Random matrix analysis of complex networks

TL;DR

This paper applies random matrix theory to analyze the eigenvalue spectra of different complex network models, revealing universal statistical behaviors and differences in long-range correlations.

Contribution

It introduces a comprehensive RMT-based spectral analysis of various complex network models, highlighting universal and distinct eigenvalue statistics.

Findings

Eigenvalue spacing distributions follow Gaussian orthogonal ensemble statistics.

Spectral rigidity $ ext{Delta}_3$ matches RMT predictions at large scales.

Small-world networks deviate from RMT predictions at smaller scales.

Abstract

We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of adjacency matrix of various model networks, namely, random, scale-free and small-world networks. These distributions follow Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via $\Delta_3$ statistic of RMT as well. It follows RMT prediction of linear behavior in semi-logarithmic scale with slope being $\sim 1/\pi^2$. Random and scale-free networks follow RMT prediction for very large scale. Small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.