在复杂网络(尤其是源于物理系统的网络)中探测结构,是横跨各科学领域的核心问题。一种常用方法是富俱乐部分析,即借助中心性度量识别重要顶点,并检验这些顶点之间的连接是否比随机预期更紧密。尽管该方法具有信息量,但它仅捕捉成对相互作用,而忽略了已知塑造诸多复杂系统结构与功能的高阶相互作用。我们提出一种超富俱乐部(hyper-rich club)分析流程,用于检验中心顶点是否通过编码高阶相互作用的超边而比随机预期更紧密地互联;该流程亦能纳入常被忽略但至关重要的方向性信息。我们在一类广泛的超图——我们称之为“一般有向超图”(general directed hypergraphs)——上开展工作;此类超图涵盖无向超图、头尾有向超图(head-and-tail directed hypergraphs)以及全序超图(totally ordered hypergraphs,后者与拓扑数据分析中的有向单纯复形相关)等特例。该定义统一了若干非等价的有向超图概念。在此类超图上,我们定义了一套超富俱乐部分析框架,其具体构造依赖于领域科学家根据研究目标所明确设定的选择;特定选择可复现图与无向超图上既有的富俱乐部定义,并首次为各类有向超图形式分别给出相应的富俱乐部定义。我们通过分析来源迥异的网络数据验证该流程的有效性:神经连接组(connectomes)、传染病传播的时间网络、诗歌网络,以及XGI超图数据库;在每种情形下,该流程均检测出标准图富俱乐部分析所遗漏的有意义结构。
Detecting structure in complex networks, especially those arising from physical systems, is a central problem across the sciences. One approach is via rich club analysis, which identifies important vertices using a centrality metric and measures whether those vertices are more tightly interconnected than expected by chance. While informative, this approach captures only pairwise interactions, missing out on higher-order ones known to shape the structure and function of many complex systems. We propose a hyper-rich club pipeline that asks whether central vertices are more tightly interconnected than expected by chance through hyperedges encoding higher-order interactions, which also enables the inclusion of important, often omitted, directional information. We work in a broad class of hypergraphs, which we call general directed hypergraphs, that includes as special cases undirected hypergraphs, head-and-tail directed hypergraphs, and totally ordered hypergraphs (a hypergraph related to directed simplicial complexes from topological data analysis). This unifies several non-equivalent notions of directed hypergraph under one definition. On these hypergraphs we define a hyper-rich club framework whose concrete construction depends on explicit choices the domain scientist fixes according to their research goals. Particular choices recover the existing rich club notions for graphs and undirected hypergraphs, and yield the first such notion for each version of directed hypergraphs. We demonstrate that the pipeline recovers meaningful structure in data by studying networks of very different origins: connectomes, temporal networks of infectious spread, networks of poems, and the XGI hypergraph database, in each case detecting structure the standard graph rich club misses.