Leader selection in multi-agent systems for smooth convergence via fast mixing

In a leader-follower multi-agent system (MAS), a set of leader nodes receive state updates directly from the network operator. The follower nodes then compute their states based on the inputs from the leader nodes. In this paper, we study the problem of selecting a set of leader nodes in order to minimize the time required for the distributed coordination law used by the MAS to converge. We first represent the convergence time of a MAS in terms of the mixing time of a random walk on the underlying network graph. We then study two leader selection problems as convex optimization problems of fast mixing. First, we formulate the problem of selecting a fixed number of leaders in order to minimize the convergence time. We then study the problem of finding the minimumsize set of leaders in order to satisfy a constraint on the convergence time. We develop leader selection algorithms based on supergradient descent methods for static network topologies as well as a MAS experiencing random link failures and a MAS that switches between predefined topologies. We compare our leader selection algorithms with random and degree-based leader selection for both static and dynamic networks through simulation study. From the simulation comparisons, we note that the convergence rate of fast mixing is faster than that of degree-based methods. We also note that the fast mixing requires smallest number of leaders to achieve a given bound on the convergence time.