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Controllability and Observability of Temporal Hypergraphs

Anqi Dong, Xin Mao, Can Chen

TL;DR

This work addresses controllability and observability for temporal hypergraphs by formulating time-varying tensor-polynomial dynamics $\dot{\mathbf{x}}(t) = \sum_{j=2}^k \mathscr{A}_j(t) \mathbf{x}(t)^{j-1}$ with linear inputs $\mathbf{B}(t)\mathbf{u}(t)$. It derives a tensor-based rank condition via a controllability matrix $\mathbf{C}(\mathbf{x},t)$ built from recursive matrices $\mathbf{M}_i$, and defines a corresponding observability framework using Lie derivatives to form an observability matrix $\mathbf{O}(\mathbf{x},t)$. The paper also discusses practical identification of the minimum number of driver nodes (MNDN) through a greedy algorithm and minimum sensor nodes (MNSN) analog, validated on synthetic and ecological networks. The results provide concrete tools for controlling and estimating dynamics in time-varying higher-order networks, with implications for stability, resource allocation, and dynamical prediction in complex systems.

Abstract

Numerous complex systems, such as those arisen in ecological networks, genomic contact networks, and social networks, exhibit higher-order and time-varying characteristics, which can be effectively modeled using temporal hypergraphs. However, analyzing and controlling temporal hypergraphs poses significant challenges due to their inherent time-varying and nonlinear nature, while most existing methods predominantly target static hypergraphs. In this article, we generalize the notions of controllability and observability to temporal hypergraphs by leveraging tensor and nonlinear systems theory. Specifically, we establish tensor-based rank conditions to determine the weak controllability and observability of temporal hypergraphs. The proposed framework is further demonstrated with synthetic and real-world examples.

Controllability and Observability of Temporal Hypergraphs

TL;DR

This work addresses controllability and observability for temporal hypergraphs by formulating time-varying tensor-polynomial dynamics with linear inputs . It derives a tensor-based rank condition via a controllability matrix built from recursive matrices , and defines a corresponding observability framework using Lie derivatives to form an observability matrix . The paper also discusses practical identification of the minimum number of driver nodes (MNDN) through a greedy algorithm and minimum sensor nodes (MNSN) analog, validated on synthetic and ecological networks. The results provide concrete tools for controlling and estimating dynamics in time-varying higher-order networks, with implications for stability, resource allocation, and dynamical prediction in complex systems.

Abstract

Numerous complex systems, such as those arisen in ecological networks, genomic contact networks, and social networks, exhibit higher-order and time-varying characteristics, which can be effectively modeled using temporal hypergraphs. However, analyzing and controlling temporal hypergraphs poses significant challenges due to their inherent time-varying and nonlinear nature, while most existing methods predominantly target static hypergraphs. In this article, we generalize the notions of controllability and observability to temporal hypergraphs by leveraging tensor and nonlinear systems theory. Specifically, we establish tensor-based rank conditions to determine the weak controllability and observability of temporal hypergraphs. The proposed framework is further demonstrated with synthetic and real-world examples.
Paper Structure (13 sections, 6 theorems, 25 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 6 theorems, 25 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraphs
  4. Tensor-based Polynomial Systems
  5. Nonlinear Controllability & Observability
  6. Controllability
  7. Tensor-based Rank Condition
  8. Controllability of Temporal Hypergraphs
  9. Observability
  10. Numerical Examples
  11. Uniform Temporal Hypergraphs
  12. Higher-order Ecological Systems
  13. Conclusion

Key Result

Proposition 1

The tensor-based polynomial time-varying control system eq:hgdl is locally weakly controllable if and only if the controllability matrix is defined as where $\textbf{M}_0(\textbf{x},t) = \textbf{B}(t)$, and for $i=1,2,\dots,n-1$, has full rank. Here, $\textbf{m}_{i-1}^{(p)}(\textbf{x},t)$ denotes the $p$th column of $\textbf{M}_{i-1}(\textbf{x},t)$ with total $l$ columns.

Figures (3)

  • Figure 1: Temporal hypergraph whose structure changes over time. The connectivity is shown by the hyperedges represented by areas with distinct colors, which changes over time ($t_0,t_1,\dots, t_n$).
  • Figure 2: 3-uniform hypergraph with time-varying weights on hyperedges.
  • Figure 3: Two higher-order temporal ecological networks with pairwise and third-order interactions.

Theorems & Definitions (16)

  • Definition 1: Adjacency tensors
  • Definition 2: Hypergraph dynamics
  • Definition 3: Local weak controllability
  • Definition 4: Local weak observability
  • Definition 5: Lie brackets
  • Proposition 1: Controllability
  • proof
  • Corollary 2: Homogeneous case
  • proof
  • Corollary 3: Linear case
  • ...and 6 more