Multiorder Laplacian for Kuramoto Dynamics with Higher-Order Interactions

Many real-world systems are characterised by higher-order interactions, where influences among units involve more than two nodes at a time, and which can significantly affect the emergence of collective behaviors. A paradigmatic case is that of synchronization, occuring when oscillators reach coherent dynamics through their mutual couplings, and which is known to display richer collective phenomena when connections are not limited to simple dyads. Here, we consider an extension of the Kuramoto model with higher-order interactions, where oscillators can interact in groups of any size, arranged in any arbitrary complex topology. We present a new operator, the multiorder Laplacian, which allows us to treat the system analytically and that can be used to assess the stability of synchronization in general higher-order networks. Our spectral approach, originally devised for Kuramoto dynamics, can be extended to a wider class of dynamical processes beyond pairwise interactions, advancing our quantitative understanding of how higher-order interactions impact network dynamics.