Collective directional switches of swarming systems with higher-order interactions.

Sudden coherent changes in the movement direction are common in animal groups; yet, the mechanism of higher-order and delayed interactions in shaping such collective switching dynamics remains poorly understood. Here, we propose a self-propelled particle model incorporating both pairwise and higher-order social interactions to study the directional switching behaviors in swarming systems, considering scenarios with and without delay. By applying a dimensional reduction method and the Fokker-Planck equation, we obtain the theoretical stationary probability density and the mean switching time. The results reveal that, without time delay, the higher-order interactions significantly increase the mean switching time, promoting stable, ordered movement states and reducing directional switches. When the time delay is introduced, the impact of higher-order interactions becomes non-monotonic. For small delays, they continue to suppress directional switching; for large delays, they instead facilitate more frequent directional switching. This non-monotonic pattern also appears in simulations on realistic social networks, underscoring the generality of the phenomenon. Our study illustrates how higher-order structures and time delays influence collective switching dynamics, highlighting the limitations of pairwise models and the necessity of considering complex interaction networks.