Random matrix analysis of network Laplacians

TL;DR

This paper applies random matrix theory to analyze the eigenvalue fluctuations of network Laplacians across different network types, revealing a transition to GOE statistics at the small-world transition.

Contribution

It demonstrates that the Laplacian eigenvalues of various networks follow GOE statistics and identifies the transition point at the small-world phase change.

Findings

Laplacian eigenvalues follow GOE statistics in various networks

Transition to GOE occurs at the small-world transition

Eigenvalue fluctuations depend on network randomness

Abstract

We analyze eigenvalues fluctuations of the Laplacian of various networks under the random matrix theory framework. Analyses of random networks, scale-free networks and small-world networks show that nearest neighbor spacing distribution of the Laplacian of these networks follow Gaussian orthogonal ensemble statistics of random matrix theory. Furthermore, we study nearest neighbor spacing distribution as a function of the random connections and find that transition to the Gaussian orthogonal ensemble statistics occurs at the small-world transition.