Synchronization of mobile autonomous agents with M-nearest-neighbor rule

This paper investigates a class of flocks with an M-nearest-neighbor rule, where each agent's neighbors are determined according to M nearest agents with M being a given integer, rather than all the agents within a fixed metric distance as in the well-known Vicsek's model. We will prove the synchronization of the model under reasonable conditions imposed only on the system's speed and the number M, but without resorting to any sort of connectivity assumptions on the trajectory of the system. In particular, we will prove that if the number M is proportional to the population size n, then for any speed v, the system will synchronize with large probability if the population size is large enough.