Synchronizing distant nodes: a universal classification of networks

TL;DR

This paper presents a universal classification framework for network synchronization based on spectral properties, simplifying the analysis of delay-coupled networks' stability.

Contribution

It introduces a universal structure of the master stability function for large delays, enabling a straightforward classification of network synchronizability.

Findings

Synchronization stability depends on spectral properties of the network topology.

The master stability function becomes rotationally symmetric for large delays.

The classification solves synchronization issues in networks with strong delays.

Abstract

Stability of synchronization in delay-coupled networks of identical units generally depends in a complicated way on the coupling topology. We show that for large coupling delays synchronizability relates in a simple way to the spectral properties of the network topology. The master stability function used to determine stability of synchronous solutions has a universal structure in the limit of large delay: it is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. This allows a universal classification of networks with respect to their synchronization properties and solves the problem of complete synchronization in networks with strongly delayed coupling.