Spectral properties of the Laplacian of a generalized Wigner matrix

TL;DR

This paper analyzes the spectral distribution of Laplacian matrices from generalized Wigner matrices with variance profiles, establishing convergence results and explicit limits for various random graph models.

Contribution

It extends spectral analysis to Laplacians with variance profiles derived from graphons, providing convergence of spectral distributions and explicit formulas in special cases.

Findings

Spectral distribution converges under graphon limits.

Explicit formulas for moments in certain cases.

Order of the maximum eigenvalue derived.

Abstract

In this article we consider the spectrum of a Laplacian matrix, also known as the Markov matrix, under the independence assumption. We assume that the entries have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution of the scaled Laplacian converge. We give an expression for the moments of the limiting measure in terms of graph homomorphisms. In some special cases we identify the limit explicitly. We also study the spectral norm and derive the order of the maximum eigenvalue. We show that our results cover Laplacian of various random graphs include inhomogeneous Erd\H{o}s- R\' enyi random graph, sparse W-random graphs, stochastic block matrices and constrained random graphs.