Chaos synchronization in networks of coupled maps with time-varying topologies

TL;DR

This paper investigates how time-varying and random topologies in complex networks influence the synchronization of coupled maps, revealing conditions under which such variations enhance or hinder synchronizability.

Contribution

It introduces a general analytical framework for understanding synchronization in networks with dynamic topologies, including new models and the Hajnal diameter as a measure.

Findings

Temporal variation can enhance synchronizability in many cases.

Random topology variations can both promote and inhibit synchronization.

A spanning tree in the union of graphs is crucial for synchronization.

Abstract

Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.