Algorithms for leader selection in stochastically forced consensus networks

TL;DR

This paper develops algorithms to optimally select leader nodes in stochastic consensus networks, minimizing deviation from consensus, with applications in control and sensor networks, using convex relaxations and semidefinite programming.

Contribution

It introduces convex relaxations and a specialized semidefinite programming approach for leader selection, addressing nonconvexities in stochastic network control.

Findings

Convex relaxations provide lower bounds on optimal leader sets.

Greedy algorithms effectively identify near-optimal leaders.

Semidefinite programming scales well to large networks.

Abstract

We are interested in assigning a pre-specified number of nodes as leaders in order to minimize the mean-square deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networks. For networks with leaders subject to noise, we show that the Boolean constraints (a node is either a leader or it is not) are the only source of nonconvexity. By relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound. For networks with leaders that perfectly follow their desired trajectories, we identify an additional source of nonconvexity in the form of a rank constraint. Removal of the rank constraint and relaxation of the Boolean constraints yields a semidefinite program for which we develop a customized algorithm well-suited for large networks. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms.