Synchronization in higher-order networks of Kuramoto oscillators

This paper examines the synchronization of phase oscillators in higher-order networks, focusing on two representative frameworks: hypergraphs and simplicial complexes. Extending the classical Kuramoto model beyond pairwise interactions, we incorporate higher-order interactions to elucidate how coupling strength distributions and network topology jointly influence synchronization dynamics. We find that higher-order interactions exert a dual regulatory effect on synchronizability: depending on the parameter regime, they may either enhance or suppress global synchronization. This result challenges the widely held view that higher-order interactions universally promote synchronization. Despite their distinct mathematical constructions, hypergraphs and simplicial complexes exhibit similar dependencies on low-order coupling strength in shaping synchronization patterns. Within the framework of linear stability theory, a multi-order Laplacian formalism is given to analytically characterize the stability of synchronous states induced by higher-order topological structures. To connect microscopic network architecture with macroscopic dynamical behavior, we further introduce a higher-order structural overlap measure. Numerical simulations demonstrate that the coupling allocation ratio, topological filling rate, and structural overlap play crucial roles in determining synchronization dynamics. These findings provide a unified physical understanding of synchronization in higher-order networks and offer insights into the emergence and control of collective dynamics in complex systems.