论文标题
锦标赛的复合体，方向性过滤和持续的同源性
Complexes of Tournaments, Directionality Filtrations and Persistent Homology
论文作者
论文摘要
在组合文学中，完整的挖掘被称为锦标赛。我们认为我们称之为“ tournaplexes”的半动画综合体系列，其简单是锦标赛。特别是，给定一个Digraph $ \ MATHCAL {G} $，我们将其与“ Flag Tournaplex”相关联，它是一个tournaplex，其中包含$ \ Mathcal {g} $的定向标志复合物，也是对未指示的群体的几何实现。我们定义了循环中的几种过滤，并利用了持久同源性，我们观察到，旗帜循环提供了比定向的旗帜复合体更优质地区分图形动力学的方法。然后，我们通过将这些想法应用于蓝色大脑项目对大鼠新皮层的数字重建产生的图形数据来证明这些想法的力量。
Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph $\mathcal{G}$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of $\mathcal{G}$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.
