Cluster synchronization on hypergraphs

We study cluster synchronization (a type of synchronization where different groups of oscillators in the system follow distinct synchronized trajectories) on hypergraphs, where hyperedges correspond to higher order interactions between the nodes. Specifically, we focus on how to determine admissible synchronization patterns from the hypergraph structure by clustering its nodes based on the input they receive from the rest of the system, and how the hypergraph structure together with the pattern of cluster synchronization can be used to simplify the stability analysis. We formulate our results in terms of external equitable partitions but show how symmetry considerations can also be used. In both cases, our analysis requires considering the partitions of hyperedges into edge clusters that are induced by the node clusters. This formulation in terms of node and edge clusters provides a general way to organize the analysis of dynamical processes on hypergraphs. Our analysis here enables the study of detailed patterns of synchronization on hypergraphs beyond full synchronization and extends the analysis of cluster synchronization to beyond purely dyadic interactions.

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