[Paper Review] Joint Centrality Distinguishes Optimal Leaders in Noisy Networks

Katherine Fitch, Naomi Ehrich Leonard|arXiv (Cornell University)|Jul 7, 2014

Distributed Control Multi-Agent Systems29 references4 citations

TL;DR

This paper introduces joint centrality, a novel graph-theoretic measure that identifies optimal leader sets in noisy multi-agent networks to minimize system error in tracking external signals. By expressing total system error in terms of network topology, the authors prove that noise-free leaders with maximal joint centrality yield minimal steady-state variance, with optimal configurations balancing high information centrality and graph coverage—especially in structured graphs like cycles, where uniform leader distribution minimizes error.

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 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 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. In the case of multiple leaders, we show that the nodes in the optimal 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.

Motivation & Objective

• To address the optimal leader selection problem in multi-agent networks tracking an external signal under stochastic disturbances.
• To develop a computationally efficient alternative to greedy algorithms for leader selection by leveraging network graph properties.
• To define a new centrality measure—joint centrality—that quantifies the collective influence of a leader set on system performance.
• To prove that maximal joint centrality corresponds to minimal total system error in both noise-free and noisy leader scenarios.
• To explicitly solve for optimal leader sets in special graph structures, such as cycle graphs with uniformly distributed leaders.

Proposed method

• Derives total system error as a function of the network graph's Laplacian and leader set configuration, using the inverse of the follower submatrix to quantify error.
• Introduces joint centrality as a measure of collective influence, derived from the trace of the inverse of the follower submatrix in the system's covariance matrix.
• Uses spectral graph theory to analyze eigenvalues of tridiagonal blocks in cycle graphs, linking error minimization to uniform distribution of follower segments.
• Applies resistance distance and matrix perturbation theory to express system error in terms of node pair distances and leader positions.
• Employs optimization techniques, including partial derivatives and rounding of continuous solutions, to identify discrete optimal leader sets in path and cycle graphs.
• Validates results through analytical proofs and illustrative examples on cycle and path graphs, showing convergence to optimal configurations.

Experimental results

Research questions

• RQ1 How can the optimal leader set be selected in a network to minimize steady-state variance in tracking an external signal?
• RQ2 What graph-theoretic property determines the collective influence of a set of leaders on system performance in noisy networks?
• RQ3 Does the optimal leader set for l leaders necessarily include the optimal set for m leaders (m < l), and how can this be avoided?
• RQ4 Can explicit solutions for optimal leader placement be derived in structured graphs like cycles and paths?
• RQ5 How does joint centrality compare to traditional centrality measures like information centrality in predicting optimal leader performance?

Key findings

• For a single leader, the optimal leader is the node with maximal information centrality, regardless of whether the leader is noise-free or noisy.
• In the case of multiple noise-free leaders, optimal performance is achieved when leaders balance high individual information centrality with broad coverage of the network graph.
• On a cycle graph with m leaders and n nodes where n/m is an integer, the optimal leader set is uniformly distributed with equal geodesic distances of p = n/m between adjacent leaders.
• The total system error is minimized when the follower segments between leaders are of equal length, which occurs when leaders are evenly spaced around the cycle.
• For path graphs, the optimal two-leader configuration is found by minimizing a quadratic form of the resistance distance, yielding leaders at approximately s₁ = round(n/5 + 0.5) and s₂ = round(4n/5 + 0.5).
• Joint centrality provides a computationally efficient alternative to brute-force or iterative greedy algorithms for leader selection, reducing complexity by directly linking system error to graph structure.