Spectra of random graphs with arbitrary expected degrees

TL;DR

This paper derives exact spectral properties of large random graphs with arbitrary expected degree distributions, revealing how hubs create isolated eigenvalues and localized eigenvectors.

Contribution

It provides a complete, exact method for calculating spectra of such graphs and analyzes the impact of hubs on spectral properties.

Findings

Hubs produce isolated eigenvalues outside the main spectral band.

Eigenvectors associated with hubs are strongly localized.

Numerical results confirm the analytical spectral expressions.

Abstract

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit of large network size and large vertex degrees. We also study the effect on the spectra of hubs in the network, vertices of unusually high degree, and show that these produce isolated eigenvalues outside the main spectral band, akin to impurity states in condensed matter systems, with accompanying eigenvectors that are strongly localized around the hubs. We also give numerical results that confirm our analytic expressions.