Eigenvalue ratio statistics of complex networks: Disorder versus randomness

Abstract

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (wc) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate wc with the time taken by maximum entropy random walker to reach the steady-state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behaviour.

Spacing ratio statistics of multiplex directed networks

“…Since average mean spacing becomes insignificant in calculating spacing ratios, no unfolding is required. Over the years, the spacing ratio has been widely used to study the spectral statistics of several many-body systems, including complex networks [7][8][9].…”