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

Matteo Lodi,Francesco Sorrentino,Marco Storace

doi:10.1038/s41467-021-24363-7

Matteo Lodi, Francesco Sorrentino + Show 1 more

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https://doi.org/10.1038/s41467-021-24363-7

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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.

Highlights

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
The weighted adjacency matrix Ak embeds information about the arrow set Ek, and the weight set Wk: Akij 2 Wk is the weight of the link going from vj to vi; Akij 1⁄4 0 if there is no arrow from vj to vi and Akij belongs to Wk otherwise
If there is an arrow with a certain weight going from vi to vj there is an arrow with the same weight going from vj to vi; in this case, Ak is a symmetric matrix and the underlying graph is undirected

Summary

Introduction

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. Synchronization is a pervasive phenomenon in networks of natural and man-made dynamical systems[1], which often enables complex functions corresponding to patterns of coordinated activity Some of these systems require complete (or full) synchronization among all network components to properly function: examples include the coherent firing of neural populations[2], synchronous behaviors in networks of chemical oscillators[3], the concerted rhythmical flashing on and off of firefly swarms[4], or frequency-locked power generation[5]. In other cases, such as higher brain centers of the nervous system[6], rhythmic activity and synchronization may correspond to a pathological condition. References[18,21,22] study undirected networks and explain how to analyze the stability of any CS solution by introducing a transformation—based on the irreducible representation (IRR) of the network symmetry group—from the node coordinate system to the IRR coordinate system

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