Eigenvalue ratio statistics of complex networks: Disorder vs. Randomness

TL;DR

This paper studies how the eigenvalue ratio distribution of various complex networks transitions from GOE to Poisson statistics as diagonal disorder increases, revealing links to network dynamics and disorder strength.

Contribution

It numerically analyzes the eigenvalue ratios distribution in different network models and relates the critical disorder to network architecture and random walk dynamics.

Findings

Eigenvalue ratios follow GOE statistics without disorder.

Adding disorder causes a transition to Poisson statistics.

Critical disorder depends on network randomness and affects network dynamics.

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, Erd\H{o}s-R\'enyi 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 the 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.