Network Synchronization, Diffusion, and the Paradox of Heterogeneity

TL;DR

This paper investigates how heterogeneity in network degree distribution affects synchronization of oscillators, revealing that weighted and directed networks optimize synchronizability regardless of degree distribution or size.

Contribution

It introduces a diffusive process framework linking network communication to synchronization stability, proposing weighted, directed networks as optimal for synchronization.

Findings

Maximum synchronizability occurs with weighted, directed networks.

Synchronizability depends only on mean degree, not degree distribution or size.

Numerical results confirm theoretical predictions for small-world and scale-free networks.

Abstract

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed, and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.