On Recurrence of Graph Connectivity in Vicsek's Model of Motion Coordination for Mobile Autonomous Agents

In this paper we complete the analysis of Vicsek's model of distributed coordination among kinematic planar agents. The model is a simple discrete time heading update rule for a set of kinematic agents (or self-propelled particles as referred to by Vicsek) moving in a finite plane with periodic boundary conditions. Contrary to existing results in the literature, we do not make any assumptions on connectivity but instead prove that under the update scheme, the network of agents stays jointly connected infinitely often for almost all initial conditions, resulting in global heading alignment. Our main result is derived using a famous theorem of Hermann Weyl on equidistribution of fractional parts of sequences. We also show that the Vicsek update scheme is closely related to the Kuramoto model of coupled nonlinear oscillators.