Empirical spectral distributions of sparse random graphs

TL;DR

This paper investigates the spectral distribution of sparse random graphs generated by the configuration model, showing convergence to a limit described by free probability and establishing connections with classical random matrix laws.

Contribution

It provides a new limit law for the spectral distribution of sparse graphs, linking it to free multiplicative convolution and classical random matrix results.

Findings

Spectral distribution converges to a free multiplicative convolution of the degree distribution and semicircle law.

Continuity of the spectral density away from zero is established.

An effective procedure for determining the support of the limiting distribution is developed.

Abstract

We study the spectrum of a random multigraph with a degree sequence ${\bf D}_n=(D_i)_{i=1}^n$ and average degree $1 \ll \omega_n \ll n$, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of $\omega_n^{-1} {\bf D}_n $ converges weakly to a limit $\nu$, under mild moment assumptions (e.g., $D_i/\omega_n$ are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to $\nu\boxtimes \sigma_{\rm sc}$, the free multiplicative convolution of $\nu$ with the semicircle law. Relating this limit with a variant of the Marchenko--Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous Erd\H{o}s-R\'enyi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.