ON THE CRITICAL EXPONENT FOR FLOCKS UNDER HIERARCHICAL LEADERSHIP
Abstract
Very recently, a model for flocking was introduced by Cucker and Smale together with a proof of convergence. This proof established unconditional convergence to a common velocity provided the interaction between agents was strong enough and conditional convergence otherwise. The strength of the interaction is measured by a parameter β ≥ 0 and the critical value at which unconditional convergence stops holding is β = 1/2. This model was extended by Shen to allow for a hierarchical leadership structure among the agents and similar convergence results were proved. But, for discrete time, unconditional convergence was proved only for 
(k being the number of agents). In this note we improve on this result showing that unconditional convergence holds indeed for β < 1/2.
