The spectral density of dense random networks and the breakdown of the Wigner semicircle law

TL;DR

This paper investigates how network topology influences the spectral density of dense random networks, revealing conditions under which the Wigner semicircle law breaks down and identifying universal spectral behaviors.

Contribution

It introduces a relation between eigenvalue moments and degree variance, providing a new classification of spectral density behaviors in dense networks.

Findings

Spectral density depends on degree fluctuations in dense networks.

Exponential degree distributions lead to a logarithmic singularity in spectral density.

A criterion for the breakdown of the Wigner semicircle law is established.

Abstract

Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree distributions, we show that the spectral density of the adjacency matrices of dense random networks is determined by the strength of the degree fluctuations. In particular, the eigenvalue distribution of dense networks with an exponential degree distribution is governed by a simple equation, from which we uncover a logarithmic singularity in the spectral density. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for dense random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviours of the spectral density in the dense limit. Our theoretical findings should lead to important insights on the mean-field behaviour of models defined on graphs.