Effects of changing the weights of arcs on the consensus convergence rate of a leader–follower multi-agent system

For a first-order leader–follower multi-agent system (MAS), its consensus convergence rate is determined by the algebraic connectivity (the smallest real part of nonzero eigenvalues) of the corresponding directed graph of its interaction topology. In this paper, effects of changing the weights of arcs among the followers on the algebraic connectivity are investigated for a leader–follower topology with a weighted strongly connected directed graph as the followers’ interaction topology. If the weight of one arc decreases (increases), the algebraic connectivity increases (decreases) if and only if the entry of the Fiedler vector corresponding to its head is smaller than that of its tail. For arcs with a common head, the arc whose tail corresponds to the largest (smallest) entry of the Fiedler vector improves the algebraic connectivity most if the weight of one of these arcs decreases (increases). A necessary and sufficient condition for improving the algebraic connectivity is also proposed for decreasing (increasing) the weights of multiple arcs by the entries of the Fiedler vector corresponding to the vertices of the arcs and the amounts of the weight changes. Moreover, a method of choosing an optimal set of arcs that improve the algebraic connectivity most is proposed if the changing weights are given. Finally, several numerical experiments are given to illustrate the theoretical results.