There is increasing focus on analyzing data represented as hypergraphs, which are better able to express complex relationships amongst entities than are graphs. Much of the critical information about hypergraph structure is available only in the intersection relationships of the hyperedges, and so forming the "intersection complex" of a hypergraph is quite valuable. This identifies a valuable isomorphism between the intersection complex and the "concept lattice" formed from taking the hypergraph's incidence matrix as a "formal context": hypergraphs also generalize graphs in that their incidence matrices are arbitrary Boolean matrices. This isomorphism allows connecting discrete algorithms for lattices and hypergraphs, in particular s-walks or s-paths on hypergraphs can be mapped to order theoretical operations on the concept lattice. We give new algorithms for formal concept lattices and hypergraph s-walks on concept lattices. We apply this to a large real-world dataset and find deep lattices implying high interconnectivity and complex geometry of hyperedges.

翻译：近年来，分析以超图形式表示的数据日益受到关注，相比普通图，超图能更有效地表达实体间的复杂关系。超图结构中大量关键信息仅存在于超边之间的交集关系中，因此构建超图的“交集复形”具有重要价值。这揭示了交集复形与“概念格”（通过将超图的关联矩阵视为“形式背景”而生成）之间的一种有价值同构关系：超图对图的推广也体现在其关联矩阵是任意布尔矩阵。该同构关系使得超图与格的离散算法得以建立联系，特别是超图上的s-游走或s-路径可映射至概念格上的序理论操作。我们提出了形式概念格及概念格上超图s-游走的新算法。将该方法应用于大规模真实数据集，发现了深层概念格结构，这表明超边具有高度互连性与复杂几何特性。