Deformed Gaussian Orthogonal Ensemble description of Small-World networks

TL;DR

This paper uses a Deformed Gaussian Orthogonal Ensemble (DGOE) model to analyze the spectral transition from regular to chaotic behavior in Small-World networks, demonstrating DGOE's effectiveness in network physics.

Contribution

It introduces the DGOE as a natural extension of GOE to model spectral correlations in Small-World networks, especially during the transition to chaos.

Findings

SW networks follow GOE statistics within certain eigenvalue ranges

DGOE accurately models spectral correlations beyond GOE regimes

DGOE proves useful in understanding network spectral behavior

Abstract

The study of spectral behavior of networks has gained enthusiasm over the last few years. In particular, Random Matrix Theory (RMT) concepts have proven to be useful. In discussing transition from regular behavior to fully chaotic behavior it has been found that an extrapolation formula of the Brody type can be used. In the present paper we analyze the regular to chaotic behavior of Small World (SW) networks using an extension of the Gaussian Orthogonal Ensemble. This RMT ensemble, coined the Deformed Gaussian Orthogonal Ensemble (DGOE), supplies a natural foundation of the Brody formula. SW networks follow GOE statistics till certain range of eigenvalues correlations depending upon the strength of random connections. We show that for these regimes of SW networks where spectral correlations do not follow GOE beyond certain range, DGOE statistics models the correlations very well. The analysis performed in this paper proves the utility of the DGOE in network physics, as much as it has been useful in other physical systems.