, 1, the value are simultaneously diagonalizable. The matrix equation i1,...,iD 1 D of the third matrix in the denominator of (27) must first in (27) then transforms into a system of equations that be computed. Subsequently, the system of equations may describes the eigenvalues, which appears in (45) of be found by noting that all matrices in equation (16) may Corollary 1. Remark 1 describes use of an iterative ap- be simultaneously diagonalized by orthogonal matrices proach to find the solution. For added clarity, please note of eigenvectors. Each distinct eigenvalue of C (z) will that i , . . . , i = 0, 1 index the parameters α (z) 1 D i1,...,iD result in an equation that the coefficients must satisfy. Let while j , . . . , j = 0, 1 index the equations that describe 1 D β be defined as in the proof of Theorem 2. Summing the the parameters. variance over each row yields Corollary 1 (Stieltjes Transform for D-Lattice Per- colation) The Stieltjes transform of the deterministic N N X 2 X 2 equivalent distribution function Fn specified in Theorem Css(z)E Hjs = Css(z)E Hjs 1 for the empirical spectral distribution of W (G ) is s=1 kβ(s)−β(j)k =1 perc 0 (48) given by D ! p(1−p) X = (M −1) α (z) γ2 d 1,...,1 SFN (z) = α1,...,1 (z) , Im (z) 6= 0 (44) d=1

8 and, therefore, that for the fixed point α. While convergence of this process is not proven, it appears to work well in practice. N !l,j=N X 2 δlj Css (z)E Hjs = Figure 2 and Figure 3 provide example deterministic s=1 l,j=1 (49) equivalents for the empirical spectral distribution of D ! scaled adjacency matrices for percolation models of p (1 − p) X (Md − 1) α1,...,1 (z) IN . lattice graphs with different lattice characteristics and γ2 d=1 link inclusion parameters as calculated using Corollary 1 Note that C(z), B, and the matrix computed in (49) and the methods in Remark 1. The corresponding density are simultaneously diagonalizable by Kronecker product function is also computed, and the expected empirical spectral distributions and densities are shown for com- properties. Let λdi (0) be the eigenvalue of Ydi cor- d d parison. Asymptotically, as each lattice dimension size responding to eigenvector v = 1. Let λdi (1) be the d grows without bound, the area of the largest region of the eigenvalue of Ydid corresponding to any other eigenvector orthogonal to v = 1. Hence, density function approaches 1, and the area of the other regions approach 0. Thus, Theorem 1 does not guar-   Md − 1 id = 0, jd = 0 antee the deterministic distribution function reflects the λdid (jd) = −1 id = 0, jd = 1 . (50) empirical spectral distribution in the regions of smaller  1 id = 1 increase. Nevertheless, the observed characteristics seem to provide a good approximation. The eigenvalues of C (z) are indexed by j1, . . . , jD = 0, 1 and are given by Finally, Theorem 3 describes the relationship between the empirical spectral distribution of the row-normalized 1 D −1 X Y adjacency matrix Ab(Gperc) = ∆ (Gperc) A (Gperc), α (z) λ (j ) (51) i1,...,iD did d where ∆ (Gperc) is the diagonal matrix of node degrees, i1,...,iD =0 d=1 and the empirical spectral distribution of the scaled 1 The corresponding eigenvalues of B are given by adjacency matrix W (Gperc) = γ A (Gperc). Because the row-normalized adjacency matrix and the symmet- D ! p X rically normalized adjacancy matrix are similar, the λ (j ) (52) γ d0 d result also applies to both of those matrices. With d=1 some manipulation, it can also be used to describe The corresponding eigenvalues of the matrix in (49) are the row-normalized Laplacian and the symmetrically given by normalized Laplacian. The theorem shows that the Levy´ √ D ! metric distance dL F √ ,F γW between F √ and p (1 − p) X p γAb p γAb (M − 1) α (z) . (53) F√ approaches 0 asymptotically. The proof follows γ2 d 1,...,1 γW d=1 that of Lemma 5 of [13] almost identically, with only Of course, the corresponding eigenvalue of zI is z. some complications in applying Lemma 2.3 of [17] to Through applying this diagonalization to the equa- bound the matrix entry differences. Consequently, the tion (16), the system of 2D equations indexed by deterministic equivalent FN computed in Corollary 1 can D be used to approximate the empirical spectral distribution j1, . . . , jD = 0, 1 that the 2 coefficients αi ,...,i 1 D of A as well. indexed by i1, . . . , iD = 0, 1 satisfy is described by b equation (45). 1 Theorem 3 Let W (Gperc) = γ A (Gperc) be the scaled adjacency matrix of Gperc (Glat, p) for scale factor PD Remark 1 (Iterative Computation of Coefficients) Com- γ = d=1 (Md − 1) with empirical spectral distribution F , and let A (G ) = ∆−1 (G ) A (G ) be the putation of the values of αi1,...,iD for i1, . . . , iD = 0, 1 W b perc perc perc D can be approached iteratively. Let the 2 dimensional row-normalized adjacency matrix of Gperc (Glat, p) with empirical spectral distribution F , where ∆ (G ) is vector α collect the values of αi1,...,iD for i1, . . . , iD = Ab perc 0, 1. Then the system of equations in (45) can be the diagonal matrix of node degress. Also let dL (·, ·) be written as Xα = g (α) where X is the invertible the Levy´ distance metric. Assume that all of the lattice matrix describing the left side of the equation and g is dimension sizes increase without bound as N → ∞. the function describing the right side of the equation. Then, Selecting an initial guess α and iterating such that 0 lim d F √ ,F√ = 0. (54) −1 L p γAb γW αn+1 = X g (αn) is a possible approach to search N→∞

9 Fig. 2: For a graph percolation model Gperc based on a two dimensional lattice with size 20 nodes × 40 nodes and link inclusion probability p = .7, the expected empirical spectral distribution E[FWN ] of the scaled adjacency 1 4 matrix WN = γ A (Gperc) was computed over 10 Monte-Carlo trials (black dotted line, top left). The deterministic distribution FN computed through Corollary 1 was also computed (blue line, top left). The corresponding density functions appear on the top right with labeled sections of interest more closely examined on the bottom row.

