Synchronization of Topological Signals on Simplicial Complexes With Higher-Dimensional Simplices
Kezan Li, Tian Shi, Xiang Li
Abstract
This article investigates a higher-order dynamical network on simplicial complexes, and the evolution of topological signals residing on simplices (e.g. nodes, links, triangles, etc) is governed by coupled higher-dimensional linear oscillators. We uncover the stability conditions of cluster synchronization of both the higher-order network and its projected systems corresponding to solenoidal and irrotational component of the dynamics, and the synchronization phenomena of the higher-order network and its projected systems can co-exist (i.e., symbiotic synchronization). In the framework of simplicial complexes, this study indicates that there are essential differences between the collective behaviours of phase oscillators and higher-dimensional oscillators.