Spectra of Modular Random Graphs

TL;DR

This paper analyzes the spectral properties of symmetric random matrices on modular graphs, revealing how inter-module connectivity influences spectral densities, including semicircular and triangular shapes, with analytical and numerical validation.

Contribution

It provides new analytical insights into the spectra of modular random graphs with different inter-module connectivity regimes, extending understanding beyond fully connected modules.

Findings

Spectral densities can be semicircular or triangular depending on connectivity.

Analytical results match well with numerical simulations.

Different inter-module connectivities lead to distinct spectral behaviors.

Abstract

We compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure. Modules are initially introduced as fully connected sub-units of a graph. By contrast, inter-module connectivity is taken to be incomplete. Two different types of inter-module connectivity are considered, one where the number of intermodule connections per-node diverges, and one where this number remains finite in the infinite module-size limit. In the first case, results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules. In the second case, matters can be more involved, and depend in detail on inter-module connectivities. For suitable parameters we even find near-triangular shaped spectral densities, similar to those observed in certain scale-free networks, in a system of consisting of just two coupled modules. Analytic results are presented for the infinite module-size limit; they are well corroborated by numerical simulations.