---
title: Volumetric Controllability Score (VCS)
url: https://www.emergentmind.com/topics/volumetric-controllability-score-vcs
type: topic
---

# Volumetric Controllability Score (VCS)

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 [2205.03032][2510.13354][2503.08011][2601.10260][2408.03023].

## 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 [2205.03032][2408.03023].

## 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 $\log\det$ [2205.03032].

- **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 [2408.03023][2510.13354].
- **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 [2408.03023].
- **Unstable systems**: For non-Hurwitz $A$, the finite-horizon Gramian is used, and under a spectral gap condition uniqueness persists for almost all $T$ [2601.10260].
- **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 [2503.08011].

## 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 [2205.03032][2408.03023].

- **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 [2408.03023].
- **Scalability**: For large $n$, use of low-rank Lyapunov solvers (e.g., CF-ADI) is recommended [2205.03032].

## 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 [2601.10260][2408.03023].
- For symmetric networks, VCS trivially recovers the uniform distribution, while AECS is generally nonuniform [2408.03023].

## 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 [2510.13354].

- **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 [2510.13354].
- **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 [2601.10260].

**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 [2503.08011].

## 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 [2408.03023]):

| 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 [2205.03032].
- **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 [2205.03032].
- **Practical usage**: For small $T$ the solution is uniform, while for large $T$ and Hurwitz $A$, VCS becomes $T$-independent [2408.03023]. 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 [2205.03032][2510.13354][2503.08011][2601.10260][2408.03023].

Source: https://www.emergentmind.com/topics/volumetric-controllability-score-vcs