Effects of clustering heterogeneity on the spectral density of sparse networks
arXiv:2404.08152 · doi:10.1103/PhysRevE.110.054307
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.
13 pages, 7 figures
References in corpus (11)
- Random graphs with clustering
- Graph spectra and the detectability of community structure in networks
- Random graphs containing arbitrary distributions of subgraphs
- Introduction to Random Matrices - Theory and Practice
- Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
- The robustness of interdependent clustered networks
- Spectra of random graphs with arbitrary expected degrees
- On the localization transition in symmetric random matrices
- Spectra of random networks with arbitrary degrees
- Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties
- Multifractality and statistical localization in highly heterogeneous random networks