Synchronization in dynamical systems coupled via multiple directed networks

TL;DR

This paper investigates how multiple directed networks can collectively induce synchronization in coupled dynamical systems, highlighting the importance of network topology and the combined graph structure.

Contribution

It demonstrates that the union of multiple directed networks can achieve synchronization even when individual networks cannot, based on the combined graph's properties.

Findings

Synchronization depends on the combined graph topology.

A directed graph with a spanning tree in its reversal ensures synchronization.

Multiple networks can compensate for weak coupling in individual networks.

Abstract

We study synchronization and consensus in a group of dynamical systems coupled via multiple directed networks. We show that even though the coupling in a single network may not be sufficient to synchronize the systems, combination of multiple networks can contribute to synchronization. We illustrate how the effectiveness of a collection of networks to synchronize the coupled systems depends on the graph topology. In particular, we show that if the graph sum is a directed graph whose reversal contains a spanning directed tree, then the network synchronizes if the coupling is strong enough. This is intuitive as there is a root node that influence every other node via a set of edges where each edge in the set is in one of the networks.