Optimal synchronization of higher-order Kuramoto model on hypergraphs

Complex networks play a crucial role in the study of collective behavior, encompassing the analysis of dynamical properties and network topology. In real-world systems, higher-order interactions among multiple entities are widespread and significantly influence collective dynamics. Here, we extend the synchronization alignment function framework to hypergraphs of arbitrary order by leveraging the multi-order Laplacian matrix to encode higher-order interactions. Our findings reveal that the upper bound of synchronous behavior is determined by the maximum eigenvalue of the multi-order Laplacian matrix. Furthermore, we decompose the contribution of each hyperedge to this eigenvalue and utilize it as a basis for designing an eigenvalue-based topology modification algorithm. This algorithm effectively enhances the upper bound of synchronous behavior without altering the total number of higher-order interactions. Our study provides new insights into dynamical optimization and topology tuning in hypergraphs, advancing the understanding of the interplay between higher-order interactions and collective dynamics.