Synchronization on community networks

TL;DR

This paper introduces a model for community networks with tunable strength, analyzing how community structure impacts network synchronizability, revealing a linear eigenvalue relationship and effects on Kuramoto model synchronization.

Contribution

The study presents a new network model with adjustable community strength and analytically links community strength to eigenvalues affecting synchronization.

Findings

Stronger community structure weakens network synchronizability.

A linear relationship exists between eigenvalues and community strength.

Very strong communities can impair global synchronization, with effects diminishing beyond a certain community strength.

Abstract

In this Letter, we propose a growing network model that can generate scale-free networks with a tunable community strength. The community strength, $C$, is directly measured by the ratio of the number of external edges to internal ones; a smaller $C$ corresponds to a stronger community structure. According to the criterion obtained based on the master stability function, we show that the synchronizability of a community network is significantly weaker than that of the original Barab\'asi-Albert network. Interestingly, we found an unreported linear relationship between the smallest nonzero eigenvalue and the community strength, which can be analytically obtained by using the combinatorial matrix theory. Furthermore, we investigated the Kuramoto model and found an abnormal region ($C\leq 0.002$), in which the network has even worse synchronizability than the uncoupled case (C=0). On the other hand, the community effect will vanish when $C$ exceeds 0.1. Between these two extreme regions, a strong community structure will hinder global synchronization.