Dynamics-induced activity patterns of active-inactive clusters in complex networks

TL;DR

This paper explores how complex networks can exhibit diverse activity patterns, including active and inactive clusters, through symmetry breaking and dynamics-induced mechanisms, extending understanding beyond symmetry-based cluster analysis.

Contribution

It introduces a novel framework for identifying active-inactive cluster patterns in networks without relying on permutation symmetries, using symmetry breaking and stability analysis.

Findings

Active-inactive cluster patterns can be generated through symmetry breaking.

Inactive clusters can be purely dynamics-induced, not symmetry-based.

The stability of patterns depends on coupling strength and intercluster weights.

Abstract

Synchrony patterns describe network states in which nodes of a coupled dynamical system are grouped into clusters based on synchronization between nodes. Beyond simple synchrony, synchronized clusters may also exhibit active or inactive states, and the collection of all such clusters constitutes an activity pattern. Although these patterns may arise naturally in networks with permutation symmetries, the requirement of symmetries imposes a restrictive and often unrealistic assumption, as many real-world networks lack such symmetries. In this work, we present synchrony patterns of coexisting active-inactive clusters that cannot be identified through symmetries. Considering dynamical systems in which intrinsic dynamics and coupling functions are odd functions in phase space, we identify all possible patterns a network can exhibit through symmetry breaking of identically synchronized clusters. The symmetry breaking of invariant clusters generates antisynchronized clusters, allowing active-inactive clusters to coexist. We show that while active clusters are external equitable partitions, inactive clusters can be purely dynamics-induced. Starting with a symmetry-broken state, we show that the existence of different invariant patterns is a function of coupling strength and intercluster weights. Finally, by combining synchronization manifolds with the Laplacian eigenvectors, we identify transversal perturbations for these patterns and present a stability analysis.