U-centrality: A Network Centrality Measure Based on Minimum Energy Control for Laplacian Dynamics
Xinran Zheng, Leonardo Massai, Massimo Franceschetti, Behrouz Touri
TL;DR
This work introduces U-centrality, a dynamic, task-aware centrality measure derived from minimum-energy control for Laplacian dynamics, where the centrality of node $i$ is quantified by the $\ell_2$ distance between the terminal state when only node $i$ is controlled and the consensus state. The framework yields closed-form energy expressions and terminal states, with $E_i = \dfrac{c^2}{t_f}$ for single-node control and $\boldsymbol{x}_{fi} = \dfrac{c}{t_f} \int_0^{t_f} e^{-L\tau} \mathbf{e}_i d\tau$, enabling a nodewise ranking. In the short-time limit ($t_f \approx 0$), U-centrality coincides with degree centrality, while in the long-time limit ($t_f \gg 0$) it aligns with Laplacian inverse centrality (via $L^{\dagger}$); for trees, explicit relationships between $L^{\dagger}$ entries and graph distances illuminate node peripherality. Numerical experiments on trees, road networks, and social networks corroborate these theoretical insights, showing a smooth interpolation between classic structural measures and current-flow-based metrics across time scales.
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.