Equitability and explosive synchronisation in multiplex and higher-order networks

TL;DR

This paper identifies conditions under which equitability is necessary and sufficient for cluster synchronisation in complex multiplex and higher-order networks, explaining the prevalence of explosive synchronisation.

Contribution

It introduces a linear independence condition that clarifies when equitability induces cluster synchronisation in general dynamical systems with complex interactions.

Findings

Equitability is necessary and sufficient for cluster synchronisation under certain conditions.

Explains why explosive synchronisation is more common than cluster synchronisation in complex networks.

Provides a framework for designing systems with specific synchronisation patterns.

Abstract

Cluster synchronisation is a key phenomenon observed in networks of coupled dynamical units. Its presence has been linked to symmetry and, more generally, to equability of the underlying pattern of interactions between dynamical units. However, it is not known under which conditions equitability-induced synchronisation is the only cluster synchronisation that can occur on a particular system. Here, we reveal a natural linear independent condition such that equitability becomes necessary, and sufficient, for the existence of cluster synchronised solutions on a very general dynamical system which allows multiplex or higher-order, arbitrarily weighted interactions. Our results explain the ubiquity of explosive synchronisation, as opposed to cluster synchronisation, in multiplex and higher-order networks: equitability imposes additional constraints that must be simultaneously satisfied on the same set of nodes. Our results have significant implications for the design of complex dynamical systems of coupled dynamical units with arbitrary cluster synchronisation patterns and coupling functions.