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

TL;DR

This paper introduces a resistance distance-based method to optimally select leader nodes in noisy consensus networks, linking network coherence to electrical resistance distances and providing analytical solutions for specific graph classes.

Contribution

It establishes a novel relationship between network coherence and resistance distances, enabling closed-form solutions for optimal leader placement in certain graph structures.

Findings

Derived closed-form coherence expressions based on resistance distances.

Established the relationship between leader placement and network performance.

Provided analytical solutions for optimal leader selection in specific graphs.

Abstract

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