Proof: where maxl |δi,d,l| = o(1). Thus, note that

Following the proof methods of Lemma 5 of [13], the Md X row sums ∆ii are first computed as γ (p + i) where Ql,s (i, d) = (Md − 1) (p + κi,d,l) (56) = maxi |i| = o (1). Subsequently, the bound on s=1 the cubed Levy´ distance found in Lemma 2.3 of [18] where (p + δi,d,l − 1) /Md ≤ κi,d,l ≤ (p + δi,d,l) /Md is evaluated and shown to asymptotically approach 0. so maxl |κi,d,l| = o(1). Hence, ∆ii may be computed Consequently, the result holds. The first of these steps is as less straightforward than the equivalent in [13]. N D M X X Xd Let β (x, d) be defined as before. For a given node ∆ii = Aij = Qβ(i,d),s (i, d) index i and lattice dimension d, form a Md × Md j=1 d=1 s=1 matrix Q (i, d) from all entries Alj such that β (l, k) = D (57) X β (j, k) = β (i, k) for k 6= d. Note that this matrix is = (Md − 1) p + κi,d,β(i,d) symmetric with zeros on the diagonal and identically d=1 distributed Bernoulli random variables with parameter = γ (p + i) p for all entries off the diagonal. These variables are where independent, except for symmetry relations. Form a D 1 X Md × Md matrix R (i, d) equal to Q (i, d) on the off = (M − 1) κ . (58) i γ d i,d,β(i,d) diagonal but with an identical Bernoulli random variable d=1 added to the diagonal element. By Lemma 2.3 of [17], Note that

max |i| ≤ max max max |κi,d,l| (59) i i d l M Xd R (i,d) = M (p+δ ) so = maxi |i| = o (1). By Lemma 2.3 of [18], l,s d d,l (55) s=1 1 √ √ 2 √ √ dL Fp γA,F γW ≤ γW − p γAb . (60) = (Md−1)(p+δi,d,l)+(p+δi,d,l) b n F

10 Fig. 3: For a graph percolation model Gperc based on a two dimensional lattice with size 15 nodes × 20 nodes ×

25 nodes and link inclusion probability p = .8, the expected empirical spectral distribution E[FWN ] of the scaled 1 4 adjacency matrix WN = γ A (Gperc) was computed over 10 Monte-Carlo trials (black dotted line, top left). The deterministic distribution FN computed through Corollary 1 was also computed (blue line, top left). The corresponding density functions appear on the top right with labeled sections of interest more closely examined on the bottom row.

1 Note that Wij = γ Aij and that for instance, in polynomial ﬁlter design for graph signal p processing. This paper examines the empirical spectral pAbij = Aij distribution of the scaled adjacency matrix eigenvalues ∆ii p for a random graph model formed by independently in- = Aij cluding or excluding each link of a D-dimensional lattice γ (p + i) 1 (61) supergraph, a percolation model. A stochastic canonical = Aij (1 + O (i)) equations theorem provided by Girko can be applied to γ 1 ﬁnd a sequence of deterministic functions that asymptot- = A (1 + O ()) ically approximates the empirical spectral distribution as γ ij the number of nodes increases. Through this theorem, the Hence, Stieltjes transforms of these deterministic distributions

2 N can be found by solving a certain system of equations. √ √ γ X 1 2 2 γW − p γAb = AijO The primary contributions of this paper focus on solv- F n γ2 i,j=1 ing this system of equations for the random network (62) N model studied with arbitrary lattice and link inclusion 1 X = A2 O 2 → 0 parameters. Specifically, Theorem 2 finds the form of the nγ ij i,j=1 matrix that satisfies the system of equations, resulting in D because = o (1). Therefore, it has been shown that a matrix with 2 parameters. Subsequently, Corollary 1 derives a system of 2D equations that describe these d F √ ,F√ → 0. L p γAb γW parameters, significantly reducing the original system of equations. The resulting distributions can also be related IV. CONCLUSION to the empirical spectral distribution of the symmetrically The eigenvalues of matrices that encode network in- normalized adjacency matrix, row normalized adjacency formation, such as the adjacency matrix and the Lapla- matrix, symmetrically normalized Laplacian, and row- cian, provide important information that can be used, normalized Laplacian. Simulations for sample model

11 parameters validate the results. While the methods used [17] C. Bordenave, P. Caputo, and D. Chafa¨ı. Spectrum of Large Ran- guarantee that the computed deterministic distribution dom Reversible Markov Chains: Two Examples, Latin American Journal of Probability and Mathematical Statistics, vol. 7, pp. function asymptotically captures the behavior of the bulk 41-64, 2010. of the eigenvalues, no guarantees are made about regions [18] Z. Bai. Methodologies in Spectral Analysis of Large Random where the fraction of eigenvalues asymptotically van- Matrices, A Review, Statistica Sinica, vol. 9, no. 3, pp. 611-677, July 1999. ishes. Although the observed correspondence between the computed and simulated distributions appears good, future efforts could include a more precise characterization. Additional topics for possible expansion could also include evaluation of the degree to which ﬁlter design beneﬁts from the distribution information gained as well as analysis of additional random network models of interest.

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