Complex systems frequently exhibit multi-way, rather than pairwise, interactions. These group interactions cannot be faithfully modeled as collections of pairwise interactions using graphs, and instead require hypergraphs. However, methods that analyze hypergraphs directly, rather than via lossy graph reductions, remain limited. Hypergraph motif mining holds promise in this regard, as motif patterns serve as building blocks for larger group interactions which are inexpressible by graphs. Recent work has focused on categorizing and counting hypergraph motifs based on the existence of nodes in hyperedge intersection regions. Here, we argue that the relative sizes of hyperedge intersections within motifs contain varied and valuable information. We propose a suite of efficient algorithms for finding triplets of hyperedges based on optimizing the sizes of these intersection patterns. This formulation uncovers interesting local patterns of interaction, finding hyperedge triplets that either (1) are the least correlated with each other, (2) have the highest pairwise but not groupwise correlation, or (3) are the most correlated with each other. We formalize this as a combinatorial optimization problem and design efficient algorithms based on filtering hyperedges. Our experimental evaluation shows that the resulting hyperedge triplets yield insightful information on real-world hypergraphs. Our approach is also orders of magnitude faster than a naive baseline implementation.

翻译：复杂系统常常表现出多路而非成对交互作用。这些群体交互无法通过图结构忠实地建模为成对交互的集合，而需要使用超图。然而，直接分析超图而非通过有损图简化方法的技术仍然有限。超图基序挖掘在此方面具有潜力，因为基序模式构成了无法用图表达的大型群体交互的基本构建块。近期研究集中于基于超边交叉区域中节点的存在性对超图基序进行分类和计数。本文提出，超边交叉的相对尺寸在基序中包含了多样且有价值的信息。我们设计了一套高效算法，通过优化交叉模式尺寸来发现超边三元组。该公式揭示了有趣的局部交互模式，能够找到（1）彼此相关性最低、（2）具有最高成对相关性但非群体相关性、或（3）彼此相关性最高的超边三元组。我们将此形式化为组合优化问题，并设计了基于超边筛选的高效算法。实验评估表明，所得超边三元组能从真实世界超图中提取富有洞见的信息。此外，我们的方法在速度上比朴素基线实现快数个数量级。