The number of leaders needed for consensus

In order to have a self-organized multi-agent system exhibit some expected collective behavior, it is necessary that some agents (called leaders) are informed about the expected behavior and able to intervene. Then a fundamental question is: how many such leaders are needed? Naturally the answer depends on the model used. In this paper the model proposed by Vicsek et al. is used for the analysis. By analyzing the system dynamics and estimating the characteristics of the initial neighbor graph, a lower bound on the ratio of leaders is provided that guarantees the expected consensus being reached almost surely.