  In a leader-follower multi-agent system (MAS), the leader agents act as
control inputs and influence the states of the remaining follower agents. The
rate at which the follower agents converge to their desired states, as well as
the errors in the follower agent states prior to convergence, are determined by
the choice of leader agents. In this paper, we study leader selection in order
to minimize convergence errors experienced by the follower agents, which we
define as a norm of the distance between the follower agents' intermediate
states and the convex hull of the leader agent states. By introducing a novel
connection to random walks on the network graph, we show that the convergence
error has an inherent supermodular structure as a function of the leader set.
Supermodularity enables development of efficient discrete optimization
algorithms that directly approximate the optimal leader set, provide provable
performance guarantees, and do not rely on continuous relaxations. We formulate
two leader selection problems within the supermodular optimization framework,
namely, the problem of selecting a fixed number of leader agents in order to
minimize the convergence error, as well as the problem of selecting the
minimum-size set of leader agents to achieve a given bound on the convergence
error. We introduce algorithms for approximating the optimal solution to both
problems in static networks, dynamic networks with known topology
distributions, and dynamic networks with unknown and unpredictable topology
distributions. Our approach is shown to provide significantly lower convergence
errors than existing random and degree-based leader selection methods in a
numerical study.
