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We study the performance of leader-follower noisy consensus networks, and inparticular, the relationship between this performance and the locations of theleader nodes. Two types of dynamics are considered (1) noise-free leaders, inwhich leaders dictate the trajectory exactly and followers are subject toexternal disturbances, and (2) noise-corrupted leaders, in which both leadersand followers are subject to external perturbations. We measure the performanceof a network by its coherence, an $H_2$ norm that quantifies how closely thefollowers track the leaders' trajectory. For both dynamics, we show arelationship between the coherence and resistance distances in an a electricalnetwork. Using this relationship, we derive closed-form expressions forcoherence as a function of the locations of the leaders. Further, we giveanalytical solutions to the optimal leader selection problem for severalspecial classes of graphs.

A Resistance Distance-Based Approach for Optimal Leader Selection in Noisy Consensus Networks