Spectral Statistics of Lattice Graph Structured, Non-uniform Percolations

TL;DR

This paper derives deterministic spectral distribution approximations for random lattice graphs with non-uniform percolation, aiding graph filter design by understanding spectral properties under stochastic structural influences.

Contribution

It introduces a novel application of Girko's stochastic canonical equations to analyze spectral distributions of non-uniformly percolated lattice graphs, providing new theoretical insights.

Findings

Deterministic equivalents approximate spectral distributions effectively.

Simulations confirm theoretical predictions for various parameters.

Results facilitate improved graph filter design under stochastic network conditions.

Abstract

Design of filters for graph signal processing benefits from knowledge of the spectral decomposition of matrices that encode graphs, such as the adjacency matrix and the Laplacian matrix, used to define the shift operator. For shift matrices with real eigenvalues, which arise for symmetric graphs, the empirical spectral distribution captures the eigenvalue locations. Under realistic circumstances, stochastic influences often affect the network structure and, consequently, the shift matrix empirical spectral distribution. Nevertheless, deterministic functions may often be found to approximate the asymptotic behavior of empirical spectral distributions of random matrices. This paper uses stochastic canonical equation methods developed by Girko to derive such deterministic equivalent distributions for the empirical spectral distributions of random graphs formed by structured, non-uniform percolation of a D-dimensional lattice supergraph. Included simulations demonstrate the results for sample parameters.