Volumetric Controllability Score (VCS)

• VCS is a metric that defines node controllability by optimally allocating actuator energy to maximize the ellipsoid volume of reachable states in linear time-invariant networks.
• It is computed through a strictly convex program minimizing the negative log-determinant of the controllability Gramian, ensuring unique solutions under mild conditions.
• The framework extends to targeted, unstable, and infinite-dimensional systems, integrating dynamics-aware network analysis with traditional centrality measures.

The Volumetric Controllability Score (VCS) is a convex-optimization-based metric for quantifying node importance and control centrality in linear time-invariant (LTI) networks. VCS measures, for a given energy constraint, how the allocation of actuators across network nodes can maximize the volume of the reachable state set, thus encoding the system’s global controllability structure. It is defined as the unique solution (under mild conditions) of a strictly convex program involving the determinant of the controllability Gramian under a “virtual input” budget allocation. The VCS forms a foundation for dynamics-aware network analysis and has been extended to target-node settings, unstable or infinite-dimensional systems, and large-scale computational scenarios (Sato et al., 2022, Sato, 15 Oct 2025, Nakabe et al., 11 Mar 2025, Umezu et al., 15 Jan 2026, Sato et al., 2024).

1. Formal Definition and Geometric Interpretation

Given a continuous-time, linear time-invariant system

$$\dot{x}(t) = A x(t) + B u(t), \quad x\in\mathbb{R}^n,\, u\in\mathbb{R}^n$$

with “diagonal input” structure $B = \mathrm{diag}(\sqrt{p_1},\dots,\sqrt{p_n})$, the infinite- or finite-horizon controllability Gramian is

$$W_\mathrm{con}(p, T) = \int_0^T e^{At} B B^\top e^{A^\top t}\,dt = \sum_{i=1}^n p_i W_i(T)$$

where $W_i(T)$ is the Gramian for single-node actuation. Under $A$ Hurwitz and $T=\infty$, $W_\mathrm{con}(p) \succ 0$ is the unique solution to a Lyapunov equation.

The Gramian defines an ellipsoid of states reachable with unit-energy inputs: $$\mathcal{E} = \{ x_f \mid x_f^\top W_\mathrm{con}(p, T)^{-1} x_f \leq 1 \}$$ whose volume is proportional to $\sqrt{\det W_\mathrm{con}(p, T)}$. Thus, $\det W_\mathrm{con}(p, T)$ quantifies the control access to the state space.

The VCS is the optimal distribution $p^*\in\Delta$ (where $\Delta = \{p\ge 0:\sum p_i=1\}$) that maximizes the ellipsoid volume, i.e., minimizes $-\log\det W_\mathrm{con}(p, T)$. Formally,

$$\min_{p\in\Delta}\ -\log\det W_\mathrm{con}(p,T) \quad \text{subject to}\ W_\mathrm{con}(p,T) \succ 0$$

The VCS vector $p^*$ assigns to each node $i$ a centrality value indicating its optimal share of control “resources” to maximize controllability volume (Sato et al., 2022, Sato et al., 2024).

2. Theoretical Properties and Uniqueness

The VCS objective $f(p) = -\log\det W_\mathrm{con}(p, T)$ is strictly convex over the feasible set where $W_\mathrm{con}(p,T) \succ 0$, due to the linearity of $W_\mathrm{con}$ in $p$ and the matrix concavity of $\log\det$ (Sato et al., 2022).

• Existence: Compactness of the simplex $\Delta$ and continuity of $f$ ensure that a minimizer exists, as any sublevel set $\{p \mid f(p)\le f(p^{(0)})\} \cap \Delta$ is compact.
• Uniqueness: For almost all $T>0$ and for generically chosen $A$, the Gramian blocks $\{W_i(T)\}$ are linearly independent, guaranteeing strict convexity and uniqueness of the optimizer (Sato et al., 2024, Sato, 15 Oct 2025).
• Special cases: For symmetric $A$, the solution is always uniform $p^* = (1/n)\mathbf{1}$; for skew-symmetric $A$, VCS and AECS coincide and are uniform (Sato et al., 2024).
• Unstable systems: For non-Hurwitz $A$, the finite-horizon Gramian is used, and under a spectral gap condition uniqueness persists for almost all $T$ (Umezu et al., 15 Jan 2026).
• Infinite-dimensional extension: Under appropriate commutativity and regularity conditions, existence and uniqueness hold for VCS posed on general separable Hilbert spaces, with the objective defined in terms of the product of the $n$ largest Gramian eigenvalues (Nakabe et al., 11 Mar 2025).

3. Algorithmic Computation

VCS is solved using a projected-gradient algorithm on the standard simplex, with each iteration of the form

$$p^{(k+1)} = \Pi_\Delta \!\bigl(p^{(k)} - \alpha^{(k)} \nabla f(p^{(k)})\bigr)$$

where $$[\nabla f(p)]_i = -\mathrm{tr}\left(W_\mathrm{con}(p, T)^{-1} W_i(T)\right)$$ and $\Pi_\Delta$ denotes Euclidean projection. Stepsize $\alpha^{(k)}$ is selected by an Armijo-type backtracking line search (Sato et al., 2022, Sato et al., 2024).

