Optimizing coherence in 1-D noisy consensus networks with noise-free leaders

We study the problem of K-leader selection in ring networks and path networks where leader nodes dictate the desired system state and follower nodes are governed by noisy consensus dynamics. We measure the performance of a network by its coherence, an ℋ
₂
norm that quantifies how closely the followers track the leaders' trajectory. By leveraging the connection between network coherence in leader-follower systems and the commute time of a simple random work, we derive analytical expressions for coherence in path and ring graphs in terms of the leader locations, and we identify the optimal leader placement. We then present distributed algorithms in which agents collaborate to identify and transfer leadership roles to these optimal leader locations, thus optimizing the coherence of the leader-follower system.