Performance of leader-follower networks in directed trees and lattices

We study the performance of externally forced leader-follower networks in directed trees and lattices. By exploiting the lower triangular structure of Laplacian matrices of both classes of graphs, we derive explicit formulae for the transfer function from disturbances to the states of the nodes. For directed trees, we show that the worst-case componentwise amplification of disturbances is achieved at zero temporal frequency and that it is a convex function of edge weights. For directed 1D and 2D lattices, we study the steady-state variance distribution in networks with leaders placed on the boundary. We show that as one moves away from leaders, the variance of the followers scales as a square-root function of node indices in 1D lattices and as a logarithmic function along the diagonal nodes in 2D lattices.