Synchronization in adaptive higher-order networks

Many natural and human-made complex systems feature group interactions that adapt over time in response to their dynamic states. However, most of the existing adaptive network models fall short of capturing these group dynamics, as they focus solely on pairwise interactions. In this study, we employ adaptive higher-order networks to describe these systems by proposing a general framework by incorporating both adaptivity and group interactions. We demonstrate that global synchronization can exist in those complex structures and we provide the necessary conditions for the emergence of a stable synchronous state. We first study the setting in which both pairwise and higher-order interactions are allowed, but only the former adapt in time. We then extend this framework by including higher-order adaptive interactions. In both analyzed settings, we show that the necessary condition is strongly related to the master stability equation, allowing to separate the dynamical and structural properties. We illustrate our theoretical findings through the relevant examples involving adaptive higher-order networks of coupled generalized Kuramoto oscillators with phase lag, coupled with an all-to-all and a nonlocal ring-like structure. We also show that the interplay of group interactions and adaptive connectivity results in the formation of stability regions that can induce transitions between synchronization and desynchronization. Our findings also reveal that the introduction of higher-order adaptation significantly alters the synchronization stability compared to the case with constant higher-order interactions.