Optimal controlling node problem for multi-agent systems via Alt-PageRank

In this paper, we deal with problems of finding optimal input nodes of multi-agent systems in the sense of convergence rates of the all agents´ states to an identical value. We consider two cases; Weakly Controlled Systems and Directly Controlled Systems, respectively, and show that in the both cases, the optimal nodes can be explicitly given by using an alternative index of PageRank, called Alt-PageRank, which we originally propose in this paper. Furthermore, we study controllability of both WCS and DCS. We obtain the same condition about controllability of both WCS and DCS, which is characterized by left eigenvectors of graph Laplacian.