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.