• Per-iteration complexity: Dominated by computation and inversion (or factorization) of $W_\mathrm{con}(p, T)$, typically $O(n^3)$.
• Projection: Implemented efficiently ($O(n\log n)$) by thresholding and sorting.
• Convergence: Under strict convexity, convergence to the unique VCS solution is linear in the norm, provided appropriate stepsize bounds (Sato et al., 2024).
• Scalability: For large $n$, use of low-rank Lyapunov solvers (e.g., CF-ADI) is recommended (Sato et al., 2022).

4. Relationship to Other Centrality Measures

VCS is contrasted with the Average Energy Controllability Score (AECS), which minimizes $\mathrm{tr}(W_\mathrm{con}(p,T)^{-1})$ and reflects the mean minimum energy for state transfer.

Measure	Optimization Objective	Emphasis	Sensitivity
VCS	$-\log\det W_\mathrm{con}(p,T)$	Reachable volume	Uniformly spreads input across all directions; penalizes near-singular directions
AECS	$\mathrm{tr}(W_\mathrm{con}^{-1})$	Average control energy	Sensitive to easy-to-control directions; may neglect hard-to-reach modes

• For stable $A$ and large T, both criteria yield well-posed and unique solutions, but VCS is fundamentally more conservative, enforcing controllability across all subspaces, while AECS may neglect “hard” directions (Umezu et al., 15 Jan 2026, Sato et al., 2024).
• For symmetric networks, VCS trivially recovers the uniform distribution, while AECS is generally nonuniform (Sato et al., 2024).

5. Extensions: Targeted, Infinite-Dimensional, and Unstable Systems

Targeted VCS: For systems with actuator constraints targeting specific nodes or outputs, the Target VCS optimizes the Gramian associated with the target outputs, using an analogous convex program on a reduced simplex (Sato, 15 Oct 2025).

• Reduced surrogate: For weak target–non-target coupling and short time horizons, a projected model restricted to targets yields accurate approximations, with nonasymptotic error bounds (Sato, 15 Oct 2025).
• Long-horizon sensitivity: The accuracy of the surrogate degrades for large horizons, with VCS being more sensitive than AECS.

Infinite-horizon and Unstable Dynamics: For unstable or non-diagonalizable $A$, the Gramian is regularized via scaling (block-diagonalizing $A$ and applying time-dependent scalings) so that VCS can be computed in the $T\to\infty$ limit and is unique when the stable part exists and blocks are linearly independent (Umezu et al., 15 Jan 2026).

Infinite-dimensional networks: VCS generalizes to separable Hilbert spaces, and the objective becomes the sum of logs of the $n$ largest Gramian eigenvalues, with existence and uniqueness guaranteed under weak structural assumptions. In practice, truncation and projection onto finite subspaces are used for computation (Nakabe et al., 11 Mar 2025).

6. Empirical Performance, Application, and Limitations

Empirical results on brain structural networks reveal that VCS highlights nodes associated with sensory and emotional processing, while AECS favors cognitive and motor hubs. Correlation analysis demonstrates that VCS and AECS are complementary and less aligned with standard centrality metrics such as degree, betweenness, and PageRank (see summary correlations in (Sato et al., 2024)):

Metric Pair	Correlation with AECS	Correlation with VCS
Indegree/Outdegree	+0.64	–0.30/–0.27
Betweenness	+0.68	–0.26
PageRank	+0.64	–0.29
Average Controllability Score	–0.60	+0.84

• Algorithmic scalability: VCS solvers based on projected gradient are substantially faster than interior-point methods, being tractable for $n\gtrsim 10^3$, but still bottlenecked by Gramian factorization (Sato et al., 2022).
• Assumptions: Gramian-based VCS requires full knowledge of $A$ and is sensitive to model errors. Extensions to data-driven or partially known $A$ remain open (Sato et al., 2022).
• Practical usage: For small $T$ the solution is uniform, while for large $T$ and Hurwitz $A$, VCS becomes $T$-independent (Sato et al., 2024). In targeted or high-dimensional settings, surrogate models and low-rank methods are essential.

7. Summary and Research Directions

VCS provides a strictly convex, well-posed, and interpretable measure of node controllability centrality in dynamical networks. Its geometric foundation is tied to maximal reachable volume under input constraints, forming a mathematically rigorous basis for node selection. Extensions accommodate target control, unstable and infinite-dimensional dynamics, and empirical results confirm its sensitivity to global dynamics and structural bottlenecks. Limitations involve computational scaling, necessity of precise system knowledge, and the breakdown of simplifications for large time horizons or unstable dynamics.

Active research directions include scalable and incomplete-data variants, robustification for model errors, and systematic integration with other network centrality paradigms (Sato et al., 2022, Sato, 15 Oct 2025, Nakabe et al., 11 Mar 2025, Umezu et al., 15 Jan 2026, Sato et al., 2024).