Kinetic description of swarming dynamics with topological interaction and emergent leaders

In this paper, we present a model describing the collective motion of birds. We explore the dynamic relationship between followers and leaders, wherein a select few agents, known as leaders, can initiate spontaneous changes in direction without being influenced by external factors like predators. Starting at the microscopic level, we develop a kinetic model that characterizes the behaviour of large crowds 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 neighbours search algorithm with a binary tree. Lastly, we conduct various numerical experiments for different scenarios to validate the algorithm's effectiveness and investigate collective dynamics in both two and three dimensions.