Structural Controllability of Large-Scale Hypergraphs

TL;DR

This paper develops a framework for understanding and ensuring controllability in large-scale hypergraphs by extending classical control concepts to higher-order interactions, providing scalable algorithms and theoretical bounds.

Contribution

It introduces a structural controllability framework for hypergraphs using polynomial dynamical systems, extending classical notions and providing scalable driver node selection methods.

Findings

Established hypergraph controllability criteria based on topology.

Derived a lower bound on the number of driver nodes needed.

Demonstrated scalability through numerical experiments on large 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.