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

TL;DR

This paper introduces U-centrality, a novel network centrality measure based on minimum energy control of Laplacian dynamics, capturing node importance in dynamic, task-specific contexts and bridging structural and dynamical centrality concepts.

Contribution

It proposes U-centrality, a dynamic, task-aware centrality measure derived from optimal control theory, unifying structural and dynamical network analysis approaches.

Findings

U-centrality interpolates between degree and current-flow closeness centralities.

The measure is rooted in minimum energy control of Laplacian dynamics.

U-centrality effectively captures node importance in dynamic environments.

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.