Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structural Controllability of Large-Scale Hypergraphs

Joshua Pickard, Xin Mao, Can Chen

Abstract

Controlling real-world networked systems, including ecological, biomedical, and engineered networks that exhibit higher-order interactions, remains challenging due to inherent nonlinearities and large system scales. Despite extensive studies on graph controllability, the controllability properties of hypergraphs remain largely underdeveloped. Existing results focus primarily on exact controllability, which is often impractical for large-scale hypergraphs. In this article, we develop a structural controllability framework for hypergraphs by modeling hypergraph dynamics as polynomial dynamical systems. In particular, we extend classical notions of accessibility and dilation from linear graph-based systems to polynomial hypergraph dynamics and establish a hypergraph-based criterion under which the topology guarantees satisfaction of classical Lie-algebraic and Kalman-type rank conditions for almost all parameter choices. We further derive a topology-based lower bound on the minimum number of driver nodes required for structural controllability and leverage this bound to design a scalable driver node selection algorithm combining dilation-aware initialization via maximum matching with greedy accessibility expansion. We demonstrate the effectiveness and scalability of the proposed framework through numerical experiments on hypergraphs with tens to thousands of nodes and higher-order interactions.

Structural Controllability of Large-Scale Hypergraphs

Abstract

Controlling real-world networked systems, including ecological, biomedical, and engineered networks that exhibit higher-order interactions, remains challenging due to inherent nonlinearities and large system scales. Despite extensive studies on graph controllability, the controllability properties of hypergraphs remain largely underdeveloped. Existing results focus primarily on exact controllability, which is often impractical for large-scale hypergraphs. In this article, we develop a structural controllability framework for hypergraphs by modeling hypergraph dynamics as polynomial dynamical systems. In particular, we extend classical notions of accessibility and dilation from linear graph-based systems to polynomial hypergraph dynamics and establish a hypergraph-based criterion under which the topology guarantees satisfaction of classical Lie-algebraic and Kalman-type rank conditions for almost all parameter choices. We further derive a topology-based lower bound on the minimum number of driver nodes required for structural controllability and leverage this bound to design a scalable driver node selection algorithm combining dilation-aware initialization via maximum matching with greedy accessibility expansion. We demonstrate the effectiveness and scalability of the proposed framework through numerical experiments on hypergraphs with tens to thousands of nodes and higher-order interactions.
Paper Structure (23 sections, 7 theorems, 32 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 7 theorems, 32 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensors and Hypergraphs
  4. Controllability of Polynomial Systems and Hypergraphs
  5. Linear Structural Controllability
  6. Structural Controllability of Hypergraphs
  7. Limitations Imposed by System Identification
  8. Hypergraph Structure for Polynomial Controllability
  9. Directed Hypergraph Walks
  10. Accessibility and Dilations
  11. Structural Controllability of Polynomial Systems
  12. Identification of Minimum Number of Driver Nodes
  13. Hypergraph Matching-Based Lower Bound
  14. Matching-Augmented Driver Node Selection
  15. Numerical Experiments
  16. ...and 8 more sections

Key Result

Proposition 1

The homogeneous polynomial system with linear input eq: polynomial with linear inputs is strongly controllable if and only if the Lie algebra spanned by the set of vector fields $\{\textbf{f},\textbf{b}_{1},\textbf{b}_{2},\dots,\textbf{b}_{m}\}$ is full rank, where $\textbf{b}_{j}$ are the $j$th col

Figures (4)

  • Figure 1: Examples of hypergraph walk, accessibility, and dilation. This hypergraph is not structurally controllable. Hyperedges are uniquely colored, and nodes are shaded to match the color of their incident hyperedge head. Hyperedges are numbered according to the order they may appear in a walk originating at the control node. The black and dark purple nodes are inaccessible, and the light purple nodes form a dilation. The black node is inaccessible as it is not an input node and lacks an incident hyperedge head. Consequently, the dark purple node is also inaccessible; its incident hyperedge head cannot be traversed because the black node is inaccessible. The light purple nodes form a dilation since there is only a single hyperedge head for two nodes.
  • Figure 2: Driver node selection on small-scale hypergraphs. Comparison of four methods for selecting driver nodes in 4-uniform hypergraphs of varying size $n$ and density $\alpha$. Our proposed method (MaG), pure greedy selection, linear matching, and brute force search (optimal).
  • Figure 3: Scaling of MaG, matching, and greedy based control node selection for large-scale hypergraphs. Runtime is averaged over three trials per hypergraph size.
  • Figure 4: Structural properties of driver nodes in structured hypergraph topologies. Comparison of driver node characteristics across scale-free (red), clustered (green), and small-world (orange) hypergraph networks with $n=100$ varying densities. Solid lines show means over 10 independent realizations; shaded regions indicate standard deviations. Points represent individual trials.

Theorems & Definitions (16)

  • Proposition 1: jurdjevic1997geometric
  • Proposition 2: chen2021controllability
  • Proposition 3: lee1967foundations
  • Proposition 4: lin1974structural
  • Proposition 5
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • ...and 6 more