Information centrality and optimal leader selection in noisy networks

We consider the leader selection problem in which a system of networked agents, subject to stochastic disturbances, uses a decentralized coordinated feedback law to track an unknown external signal, and only a limited number of agents, known as leaders, can measure the signal directly. The optimal leader selection minimizes the total system error by minimizing the steady-state variance about the external signal, equivalent to an H₂ norm of the linear stochastic network dynamics. Efficient greedy algorithms have been proposed in the literature for similar optimal leader selection problems. In contrast, we seek systematic solutions. We prove that the single optimal leader is the node in the network graph with maximal information centrality. In the case of two leaders, we prove that the optimal pair maximizes a joint centrality, which depends on the information centrality of each leader and how well the pair covers the graph. We apply these results to solve explicitly for the optimal single leader and the optimal pair of leaders in special classes of network graphs. To generalize we compute joint centrality for m leaders.