Uniqueness Analysis of Controllability Scores and Their Application to Brain Networks

Assessing centrality in network systems is critical for understanding node importance and guiding decision-making processes. In dynamic networks, incorporating a controllability perspective is essential for identifying key nodes. In this paper, we study two control theoretic centrality measures -- the Volumetric Controllability Score (VCS) and Average Energy Controllability Score (AECS) -- to quantify node importance in linear time-invariant network systems. We prove the uniqueness of VCS and AECS for almost all specified terminal times, thereby enhancing their applicability beyond previously recognized cases. This ensures their interpretability, comparability, and reproducibility. Our analysis reveals substantial differences between VCS and AECS in linear systems with symmetric and skew-symmetric transition matrices. We also investigate the dependence of VCS and AECS on the terminal time and prove that when this parameter is extremely small, both scores become essentially uniform. Additionally, we prove that a sequence generated by a projected gradient method for computing VCS and AECS converges linearly to both measures under several assumptions. Finally, evaluations on brain networks modeled via Laplacian dynamics using real data reveal contrasting evaluation tendencies and correlations for VCS and AECS, with AECS favoring brain regions associated with cognitive and motor functions, while VCS emphasizes sensory and emotional regions.