U-centrality: A Network Centrality Measure Based on Minimum Energy Control for Laplacian Dynamics
This work provides a principled tool for network analysis in dynamic environments, bridging structural and dynamical approaches, but it is incremental as it builds on existing centrality measures.
The authors tackled the problem of quantifying node importance in networks by proposing U-centrality, a dynamic, task-aware measure based on optimal control theory for Laplacian dynamics, which interpolates between degree centrality and current-flow closeness centrality.
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.