Fibration Symmetries and Cluster Synchronization in Multi-Body Systems

Based on recent advances in fibration symmetry theory, we investigate how structural symmetries influence synchronization in systems with higher-order interactions (HOI). Using bipartite graph representations, we identify a node partition in fibres, based on equivalent incidence relations in hypergraphs. We study how identical nodes with an isomorphic input set can synchronize due to structural properties under our specific model assumptions, examining the dynamical model of Kuramoto with higher-order interactions and frustration parameters. Recent works established for directed hypergraphs that balanced partitions characterize robust synchrony, invariant under all admissible dynamics, whereas our contribution isolates the case of Kuramoto dynamics and shows that synchrony under homogeneous initial conditions and natural frequencies necessarily coincides with the fibration partition. As a conclusion, let us examine situations that require adjustments to the hypergraph topology to handle redundancy or to align with a target cluster configuration, especially in the presence of noise or incomplete information. These considerations open up new questions for future investigations. Our methodology combines theoretical modeling and simulations with applications to real-world data topologies, highlighting how representational choices and local input equivalence influence synchronization behavior.