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

TL;DR

The paper addresses how symmetry in network structure, combined with odd-in-phase node dynamics and coupling, can produce stable patterns where some clusters are active while others are inactive. It develops a quotient-dynamics framework and a symmetry-based stability analysis to analyze inactive and antisynchronized clusters, extending conventional master stability concepts to these activity patterns. Key results include explicit conditions for activity-to-inactivity transitions in Van der Pol and Stuart-Landau networks and a method to decouple perturbations to assess stability. The findings reveal how antisynchronization and structural symmetry enable robust coexistence of activity states and predict pattern switching as coupling strength varies, with potential implications for understanding brain-like modular activation patterns. The approach provides a principled route to engineer and analyze coexisting activity patterns in complex networks.

Abstract

We present activity patterns consisting of active and inactive clusters of synchronized nodes in networks. We call a cluster active if nodes in it have nonzero velocity and inactive vice versa. The simultaneous invariance of active and inactive clusters poses a challenge because fluctuations from active clusters must cancel out for a desired cluster to be inactive. With the help of permutation symmetries in network topology and selecting dynamics on top such that internal dynamics and coupling functions are odd functions in the phase space, we demonstrate that such a combination of structure and dynamics exhibits (stable) invariant patterns consisting of active and inactive clusters. Symmetry breaking of synchronized clusters creates active clusters that are in antisynchrony with each other, resulting in the cancellation of fluctuations for clusters connected with these antisynchronous clusters. Furthermore, 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 have been presented for networks of Van der Pol and Stuart-Landau oscillators. We extend the master stability approach to these patterns and provide stability conditions for their existence.

Figures (3)

• Figure 1: Invariant patterns containing active-inactive clusters in a network of $6$ nodes. Synchronized nodes are shown using the same color and boundary thickness. Inactive clusters are shown in black (amplitude death state) and white (oscillation death state). If two clusters are antisynchronized with each other, they are shown in the same color, but one of them has a thick boundary line.
• Figure 2: Time series shows activity patterns for the 6-node network presented in Fig. \ref{['fig:activity_patterns']}. (a)-(c) The network of Van der Pol oscillators switch between patterns $\mathcal{P}_{9} \rightarrow \mathcal{P}_1 \rightarrow \mathcal{P}_8$ when $\sigma$ is increased, and (d)-(g) the network of Stuart-Landau oscillators switch between patterns $\mathcal{P}_8 \rightarrow \mathcal{P}_1 \rightarrow \mathcal{P}_9 \rightarrow \mathcal{P}_7$ as a function of $\sigma$.
• Figure 3: Stability analysis of activity patterns shown in Fig. \ref{['fig:time_series']}: (a) Van der Pol oscillators and (b) Stuart-Landau oscillators. A negative $\Gamma(\sigma)$ value indicates the stability of the associated invariant pattern. Each line plot corresponds to a pattern state. The vertical dashed lines show stable $\sigma$ regimes of different patterns.