Allocating Minimum Number of Leaders for Seeking Consensus over Directed Networks with Time-varying Nonlinear Multi-agents

In this paper, we consider how to determine the minimum number of leaders with allocation and how to achieve consensus over directed networks consisting of time-varying nonlinear multi-agents. Firstly, the problem of finding minimum number of leaders is formulated as a minimum spanning forest problem, i.e., finding the minimum population of trees in the network. By introducing a toll station connecting with each agent, this problem is converted to a minimum spanning tree problem. In this way, the minimum number of leaders is determined and these leaders are found locating at the roots of each tree in the obtained spanning forest. Secondly, we describe a virtual leader connected with the allocated leaders, which indicates that the number of edges connected the follower agents with the virtual leader is the least in an arbitrary directed network. This method is different from the existing consensus problem of redundant leaders or edges that connect the follower with one leader in special networks. A distributed consensus protocol is revisited for achieving final global consensus of all agents. It is theoretically shown that such a protocol indeed ensures consensus. Simulation examples in real-life networks are also provided to show the effectiveness of the proposed methodology. Our works enable studying and extending application of consensus problems in various complex networks.