Understanding the enhanced synchronization of delay-coupled networks with fluctuating topology

TL;DR

This paper develops an analytical framework to understand how the synchronization of delay-coupled networks is affected by fluctuating topologies, revealing how effective network structures emerge in different time scale regimes.

Contribution

It introduces a theory linking network topology fluctuations with synchronization stability, deriving effective topologies in fast and slow switching limits.

Findings

Effective topology in fast switching is the arithmetic mean of topologies.

In slow switching, the effective topology is the geometric mean.

Synchronization properties depend on the ratio of time scales and specific topologies.

Abstract

We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases the system can be reduced to an "effective" topology: In the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime the system shows a sensitive dependence on the ratio of time scales, and specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps.