On flocks influenced by closest neighbors
Felipe Cucker, Jiu‐Gang Dong · Mathematical Models and Methods in Applied Sciences · 2016
We prove, both for continuous time and discrete time, results establishing convergence to flocking in a model in which each agent is influenced by only a few closest neighbors. We show unconditional convergence to flocking when this number of closest neighbors is at least half of the population and, otherwise, conditional convergence. That is, convergence to flocking provided the initial positions and velocities satisfy an explicit constraint. We also show that the proportion of neighbors necessary for unconditional convergence, one-half, is sharp. Nothing less will do.