论文标题
阴影知道：最小跨度抗体和随机复合物的持久图的经验分布
The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes
论文作者
论文摘要
1985年，Frieze表明，在均匀加权图中，最小生成树（MST）的边缘重量的预期总和收敛到$ζ（3）$。最近，Hino和Kanazawa将这一结果扩展到了均匀加权的简单复合物，其中MST的作用是由其较高维度的类似物（最小跨度acycle（MSA））扮演的。我们的工作不断壮大，并描述了随机MST和随机MSA中所有权重的直方图。具体来说，我们表明他们的经验分布会根据一个称为“影子”的概念收敛到措施。图的阴影是所有缺失的横向边缘的集合，对于简单的复合物，它是相关的拓扑概括。作为推论，我们在对应于上面加权复合物的持久图中获得了类似的死亡时间主张，这是对应用拓扑的兴趣的结果。
In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to $ζ(3)$. Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.
