Small-world spectra in mean field theory

TL;DR

This paper derives analytical mean field predictions for the spectra of small-world networks, covering a spectrum from regular to random topologies, and validates these predictions with numerical results.

Contribution

It introduces a systematic mean field approach to predict the spectra of small-world networks across different topologies, bridging theory and numerical validation.

Findings

Analytic spectra predictions match numerical diagonalization results.

Predictions are valid for both undirected and directed networks.

Results apply across a range from regular to strongly random topologies.

Abstract

Collective dynamics on small-world networks emerge in a broad range of systems with their spectra characterizing fundamental asymptotic features. Here we derive analytic mean field predictions for the spectra of small-world models that systematically interpolate between regular and random topologies by varying their randomness. These theoretical predictions agree well with the actual spectra (obtained by numerical diagonalization) for undirected and directed networks and from fully regular to strongly random topologies. These results may provide analytical insights to empirically found features of dynamics on small-world networks from various research fields, including biology, physics, engineering and social science.