Leader selection for optimal network coherence

We consider the problem of leader-based distributed coordination in networks where agents are subject to stochastic disturbances, but where certain designated leaders are immune to those disturbances. Specifically, we address the effect of leader selection on the coherence of the network, defined in terms of an H2 norm of the system. This quantity captures the level of agreement of the nodes in the face of the external disturbances. We show that network coherence depends on the eigenvalues of a principal submatrix of the Laplacian matrix, and we formulate an optimization problem to select the set of leaders that results in the highest coherence. As this optimization problem is combinatorial in nature, we also present several greedy algorithms for leader selection that rely on more easily computable bounds of the H2 norm and the eigenvalues of the system. Finally, we illustrate the effectiveness of these algorithms using several network examples.