On the spectral distribution of large weighted random regular graphs

TL;DR

This paper investigates the spectral distribution of large weighted random regular graphs, establishing the existence of a unique eigendistribution and showing it differs slightly from the semi-circle law, with implications for understanding spectral measures.

Contribution

It introduces the concept of eigendistribution for weighted regular graphs and proves it differs from the semi-circle distribution, with detailed moment analysis.

Findings

The eigendistribution exists uniquely for weighted regular graphs.

It closely approximates the semi-circle law but differs in higher moments.

The difference in moments is on the order of 1/d^2, indicating small deviations.

Abstract

McKay proved that the limiting spectral measures of the ensembles of $d$-regular graphs with $N$ vertices converge to Kesten's measure as $N\to\infty$. In this paper we explore the case of weighted graphs. More precisely, given a large $d$-regular graph we assign random weights, drawn from some distribution $\mathcal{W}$, to its edges. We study the relationship between $\mathcal{W}$ and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique `eigendistribution', i.e., a weight distribution $\mathcal{W}$ such that the associated limiting spectral distribution is a rescaling of $\mathcal{W}$. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is $O(1/d^2)$). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.