Spectral Properties of Directed Random Networks with Modular Structure

TL;DR

This paper analyzes the spectral properties and eigenvector localization in directed random networks with modular structures, revealing how correlation, directionality, and community structure influence eigenvector localization and spectral distribution.

Contribution

It introduces a detailed analysis of how correlation and community structure affect spectral properties and eigenvector localization in directed networks, including real metabolic network data.

Findings

Correlated entries lead to localized eigenvectors.

Eigenstates form complex conjugate pairs as directionality increases.

Community structure causes spectrum localization, sensitive to network deformation.

Abstract

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different value of correlation among their entries. Spectra of random networks, with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, where as networks with correlated entries have localized eigenvectors. In order to understand the origin of localization we track the spectra as a function of connection probability and directionality. As connections are made directed, eigenstates start occurring in complex conjugate pairs and the eigenvalue distribution combined with the localization measure shows a rich pattern. Moreover, for a very well distinguished community structure, the whole spectrum is localized except few eigenstates at boundary of the circular distribution. As the network deviates from the community structure there is a sudden change in the localization property for a very small value of deformation from the perfect community structure. We search for this effect for the whole range of correlation strengths and for different community configurations. Furthermore, we investigate spectral properties of a metabolic network of zebrafish, and compare them with those of the model networks.