On the minimum number of neighbors needed for consensus of flocks

Abstract

This paper investigates consensus of flocks consisting of n autonomous agents in the plane, where each agent has the same constant moving speed v n and updates its heading by the average value of the k n nearest agents from it, with v n and k n being two prescribed parameters depending on n. Such a topological interaction rule is referred to as k n -nearest-neighbors rule, which has been validated for a class of birds by biologists and verified to be robust with respect to disturbances. A theoretical analysis will be presented for this flocking model under a random framework with large population, but without imposing any a priori connectivity assumptions. We will show that the minimum number of k n needed for consensus is of the order O(log n) in a certain sense. To be precise, there exist two constants C1 > C2 > 0 such that, if k n > C1 log n, then the flocking model will achieve consensus for any initial headings with high probability, provided that the speed v n is suitably small. On the other hand, if k n < C2 log n, then for large n, with probability 1, there exist some initial headings such that consensus cannot be achieved, regardless of the value of v n .