Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties

TL;DR

This paper analytically solves the resolvent equations for heterogeneous random graphs with arbitrary degree distributions, revealing how degree fluctuations influence spectral density, eigenvector extension, and local density of states.

Contribution

It provides an exact analytical solution for the resolvent equations of high-connectivity random graphs with arbitrary degree distributions, a problem previously unresolved.

Findings

Eigenvectors are all extended.

Spectral density diverges logarithmically or as a power law with large degree variance.

Local density of states exhibits a power-law tail influenced by degree variance.

Abstract

The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states. We show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the local density of states at the center of the spectrum displays a power-law tail determined by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs.