Joint Centrality Distinguishes Optimal Leaders in Noisy Networks

Abstract

We study the performance of a network of agents tasked with tracking an external unknown signal in the presence of stochastic disturbances and under the condition that only a limited subset of agents, known as leaders, can measure the signal directly. We investigate the optimal leader selection problem for a prescribed maximum number of leaders, where the optimal leader set minimizes total system error defined as steady-state variance about the external signal.In contrast to previously established greedy algorithms for optimal leader selection, our results rely on an expression of total system error in terms of properties of the underlying network graph. We demonstrate that the performance of any given set of noise-free leaders depends on their influence as determined by a new graph measure of centrality of a set. We define the joint centrality of a set of nodes in a network graph such that a noise-free leader set with maximal joint centrality is an optimal leader set. In the case of a single leader, we prove that the optimal leader is the node with maximal information centrality for both the nois-corrupted and noise-free leader cases. In the case of multiple leaders, we show that the nodes in the optimal noise-free leader set balance high information centrality with a coverage of the graph. For special cases of graphs, we solve explicitly for optimal leader sets. We illustrate with examples.

Mean-field analysis of the convergence time of message-passing computation of harmonic influence in social networks

“…This definition is equivalent to the Harmonic Influence Centrality introduced in Vassio et al (2014) and implicitly used in Acemoglu et al (2013); Yildiz et al (2013). Also other definitions have bee used to evaluate nodes as potential leaders, see for instance Lin et al (2014); Fitch and Leonard (2016).…”

Structured Decentralized Control of Positive Systems With Applications to Combination Drug Therapy and Leader Selection in Directed Networks

“…We denote the resulting lower bounds on the optimal values of the H 2 and H ∞ versions of Problem 1 with N leaders by J lb 2 (N ) and J lb ∞ (N ), respectively. 2) Upper bounds for Problem 1: If k denotes the number of subsets S j , a stabilizing candidate solution to Problem 1 can be obtained by "rounding" the solution to (14) by taking N leaders to contain the largest element from each subset S j and N − k largest remaining elements. The greedy swapping algorithm proposed in [12] can further tighten this upper bound.…”

“…The symmetric component of the Laplacian of a balanced graph, L s := 1 2 (L + L T ), is the Laplacian of an undirected network. The exact optimal leader set for an undirected network can be efficiently computed when N is either small or large [13], [14]. Since the performance of the symmetric component of a system provides an upper bound on the performance of the original system, these sets of leaders will have better performance with L than with L s for both the H 2 [48,Corollary 3] and H ∞ norms [49,Proposition 4].…”

Coherence and convergence rate in networked dynamical systems

“…From (16) we have trace(L −1 gk ) − trace(L −1 gi ) = 2S k i − nr ik . Thus for v k to be a better leader than v i for H 2 coherence, it is sufficient to have…”

“…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].…”