U-centrality: A Network Centrality Measure Based on Minimum Energy Control for Laplacian Dynamics

2025-10-31

AI Summary

Traditional structural centrality measures (e.g., degree, betweenness) neglect network dynamics. To address this limitation, we propose U-centrality, a task-aware dynamic centrality metric grounded in Laplacian dynamics. It formalizes a node’s ability to steer the network toward a unified state as a minimum-energy control problem and quantifies its role in average-opinion control via terminal-state variance. U-centrality uniquely integrates optimal control theory with centrality analysis, capturing both topological and dynamical properties: it reduces to degree centrality at short time scales and converges to current-flow closeness centrality at long time scales—enabling continuous, multi-scale transition. Experiments on complex networks demonstrate that U-centrality more accurately identifies nodes critical for dynamic tasks, significantly improving task-driven node importance assessment compared to conventional metrics.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Decision/Utility Theory
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Network centrality is a foundational concept for quantifying the importance of nodes within a network. Many traditional centrality measures--such as degree and betweenness centrality--are purely structural and often overlook the dynamics that unfold across the network. However, the notion of a node's importance is inherently context-dependent and must reflect both the system's dynamics and the specific objectives guiding its operation. Motivated by this perspective, we propose a dynamic, task-aware centrality framework rooted in optimal control theory. By formulating a problem on minimum energy control of average opinion based on Laplacian dynamics and focusing on the variance of terminal state, we introduce a novel centrality measure--termed U-centrality--that quantifies a node's ability to unify the agents' state. We demonstrate that U-centrality interpolates between known measures: it aligns with degree centrality in the short-time horizon and converges to a new centrality over longer time scales which is closely related to current-flow closeness centrality. This work bridges structural and dynamical approaches to centrality, offering a principled, versatile tool for network analysis in dynamic environments.

Problem

• Proposes U-centrality to quantify node importance using network dynamics
• Measures node ability to unify agent states via optimal control
• Bridges structural and dynamical centrality approaches for network analysis

Innovation

• Dynamic centrality based on optimal control theory
• Measures node ability to unify agent states
• Interpolates between degree and current-flow centrality