Fibration symmetries and cluster synchronization in multi-body systems

Structural symmetries influence synchronization patterns in complex systems. In particular, for graphs, the interplay between synchronization clusters for dynamical processes and classes of equivalent nodes with isomorphic input trees is established. Recent works generalized and proved for directed hypergraphs that balanced partitions characterize robust synchrony invariant under all admissible dynamics. In this work, we address undirected hypergraphs and identify node partitions based on equivalent incidence relations in a hypergraph, using its bipartite graph representations. We also show that, for the higher-order Kuramoto-Sakaguchi model with identical natural frequencies and identical initial conditions, fibres define invariant synchrony subspaces. Moreover, under a separating interaction-signature non-degeneracy condition, the trajectory-induced Kuramoto synchrony partition coincides with the fibration partition for almost all choices of the coupling parameters. We provide algorithms, code implementations, and visualization methods to compute local input equivalence and analyse its relation to synchronization behavior. Simulations with real-world and synthetic topologies illustrate how structural properties and representational choices affect cluster formation. We outline exploratory topology-adjustment procedures for target cluster configurations under noise and incomplete information, identifying directions for future investigations.