Equilibrium topology of multi-agent systems with two leaders: A zero-sum game perspective

It is typical in containment control problems to assume that there is no conflict of interest among leaders. In this paper, we consider the situation where there is conflict between leaders; namely, the leaders compete to attract followers. The strategies of each leader are defined by choosing at most k followers to propagate their information. Then, we formulate a standard two-player zero-sum game by using graph theory and matrix theory. We further prove that each player will choose exactly k followers when the game achieves a Nash equilibrium. It is noteworthy that the interaction graph here is generated from the conflict between leaders and then the Nash equilibrium point of the game corresponds to the equilibrium topology. For the case of choosing one follower, a necessary and sufficient condition for an interaction graph to be the equilibrium topology is derived. Moreover, we can obtain the equilibrium topology directly if followers' interaction graph is a circulant graph or a graph with a center vertex. Simulation examples are provided to validate the effectiveness of the theoretical results.