[论文解读] Structural Controllability of Large-Scale Hypergraphs

Joshua Pickard, Xin Mao+1 more|arXiv (Cornell University)|Mar 20, 2026

Distributed Control Multi-Agent Systems被引用 0

一句话总结

该论文通过将超图动力学建模为多项式（张量）系统，扩展可达性与扩张性，并提出可扩展的驱动节点选择算法，建立了面向大规模超图的结构可控性框架。还给出基于拓扑的驱动节点数量下界，并在大规模超图上验证该方法。

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.

研究动机与目标

• 为说明在存在高阶（超边）交互和不确定参数的现实网络中需要控制的动机
• 将超图动力学建模为多项式（张量）系统以捕捉高阶交互
• 通过张量框架将线性结构可控性的概念（可达性与扩张性）扩展到超图
• 推导关于驱动节点最小数量的基于拓扑的下界
• 提出结合最大匹配与贪婪可达性扩展的可扩展驱动节点选择算法
• 通过对大规模超图的数值实验展示可扩展性

提出的方法

• 使用 k 阶邻接张量表示超图动力学以构建齐次多项式系统
• 将李代数/卡尔曼型可控性概念扩展到张量基动力学（非线性可控性矩阵 C）
• 定义带状态边和控制边的有向超图以编码稀疏性模式
• 引入超图步、可达性和扩张性以将结构可控性推广到多项式系统
• 证明结构可控性当且仅当超图中不存在不可达节点和无扩张（并给出相应证明）
• 通过将基于最大匹配的初始化与贪婪的可达性扩展相结合，提出可扩展的驱动节点设计

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 or

实验结果

研究问题

• RQ1 如何为基于超图的多项式动力学刻画结构可控性
• RQ2 从图到超图，何以推广可达性与扩张性及其对可控性的含义
• RQ3 在超图中实现结构可控所需驱动节点数量的基于拓扑的下界是什么
• RQ4 如何设计可扩展算法，利用超图结构（最大匹配+贪婪扩展）来选择驱动节点
• RQ5 所提框架是否能在具有高阶交互的大规模超图上得到验证

主要发现

• 基于超图的结构可控性框架将经典概念扩展到由邻接张量表示的多项式动力学
• 结构可控性由相关超图中不存在不可达节点和不存在超边扩张来表征
• 作者推导出实现可控性的驱动节点数量的基于拓扑的下界
• 提出通过将扩张感知的最大匹配初始化与贪婪可达性扩展相结合的可扩展驱动节点选择算法
• 数值实验在具有数万节点和高阶交互的超图上展示了有效性与可扩展性

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).