Consensus of Flocks under M-Nearest-Neighbor Rules

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.Such a neighbor rule has been validated by biologists through experiments and the authors will prove that,similar to the Vicsek's model,such a new neighbor rule can also achieve consensus under some conditions imposed only on the system's speed and the number M,n,without resorting to any priori connectivity assumptions on the trajectory of the system.In particular,the authors will prove that if the number M is proportional to the population size n,then for any speed v,the system will achieve consensus with large probability if the population size is large enough.