Testing the Collective Properties of Small-World Networks through Roughness Scaling

TL;DR

This paper investigates the synchronization properties of small-world networks using the Edwards-Wilkinson model, revealing how the network's coupling strength affects surface roughness and synchronization in different dimensions.

Contribution

It introduces a perturbative approach to analyze the width of the stationary surface in small-world networks, highlighting anomalous scaling behaviors.

Findings

Width remains finite in both network variations.

Anomalous scaling occurs with respect to coupling strength p in low dimensions.

Results verified through numerical diagonalization.

Abstract

Motivated by a fundamental synchronization problem in scalable parallel computing and by a recent criterion for ``mean-field'' synchronizability in interacting systems, we study the Edwards-Wilkinson model on two variations of a small-worldnetwork. In the first version each site has exactly one random link of strength $p$, while in the second one each site on average has $p$ links of unit strength. We construct a perturbative description for the width of the stationary-state surface (a measure of synchronization), in the weak- and sparse-coupling limits, respectively, and verify the results by performing exact numerical diagonalization. The width remains finite in both cases, but exhibits anomalous scaling with $p$ in the latter for $d\leq 2$.