STruD：单纯形复形的桁架分解
STruD: Truss Decomposition of Simplicial Complexes
摘要：
单纯复形是图的一个推广：n元关系的集合（而不是图的边的二进制），称为单纯形。在本文中，我们开发了一种新的工具来研究单纯复形的结构：我们推广的图形概念的桁架分解复杂，并表明，这种更强大的表示产生不同的属性相比，基于图形的。然而，这种能力带来了重要的计算挑战，这些挑战来自于复合物的向下闭合特性所引起的组合爆炸。借鉴项目集挖掘和相似性搜索的思想，我们设计了一个内存感知算法，被称为STruD，它能够有效地计算单纯复形的桁架分解。STruD通过以紧凑的方式存储中间数据来使其行为适应可用内存的量。然后，我们设计了一个变量，直接计算最大桁架的n个单形。通过将STruD应用于几个数据集，我们证明了它的可扩展性，并提供了它们的结构分析。最后，我们表明，桁架分解可以被看作是一个过滤，因此它可以用来研究数据集的持久同源性，在不同的空间分辨率计算拓扑特征的方法，突出的拓扑数据分析。
英文摘要：
A simplicial complex is a generalization of a graph: a collection of n-ary relationships (instead of binary as the edges of a graph), named simplices. In this paper, we develop a new tool to study the structure of simplicial complexes: we generalize the graph notion of truss decomposition to complexes, and show that this more powerful representation gives rise to different properties compared to the graph-based one. This power, however, comes with important computational challenges derived from the combinatorial explosion caused by the downward closure property of complexes. Drawing upon ideas from itemset mining and similarity search, we design a memory-aware algorithm, dubbed STruD, which is able to efficiently compute the truss decomposition of a simplicial complex. STruD adapts its behavior to the amount of available memory by storing intermediate data in a compact way. We then devise a variant that computes directly the n simplices of maximum trussness. By applying STruD to several datasets, we prove its scalability, and provide an analysis of their structure. Finally, we show that the truss decomposition can be seen as a filtration, and as such it can be used to study the persistent homology of a dataset, a method for computing topological features at different spatial resolutions, prominent in Topological Data Analysis.
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