Structural Controllability of Large-Scale Hypergraphs

2026-03-20

AI Summary

Existing structural controllability theory struggles to address large-scale hypergraph systems characterized by high-order interactions and nonlinear dynamics. This work extends structural controllability to hypergraphs for the first time by modeling system dynamics through polynomial vector fields and integrating Lie algebraic tools with a Kalman-type rank condition. The authors derive a topology-based controllability criterion and establish a lower bound on the number of driver nodes required for control. Furthermore, they develop a scalable driver node selection algorithm that combines maximum matching with a greedy strategy. Empirical validation on hypergraphs ranging from tens to thousands of nodes demonstrates that the proposed approach substantially enhances both control efficiency and scalability in high-order networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• structural controllability
• hypergraphs
• large-scale networks
• higher-order interactions
• driver nodes

Innovation

• structural controllability
• hypergraphs
• polynomial dynamical systems
• driver node selection
• higher-order interactions