One-way dependent clusters and stability of cluster synchronization in directed networks

TL;DR

This paper investigates cluster synchronization in directed networks, revealing one-way dependencies among clusters and providing a method to analyze their stability by decomposing the problem into simpler subproblems.

Contribution

It introduces a novel approach to analyze cluster stability in directed networks by transforming the problem into an irreducible form, enabling detection of inter-cluster dependencies.

Findings

Directed networks exhibit one-way cluster dependencies.

The method simplifies stability analysis by decomposing the problem.

Applications include neural and social network examples.

Abstract

Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.