Synchronization of a group of mobile agents with supercritical connectivity radius and large population

This paper investigates the synchronization behavior of a class of mobile agents based on the Vicsek´s model. It is well-known that the connectivity of the dynamical neighbor graphs is important for synchronization of the model, but such a connectivity is defined by the system trajectories, which, in turn, are determined by the model parameters and the initial configuration of all agents. We will carry out our analysis under the assumptions that all agents are independently and uniformly distributed in [0, 1]² at the initial time, and that the interaction radius is taken as the supercritical connectivity radius O(√log n/n). By estimating the spectral gap of the average matrix of the initial neighbor graph with the supercritical connectivity radius, we will establish the synchronization condition, which is imposed on the moving speed of the agents only.