Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions

Yuanzhao Zhang,Vito Latora,Adilson E Motter

doi:10.1038/s42005-021-00695-0

Abstract

When describing complex interconnected systems, one often has to go beyond the standard network description to account for generalized interactions. Here, we establish a unified framework to simplify the stability analysis of cluster synchronization patterns for a wide range of generalized networks, including hypergraphs, multilayer networks, and temporal networks. The framework is based on finding a simultaneous block diagonalization of the matrices encoding the synchronization pattern and the network topology. As an application, we use simultaneous block diagonalization to unveil an intriguing type of chimera states that appear only in the presence of higher-order interactions. The unified framework established here can be extended to other dynamical processes and can facilitate the discovery of emergent phenomena in complex systems with generalized interactions.

Highlights

When describing complex interconnected systems, one often has to go beyond the standard network description to account for generalized interactions
The original formulation has been generalized in different directions, including hypergraphs that account for nonpairwise interactions involving three or more nodes simultaneously[7,8], multilayer networks that accommodate multiple types of interactions[9,10], and temporal networks whose connections change over time[11]
Aside from multilayer networks, for which the multiple layers naturally translate into multiple matrices[32,33], the full potential of simultaneous block diagonalization (SBD) for analyzing dynamical patterns in generalized networks is yet to be realized

Summary

Introduction

When describing complex interconnected systems, one often has to go beyond the standard network description to account for generalized interactions. Neuronal networks change over time due to plasticity and comprise both chemical and electrical interaction pathways[6] For this reason, the original formulation has been generalized in different directions, including hypergraphs that account for nonpairwise interactions involving three or more nodes simultaneously[7,8], multilayer networks that accommodate multiple types of interactions[9,10], and temporal networks whose connections change over time[11]. SBD has found applications in numerous fields such as semi-definite programming[34], structural engineering[35], signal processing[36], and quantum algorithms[37] In this Article, we develop a versatile SBD-based framework that allows the stability analysis of synchronization patterns in generalized networks, which include hypergraphs, multilayer networks, and temporal networks. This framework enables us to treat all three classes of generalized networks in a unified fashion

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