Spectral density of random graphs with topological constraints

TL;DR

This paper analyzes how topological constraints like degree correlations and community structures influence the spectral density of random graphs, providing exact solutions via the replica method.

Contribution

It introduces a unified approach to derive exact spectral densities for complex graph ensembles with topological constraints using the replica method.

Findings

Spectral densities depend significantly on degree correlations and community structures.

Exact solutions are obtained through consistency equations tailored to each ensemble.

Topological constraints markedly alter the spectral properties of the graphs.

Abstract

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting spectral density.