  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. This paper focuses on the Volumetric Controllability Score (VCS) and
Average Energy Controllability Score (AECS) as centrality measures for
selecting control nodes in linear dynamical network systems. We prove the
uniqueness of VCS and AECS for any linear time-invariant system, enhancing
their applicability beyond previously recognized cases. This ensures their
interpretability, comparability, and reproducibility across different
researchers. Our analysis reveals substantial differences between VCS and AECS
in linear systems with symmetric and skew-symmetric transition matrices.
Moreover, numerical experiments using real-world data indicate that VCS and
AECS assessments in brain networks, modeled as Laplacian dynamics, exhibit
contrasting trends.
