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Hodge Decomposition Guides the Optimization of Synchronization over Simplicial Complexes

Despite growing interest in synchronization dynamics over "higher-order" network models, optimization theory for such systems is limited. Here, we study a family of Kuramoto models inspired by algebraic topology in which oscillators are coupled over simplicial complexes (SCs) using their associated Hodge Laplacian matrices. We optimize such systems by extending the synchrony alignment function -- an optimization framework for synchronizing graph-coupled heterogeneous oscillators. Computational experiments are given to illustrate how this approach can effectively solve a variety of combinatorial problems including the joint optimization of projected synchronization dynamics onto lower- and upper-dimensional simplices within SCs. We also investigate the role of SC homology and develop bifurcation theory to characterize the extent to which optimal solutions are contained within (or spread across) the three Hodge subspaces. Our work extends optimization theory to the setting of higher-order networks, provides practical algorithms for Hodge-Laplacian-related dynamics including (but not limited to) Kuramoto oscillators, and paves the way for an emerging field that interfaces algebraic topology, combinatorial optimization, and dynamical systems.