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Target Controllability Score

Kazuhiro Sato

TL;DR

The paper introduces the Target Controllability Score (TCS), a dynamics-aware centrality metric for designated target nodes in linear networks under actuator constraints. TCS comprises target VCS and target AECS, defined as convex optimizations over the target-weight vector $p$ using the output controllability Gramian $W(p,T)$, with gradients that quantify each target’s influence on reachability and energy. To enable scalability, a reduced virtual system is constructed, and rigorous bounds link the reduced Gramian $W_{red}(p,T)$ to $W(p,T)$ via cross-coupling and the logarithmic norm $\mu(A)$, yielding horizon-dependent approximation guarantees for both VCS and AECS. Numerical experiments on 88 human-brain networks reveal that short horizons favor accurate reduced-model approximations for both scores, while long horizons see AECS maintain robust, horizon-invariant target identification, whereas VCS becomes more horizon-sensitive. The work provides a principled framework for identifying intervention targets under practical actuation constraints and offers rigorous guidance on when reduced models faithfully approximate full-system metrics.

Abstract

We introduce the target controllability score (TCS), a concept for evaluating node importance under actuator constraints and designated target objectives, formulated within a virtual system setting. The TCS consists of the target volumetric controllability score (VCS) and the target average energy controllability score (AECS), each defined as an optimal solution to a convex optimization problem associated with the output controllability Gramian. We establish the existence and uniqueness (for almost all time horizons) and develop a projected gradient method for their computation. To enable scalability, we construct a target-only reduced virtual system and derive non-asymptotic bounds showing that weak cross-coupling and a low or negative logarithmic norm of the system matrix yield accurate approximations of target VCS/AECS, particularly over short or moderate time horizons. Experiments on human brain networks reveal a clear trade-off: at short horizons, both target VCS and target AECS are well approximated by their reduced formulations, while at long horizons, target AECS remains robust but target VCS deteriorates.

Target Controllability Score

TL;DR

The paper introduces the Target Controllability Score (TCS), a dynamics-aware centrality metric for designated target nodes in linear networks under actuator constraints. TCS comprises target VCS and target AECS, defined as convex optimizations over the target-weight vector using the output controllability Gramian , with gradients that quantify each target’s influence on reachability and energy. To enable scalability, a reduced virtual system is constructed, and rigorous bounds link the reduced Gramian to via cross-coupling and the logarithmic norm , yielding horizon-dependent approximation guarantees for both VCS and AECS. Numerical experiments on 88 human-brain networks reveal that short horizons favor accurate reduced-model approximations for both scores, while long horizons see AECS maintain robust, horizon-invariant target identification, whereas VCS becomes more horizon-sensitive. The work provides a principled framework for identifying intervention targets under practical actuation constraints and offers rigorous guidance on when reduced models faithfully approximate full-system metrics.

Abstract

We introduce the target controllability score (TCS), a concept for evaluating node importance under actuator constraints and designated target objectives, formulated within a virtual system setting. The TCS consists of the target volumetric controllability score (VCS) and the target average energy controllability score (AECS), each defined as an optimal solution to a convex optimization problem associated with the output controllability Gramian. We establish the existence and uniqueness (for almost all time horizons) and develop a projected gradient method for their computation. To enable scalability, we construct a target-only reduced virtual system and derive non-asymptotic bounds showing that weak cross-coupling and a low or negative logarithmic norm of the system matrix yield accurate approximations of target VCS/AECS, particularly over short or moderate time horizons. Experiments on human brain networks reveal a clear trade-off: at short horizons, both target VCS and target AECS are well approximated by their reduced formulations, while at long horizons, target AECS remains robust but target VCS deteriorates.
Paper Structure (17 sections, 12 theorems, 118 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 17 sections, 12 theorems, 118 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Target Controllability Score
  3. Output Controllability
  4. Geometric and Energy-Based Interpretation of Output Controllability
  5. Target VCS and Target AECS
  6. Uniqueness of Target Controllability Score
  7. Approximate computation of target VCS/AECS
  8. Error and Structural Analysis between the Reduced and Output Controllability Gramians
  9. Relative (Multiplicative) Gramian Error Analysis
  10. Approximation Guarantees for Target VCS/AECS
  11. Numerical experiments using real-world human brain network data
  12. Comparison of Target and Non-Target Coupling for VCS and AECS
  13. Approximation Error Between Target and Reduced Formulations
  14. Conclusions
  15. Summary
  16. ...and 2 more sections

Key Result

Lemma 1

For any $T>0$ and any $i\in\{1,\dots,m\}$, the matrix $W_i(T)$ in output_con_Gra satisfies $W_i(T)\succeq O$ and $W_i(T)\neq O$.

Figures (8)

  • Figure 1: Comparison between the conventional and practical settings.
  • Figure 2: Illustration of the block decomposition of $A$ and the role of $A_{12}$. The block $A_{12}$ represents upward coupling from the lower-level subsystem to the higher-level subsystem. When $\|A_{12}\|$ is small and the exponential factor $\Phi_{\mu(A)}(T)$ is sufficiently small, the reduced Gramian $W_{\mathrm{red}}(p,T)$ provides a close approximation to the full output controllability Gramian $W(p,T)$.
  • Figure 3: Boxplots of the top 5 nodes: target AECS (top) and its reduced-system approximation (bottom) for $(T,m)=(1,30)$.
  • Figure 4: Boxplots of the top 5 nodes: target VCS (top) and its reduced-system approximation (bottom) for $(T,m)=(1,30)$.
  • Figure 5: Boxplots of the top 5 nodes: target AECS (top) and its reduced-system approximation (bottom) for $(T,m)=(100,30)$.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Remark 1
  • Lemma 1
  • Proof 1
  • Proposition 1
  • Remark 2
  • Lemma 2
  • Proof 2
  • Lemma 3
  • Proof 3
  • Theorem 1
  • ...and 19 more