The consensus performance of leader-follower multi-agent systems under reversing the directions of arcs

A leader-follower multi-agent system (MAS) on a digraph can reach a consensus provided that there exists a directed spanning tree in the digraph. The consensus performance of the MAS, which is characterized by the convergence rate, is associated with the Laplacian spectrum of the digraph. Moreover, the convergence rate increases with an increase on the algebraic connectivity, which is the minimal value among all real parts of nonzero Laplacian eigenvalues of the digraph. To explore the effects of graph operations on convergence rate, we consider the impacts on algebraic connectivity under reversing the directions of arcs among the network between followers in this paper. Consider reversing the direction of one arc, a necessary and sufficient condition for the increase on the algebraic connectivity is proposed based on the entries of a Fiedler vector. What's more, we give some sufficient conditions for the increase or the decrease on the algebraic connectivity under reversing the directions of multiple arcs, respectively.