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Task-Dependent Weighted Average Energy Controllability Score for Network Intervention

Kazuhiro Sato

Abstract

Controllability scores provide principled information on where intervention should be applied in large-scale network systems when explicit control design is difficult. Two representative controllability scores are the volumetric controllability score (VCS) and the average energy controllability score (AECS). While both are important, the standard AECS treats all state-transition directions uniformly. In this paper, we propose the weighted average energy controllability score (W-AECS), a task-dependent extension of AECS that incorporates a prescribed transition of interest through a weighting matrix. We show that the proposed formulation admits a control-theoretic interpretation via expected minimum-energy steering, and establish strict convexity and generic uniqueness. These results support the interpretation of W-AECS as a well-defined node-wise task-dependent intervention score. We also illustrate the proposed method on a structural brain-network dataset, where transition-dependent weighting reshapes the scoring pattern, yielding a VCS-like preference among the highest-ranked regions while preserving an overall structure distinct from both standard AECS and VCS.

Task-Dependent Weighted Average Energy Controllability Score for Network Intervention

Abstract

Controllability scores provide principled information on where intervention should be applied in large-scale network systems when explicit control design is difficult. Two representative controllability scores are the volumetric controllability score (VCS) and the average energy controllability score (AECS). While both are important, the standard AECS treats all state-transition directions uniformly. In this paper, we propose the weighted average energy controllability score (W-AECS), a task-dependent extension of AECS that incorporates a prescribed transition of interest through a weighting matrix. We show that the proposed formulation admits a control-theoretic interpretation via expected minimum-energy steering, and establish strict convexity and generic uniqueness. These results support the interpretation of W-AECS as a well-defined node-wise task-dependent intervention score. We also illustrate the proposed method on a structural brain-network dataset, where transition-dependent weighting reshapes the scoring pattern, yielding a VCS-like preference among the highest-ranked regions while preserving an overall structure distinct from both standard AECS and VCS.

Paper Structure

This paper contains 22 sections, 5 theorems, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Virtual actuation model
  4. Finite-horizon controllability Gramian
  5. Expected Minimum Energy
  6. Weighted Average Energy Controllability Scoring
  7. Existence of an optimal solution
  8. Strict convexity
  9. Uniqueness
  10. Optimization algorithm for W-AECS
  11. Numerical experiment on brain networks
  12. Purpose of the experiment
  13. Dataset, network dynamics, and time horizon
  14. Task specification: transition from a rest-related state to a VCS-favored state
  15. ROI classification used in the experiment
  16. ...and 7 more sections

Key Result

Lemma 1

Fix $T>0$ and assume that $M(T)\succ O$. Then Problem eq:opt_problem admits an optimal solution.

Figures (1)

  • Figure 1: Box plots of W-AECS for $T=100$. (Left: Top 5; Right: Bottom 5.)

Theorems & Definitions (14)

  • Example 1
  • Example 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2: Directional derivatives
  • proof
  • Theorem 1
  • proof
  • ...and 4 more