The proportion of leaders needed in the consensus decision-making

In an article of Nature in 2005, Couzin et al. put forward a discrete-time multi-agent system to investigate the mechanism of leadership and decision-making in animal groups on the move. The system consists of two kinds of agents: informed agents (called leaders) and autonomous agents (called followers). The leaders have the preferred but pertinent information about where to go, whereas the followers do not have such information, and they make movement decisions according to the local information of their corresponding neighbors. In this paper, we will consider the consensus decision-making under two cases: system with one kind of leaders and system with two different kinds of leaders. For the system with one kind of leaders, we will provide necessary condition and sufficient condition on the proportion of leaders for the expected consensus; while for the system with two kinds of leaders, we will show how the proportion of these two kinds of leaders affect the behavior of the system.