Optimal topology of multi-agent systems with two leaders: a zero-sum game perspective

It is typical to assume that there is no conflict of interest among leaders. Under such assumption, it is known that, for a multi-agent system with two leaders, if the followers' interaction subgraph is undirected and connected, then followers will converge to a convex combination of two leaders' states with linear consensus protocol. In this paper, we introduce the conflict between leaders: by choosing k followers to connect with, every leader attempts all followers converge to himself closer than that of the other. By using graph theory and matrix theory, we formulate this conflict as a standard two-player zero-sum game and give some properties about it. It is noteworthy that the interaction graph here is generated from the conflict between leaders. Interestingly, we find that to find the optimal topology of the system is equivalent to solve a Nash equilibrium. Especially for the case of choosing one connected follower, the necessary and sufficient condition for an interaction graph to be the optimal one is given. Moreover, if followers' interaction graph is a circulant graph or a graph with a center node, then the system's optimal topology is obtained. Simulation examples are provided to validate the effectiveness of the theoretical results.