How many leaders are required for consensus

Many natural and artificial systems, for example, fish school, bird flocks, robot team, are often composed of two different classes of agents: autonomous agents and leaders. The leaders have pertinent information, such as the location of the food source, while the autonomous agents make decisions according to local interactions. Then a basic problem is: how many leaders are required in order to induce all agents move with the same direction as leaders(such collective behavior is called consensus)? In this paper, we will consider a simple, but typical model proposed by Vicsek et al.(Vicsek model), in addition to the ordinary agents moving at the constant velocity with heading updated according to local rules, we add some leaders at the same velocity but with fixed heading. We will show that in order to induce the whole system reach consensus, the proportion of the leaders decreases as the number of autonomous agents increases under some conditions on the velocity v and neighborhood radius r. And the numerical examples are given to illustrate the necessity of the leaders. The proof is based on the analysis of the system dynamics and the initial neighbor graph, where multi-array martingale theorem play a key role.