Polarized networks, diameter, and synchronizability of networks

TL;DR

This paper investigates the relationship between network diameter and synchronizability, revealing that diameter alone does not determine synchronizability and that network size influences this property.

Contribution

The study constructs polarized and fixed-diameter networks to demonstrate the limited role of diameter and derives analytic estimates linking size and synchronizability.

Findings

Diameter alone does not determine synchronizability.

Larger networks have more flexible synchronizability.

Numerical results confirm the analytic estimates.

Abstract

Previous research claimed or disclaimed the role of a small diameter in the synchronization of a network of coupled dynamical systems. We investigate this connection and show that it is two folds. We first construct two classes of networks, the polarized networks and the random networks with a fixed diameter, which exhibit very different synchronizability. This shows that the diameter itself is insufficient to determine the synchronizability of networks. Secondly, we derive analytic estimates on the synchronizability of networks in terms of the diameter, and find that a larger size of network admits of a more flexible synchronizability. The analysis is confirmed by numerical results.