Synchronization in Dynamical Systems Coupled via Multiple Directed Networks

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.

I. Introduction

There has been much research activity in studying the synchronization and consensus behavior in networks of coupled dynamical systems [1]–[4]. The coupled systems model various types of networks both man-made and naturally occurring, such as social networks, transportation networks, genetic networks and neural networks. In many cases the individual systems or agents are connected via multiple networks. For instance, automobiles are connected via a transportation network, but they can also connected via cell phones (or other vehicle-to-vehicle technologies such as DSRC [5]) in a communication network. Similarly, people can be connected via multiple social networks such as Facebook and LinkedIn. The Erdős-Bacon number is the sum of the distance in the actor and in the mathematical collaboration network.