A directed hypergraph, which consists of nodes and hyperarcs, is a higher-order data structure that naturally models directional group interactions (e.g., chemical reactions of molecules). Although there have been extensive studies on local structures of (directed) graphs in the real world, those of directed hypergraphs remain unexplored. In this work, we focus on measurements, findings, and applications related to local structures of directed hypergraphs, and they together contribute to a systematic understanding of various real-world systems interconnected by directed group interactions. Our first contribution is to define 91 directed hypergraphlets (DHGs), which disjointly categorize directed connections and overlaps among four node sets that compose two incident hyperarcs. Our second contribution is to develop exact and approximate algorithms for counting the occurrences of each DHG. Our last contribution is to characterize 11 real-world directed hypergraphs and individual hyperarcs in them using the occurrences of DHGs, which reveals clear domain-based local structural patterns. Our experiments demonstrate that our DHG-based characterization gives up to 12% and 33% better performances on hypergraph clustering and hyperarc prediction, respectively, than baseline characterization methods. Moreover, we show that CODA-A, which is our proposed approximate algorithm, is up to 32X faster than its competitors with similar characterization quality.

翻译：有向超图由节点和超弧构成，是一种高阶数据结构，能够自然建模方向性群体交互（例如分子化学反应）。尽管现实世界中（有向）图的局部结构已被广泛研究，但有向超图的局部结构仍属未探索领域。本文聚焦于有向超图局部结构的度量、发现与应用，旨在系统理解由方向性群体交互互联的各类现实世界系统。首先，我们定义了91种有向超图子结构（DHGs），这些结构对构成两个相邻超弧的四个节点集合间的有向连接与重叠关系进行了不交类别划分。其次，我们开发了用于统计每种DHG出现次数的精确算法与近似算法。最后，我们利用DHG出现次数对11个现实世界有向超图及其中的个体超弧进行特征刻画，揭示了清晰的领域依赖性局部结构模式。实验表明，基于DHG的特征刻画方法在超图聚类和超弧预测任务中，相比基准方法分别提升高达12%和33%的性能。此外，我们提出的近似算法CODA-A在保持相似特征刻画质量的前提下，其运行速度比同类方法快32倍。