Effects of clustering heterogeneity on the spectral density of sparse networks

TL;DR

This paper derives exact equations for the spectral density of sparse networks with clustering, revealing how heterogeneity affects spectral symmetry and highlighting limitations of traditional approximations in complex motifs.

Contribution

It introduces a systematic method to analyze the spectral density of clustered sparse networks with arbitrary degree distributions, including heterogeneity effects.

Findings

Spectral density becomes more symmetric with increased triangle-degree fluctuations.

Traditional large-degree approximations fail in high-connectivity, regular clustered networks.

Numerical simulations confirm the theoretical spectral density predictions.

Abstract

We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effect of clustering on the spectral properties of the network adjacency matrix. In the case of heterogeneous networks, we demonstrate that the spectral density becomes more symmetric as the fluctuations in the triangle-degree sequence increase. This phenomenon is explained by the small clustering coefficient of networks with a large variance of the triangle-degree distribution. In the homogeneous case of regular clustered networks, we find that both perturbative and non-perturbative approximations fail to predict the spectral density in the high-connectivity limit. This suggests that traditional large-degree approximations may be ineffective in studying the spectral properties of networks with more complex motifs. Our theoretical results are fully confirmed by numerical diagonalizations of finite adjacency matrices.