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Controllability scores of linear time-varying network systems

Kota Umezu, Kazuhiro Sato

TL;DR

This work extends the controllability score from static LTI networks to linear time-varying (LTV) systems, enabling meaningful node centrality in temporal networks. It introduces volumetric and average-energy variants with a single global design variable $p$ over the entire horizon, proves almost-sure uniqueness and continuity with respect to time parameters, and develops a data-driven Gramian computation method that eliminates the need for full system identification. Through numerical experiments, the authors show that LTV scores can differ from LTI scores and better capture temporal dynamics and chronological order, highlighting advantages over existing generalized centralities. The proposed data-driven approach enables practical centrality assessment from experimental data and paves the way for applying the controllability score to real-world temporal networks such as brain and ecological systems.

Abstract

For large-scale network systems, network centrality based on control theory plays a crucial role in understanding their properties and controlling them efficiently. The controllability score is such a centrality index and can give a physically meaningful measure. It is originally proposed for linear time-invariant (LTI) systems, and we extend it to linear time-varying (LTV) systems in this paper. Since the controllability score is defined as an optimal solution to some optimization problem, it is not necessarily uniquely determined. Its uniqueness must be guaranteed for reproducibility and interpretability. In this paper, we show its uniqueness in almost all cases, which guarantees its use as a network centrality measure. We also prove its continuity with respect to the time parameters. In addition, we propose a data-driven method to compute it. Finally, we verify the effectiveness of the extension and examine the performance of the data-driven method through numerical experiments.

Controllability scores of linear time-varying network systems

TL;DR

This work extends the controllability score from static LTI networks to linear time-varying (LTV) systems, enabling meaningful node centrality in temporal networks. It introduces volumetric and average-energy variants with a single global design variable over the entire horizon, proves almost-sure uniqueness and continuity with respect to time parameters, and develops a data-driven Gramian computation method that eliminates the need for full system identification. Through numerical experiments, the authors show that LTV scores can differ from LTI scores and better capture temporal dynamics and chronological order, highlighting advantages over existing generalized centralities. The proposed data-driven approach enables practical centrality assessment from experimental data and paves the way for applying the controllability score to real-world temporal networks such as brain and ecological systems.

Abstract

For large-scale network systems, network centrality based on control theory plays a crucial role in understanding their properties and controlling them efficiently. The controllability score is such a centrality index and can give a physically meaningful measure. It is originally proposed for linear time-invariant (LTI) systems, and we extend it to linear time-varying (LTV) systems in this paper. Since the controllability score is defined as an optimal solution to some optimization problem, it is not necessarily uniquely determined. Its uniqueness must be guaranteed for reproducibility and interpretability. In this paper, we show its uniqueness in almost all cases, which guarantees its use as a network centrality measure. We also prove its continuity with respect to the time parameters. In addition, we propose a data-driven method to compute it. Finally, we verify the effectiveness of the extension and examine the performance of the data-driven method through numerical experiments.
Paper Structure (28 sections, 14 theorems, 51 equations, 5 figures, 5 tables, 4 algorithms)

This paper contains 28 sections, 14 theorems, 51 equations, 5 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Challenge
  4. Contribution
  5. Outline
  6. Preliminaries
  7. Notation
  8. Dynamical systems
  9. Controllability Gramian and minimum-energy control
  10. Controllability scores for LTV systems
  11. Formulation and definition
  12. Difference from the existing work
  13. Uniqueness of controllability scores
  14. Continuity of controllability scores
  15. Algorithm and data-driven computation
  16. ...and 13 more sections

Key Result

Proposition 1

Assume that the LTV system eq:ltv_input is controllable on the time interval $[0,T]$, which is equivalent to $\mathcal{W}(T)\succ O$. Then, for any desired state vector $x_{\mathrm{f}}\in\mathbb{R}^{n}$, the minimum energy required for driving the state vector from the origin at time $0$ to $x_{\mat

Figures (5)

  • Figure 1: Illustrations of the two types of network evolution
  • Figure 2: The difference between the GCS Mo2025 and ours
  • Figure 3: The snapshots of the temporal networks
  • Figure 8: The structure of the aggregated network
  • Figure 9: The error of the proposed data-driven method

Theorems & Definitions (22)

  • Proposition 1: minimum energy for control Kalman1963
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Proof 1
  • Proposition 4: SatoTerasaki2024
  • Proposition 5: SatoKawamura2025
  • Theorem 2
  • Proof 2
  • Lemma 1
  • ...and 12 more