Performance of leader-follower multi-agent systems in directed networks

We consider leader-follower multi-agent systems in which the leader executes the desired trajectory and the followers implement the consensus algorithm subject to stochastic disturbances. The performance of the leader-follower systems is quantified by the steady-state variance of the deviation of the followers from the desired trajectory. We study the asymptotic scaling of the variance in directed lattices in one, two, and three dimensions. We show that in 1D and 2D the variance of the followers' deviation increases to infinity as one moves away from the leader, while in 3D the variance remains bounded regardless of the network size. We prove that the variance of the followers scales as a square-root function of the distance to the leader in 1D and a logarithmic function in 2D lattices.