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

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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 struc- ture 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 jλjp. 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 jP (λi(Lb))j < 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 pro- of 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.
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