  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.
