Eigenvalue spectra and stability of directed complex networks

TL;DR

This paper derives analytical expressions for the eigenvalue spectra of large, directed, weighted networks with degree heterogeneity, providing insights into how network structure influences system stability.

Contribution

It extends previous spectral results by incorporating degree heterogeneity, deriving modified laws and explicit spectra for complex directed networks.

Findings

Derived closed-form eigenvalue spectra for weighted directed networks.

Modified classical random matrix laws to account for network heterogeneity.

Provided an analytical eigenvalue density for directed Barabasi-Albert networks.

Abstract

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build upon previous results, which usually only take into account the mean degree of the network, by allowing for non-trivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulae for the corrections (due to non-zero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semi-circle law, the Girko circle law and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of an directed Barabasi-Albert network. We are thus able to make general deductions about the effect of network heterogeneity on the stability of complex dynamic systems.