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

TL;DR

This work identifies all possible patterns a network can exhibit through symmetry breaking of identically synchronized clusters and shows that the existence of different invariant patterns is a function of 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.

Figures (4)

• Figure 1: Coexisting active-inactive clusters are generated from symmetry breaking of identically synchronized nodes. Starting with complete amplitude death state, pattern $\mathcal{P}_1$, the symmetry breaking generates patterns $\mathcal{P}_2-\mathcal{P}_{10}$. Synchronized nodes forming a cluster are shown using the same color and boundary thickness. If two clusters are in antisynchrony with each other, one of the clusters is shown by a thicker boundary line. Inactive clusters are shown in black.
• Figure 2: Time series shows activity patterns in an 8-node network of Stuart-Landau oscillators. (a)-(d) The network transitions through the patterns $\mathcal{P}_{1} \rightarrow \mathcal{P}_2 \rightarrow \mathcal{P}_3 \rightarrow \mathcal{P}_4$ as $\sigma$ increases. The parameters are $\lambda=-1, ~\omega=2$, $\sigma_x=-1$, $\sigma_y=0$. The links weights are $w_{18}=w_{81}=4$, while $w_{ij}=1$ for the rest of the links.
• Figure 3: Time series illustrates dynamics induced clusters, which cannot be identified through EEPs, in the amplitude death state. The parameters are $\lambda=-1, ~\omega=2$, $\sigma_x=-1$, $\sigma_y=0$. The links weights are $w_{18}=w_{81}=w_{23}=w_{32}=3$, and $w_{57}=w_{75}=2$, while $w_{ij}=1$ for the rest of the links.
• Figure 4: Stability analysis of activity patterns observed in Fig. \ref{['fig:time_series']}. A negative $\Gamma(\sigma)$ value for an invariant pattern shows its stability.

Theorems & Definitions (1)

• Proposition