Symmetries and Cluster Synchronization in Multilayer Networks

TL;DR

This paper explores how symmetries in multilayer networks influence cluster synchronization, providing a theoretical framework and experimental validation for stability analysis in complex interconnected systems.

Contribution

It introduces a method to define and analyze symmetries in multilayer networks and studies their impact on cluster synchronization stability.

Findings

Dependent layer symmetries affect stability differently than independent ones

Master Stability Function effectively assesses cluster stability

Experimental validation with electronic oscillators confirms theoretical predictions

Abstract

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries which occur in one layer and are independent of the other layers and dependent layer symmetries which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure than the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling.