Efficient, optimal k-leader selection for coherent, one-dimensional formations

Abstract

We study the problem of optimal leader selection in consensus networks with noisy relative information. The objective is to identify the set of k leaders that minimizes the formation's deviation from the desired trajectory established by the leaders. An optimal leader set can be found by an exhaustive search over all possible leader sets; however, this approach is not scalable to large networks. In recent years, several works have proposed approximation algorithms to the k-leader selection problem, yet the question of whether there exists an efficient, non-combinatorial method to identify the optimal leader set remains open. This work takes a first step towards answering this question. We show that, in one-dimensional weighted graphs, namely path graphs and ring graphs, the k-leader selection problem can be solved in polynomial time (in both k and the network size n). We give an O(n 3 ) solution for optimal k-leader selection in path graphs and an O(kn 3 ) solution for optimal k-leader selection in ring graphs.

Robustness of Leader–Follower Networked Dynamical Systems

“…In this paper, we refer to the quantityτ max = π 2λ n−|S| (Lg) as the delay threshold. The above characterization of the robustness of (5) to disturbances, based on system H 2 and H ∞ norms in (9) and (10), and the robustness of (6) to delay given by the quantity in (13), illustrates the role that the spectrum of the grounded Laplacian matrix L g plays in such robustness metrics. Thus, we analyze the spectrum of the grounded Laplacian matrix in this paper, and consequently give bounds on the above robustness metrics.…”

Optimal k-leader selection for coherence and convergence rate in one-dimensional networks

“…In ring graphs, our algorithms find the optimal leader set of size at most k in O(kn 3 ) time. Our algorithm for optimal k-leader selection for coherence first appeared in [18]. This paper extends our previous work to the leader selection problem for fast convergence.…”

Coherence and convergence rate in networked dynamical systems

“…In the context of network coherence, various papers have investigated the problem of choosing leaders to maximize coherence (minimize system H 2 norm) using different algorithms [9], [14], [15] and centrality metrics [6], [16]. There has also been an investigation of bounds on coherence and how it scales with network topology [7], [17], [18].…”