Cluster synchronization on hypergraphs

Cluster synchronization is a type of synchronization in which different groups of nodes follow distinct trajectories. It can manifest in behaviors such as chimera states and remote synchronization with wide areas of applicability from neuroscience to power grid analysis.  In contrast to the broadly analyzed case of cluster synchronization on dyadic networks, we study cluster synchronization on hypergraphs, where hyperedges correspond to higher order interactions. Importantly, we show that our analysis can not be reduced to analyzing dynamics on hypergraph projections onto dyadic networks. 

We demonstrate how to determine admissible synchronization patterns from the hypergraph structure. We also show how the hypergraph structure together with the pattern of cluster synchronization can be used to simplify the stability analysis using simultaneous block diagonalization. We formulate our results in terms of external equitable partitions but show how symmetry considerations can also be used to obtain some of the patterns. In both cases, our analysis requires considering the partitions of hyperedges into edge clusters that are induced by the node clustering. The node and edge cluster formulation provides a general way to organize the analysis of dynamical processes on hypergraphs.