Joint Centrality Distinguishes Optimal Leaders in Noisy Networks

Abstract

We study the performance of a network of agents tasked with tracking an external unknown signal in the presence of stochastic disturbances and under the condition that only a limited subset of agents, known as leaders, can measure the signal directly. We investigate the optimal leader selection problem for a prescribed maximum number of leaders, where the optimal leader set minimizes total system error defined as steady-state variance about the external signal. In contrast to previously established greedy algorithms for optimal leader selection, our results rely on an expression of total system error in terms of properties of the underlying network graph. We demonstrate that the performance of any given set of noise-free leaders depends on their influence as determined by a new graph measure of the centrality of a set. We define the joint centrality of a set of nodes in a network graph such that a noise-free leader set with maximal joint centrality is an optimal leader set. In the case of a single leader, we prove that the optimal leader is the node with maximal information centrality for the noise-corrupted and noise-free leader cases. In the case of multiple leaders, we show that the nodes in the optimal noise-free leader set balance high information centrality with a coverage of the graph. For special cases of graphs, we solve explicitly for optimal leader sets. Examples are used to illustrate.