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

Abstract

Synchrony patterns characterize network states in which nodes organize into clusters based on their synchronized dynamics. The synchronized clusters may further exhibit either active or inactive states. The simultaneous invariance of active and inactive clusters of synchronized nodes poses a dynamical constraint because fluctuations from active clusters must cancel out for a desired cluster to be inactive. By exploiting permutation symmetries in the network structure and choosing dynamics on top such that internal dynamics and coupling functions are odd functions in the phase space, we demonstrate that this combination of structure and dynamics exhibits stable invariant patterns composed of coexisting active and inactive clusters. The symmetries in a network generate active clusters that are in antisynchrony with each other, resulting in the cancellation of fluctuations for clusters connected with these antisynchronous clusters. We use full network symmetries to obtain synchronized clusters, while quotient network symmetries are used to find coexisting active-inactive states of clusters. We show that as the coupling between nodes changes, active clusters lose their activity at different coupling values, and the network transitions from one activity pattern to another. Numerical simulations are presented for networks of Van der Pol and Stuart-Landau oscillators. Finally, we extend the master stability framework to these patterns and provide stability conditions for their existence.