Topological Dirac cluster synchronization on directed hypergraphs
Topological higher-order synchronization reveals collective phenomena demonstrating how topology shapes dynamics, yet providing a general theoretical framework for designing and controlling dynamical states on higher-order networks remains a challenge of the field. Here we propose a dynamical systems framework for topological cluster synchronization on directed hypergraphs that can be used to design synchronization patterns on nodes and hyperedges of directed hypergraphs. By encoding oriented hyperedges into the hypergraph topological Dirac Hamiltonian, we obtain a spectrum whose isolated eigenstates correspond to distinct topological synchronization cluster states defined jointly on nodes and hyperedges. By selecting any isolated eigenstate, the system can be driven toward the associated dynamical state reflecting a specific partition of the hypergraph without modifying the underlying hypergraph structure. We numerically demonstrate the ability to design different topological cluster synchronization states on directed-hypergraph block models and empirical systems-including higher-order contact networks and the ABIDE functional brain network. Our results establish a general and interpretable route for controlling collective dynamics in directed higher-order systems.