Kinetic Description of Swarming Dynamics with Topological Interaction and Transient Leaders

In this paper, we present a model describing the collective motion of birds. The model introduces spontaneous changes in direction which are initialized by few agents, here referred to as leaders, whose influence acts on their nearest neighbors, in the following referred to as followers. Starting at the microscopic level, we develop a kinetic model that characterizes the behavior of large flocks with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a k-nearest neighbors search algorithm with a binary tree. Finally, we conduct various numerical experiments for different scenarios to validate the algorithm’s effectiveness and investigate collective dynamics in both two and three dimensions.