Global topological synchronization on simplicial and cell complexes

TL;DR

This paper investigates how topological structures like simplicial and cell complexes influence the global synchronization of signals defined on their elements, revealing topological obstructions and conditions for synchronization.

Contribution

It combines topology and nonlinear dynamics to identify conditions for global synchronization on higher-order networks, highlighting differences between simplicial and cell complexes.

Findings

Topological obstructions prevent odd-dimensional signals from synchronizing on simplicial complexes.

Cell complexes can overcome topological obstructions, enabling synchronization of signals of any dimension.

The study advances understanding of collective phenomena in topological signals on complex networks.

Abstract

Topological signals, i.e., dynamical variables defined on nodes, links, triangles, etc. of higher-order networks, are attracting increasing attention. However the investigation of their collective phenomena is only at its infancy. Here we combine topology and nonlinear dynamics to determine the conditions for global synchronization of topological signals defined on simplicial or cell complexes. On simplicial complexes we show that topological obstruction impedes odd dimensional signals to globally synchronize. On the other hand, we show that cell complexes can overcome topological obstruction and in some structures, signals of any dimension can achieve global synchronization.