The number of neighbors needed for consensus of flocks

This paper investigates a class of flocks with a k-nearest-neighbor rule, where each agent´s neighbors are determined according to k nearest agents with k 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 we will prove that, similar to the Vicsek´s model, consensus can be also achieved under conditions imposed only on the model parameters and initial states. We reveal that, with some speed conditions, Θ(log n) neighbors are sufficient and necessary for consensus of flocks under such a neighbor rule, where n is the population size of the flock.