Spectra of Modular and Small-World Matrices

TL;DR

This paper analyzes the spectral properties of symmetric random matrices representing modular and small-world networks, revealing how network structure influences spectral density, localized states, and satellite bands.

Contribution

It provides a comprehensive spectral analysis of modular and small-world network matrices, including Lifshitz singularities and localized states, using a cavity approach and numerical validation.

Findings

Spectral density exhibits Lifshitz type singularities near zero eigenvalues.

Modular networks show local density contributions from individual modules.

Small-world networks develop satellite bands outside the main spectral band.

Abstract

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing short-cuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz type singular behaviour of the spectral density in a localised region of small $|\lambda|$ values. In the case of modular networks, we can identify contributions local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localised states in the band-gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalisation results.