论文标题
有向图的简单$ q $ - 连接性以及用于网络分析的应用程序
Simplicial $q$-connectivity of directed graphs with applications to network analysis
论文作者
论文摘要
有向图是网络无处不在的模型，他们生成的拓扑空间（例如有向的标志复合物）已成为应用拓扑中有用的对象。简单是由定向集团形成的。我们将Atkin的$ Q $ - 连接性理论扩展到了定向简单的情况。这导致了预订，其中简单是通过简单序列相关的，这些简单相对于所选择的面部图指定的方向共享$ q $ face。我们利用预订和拓扑空间之间的Alexandroff等效性来引入针对有向图的新拓扑空间，从而使新的同质副本类型与Simplicial同源性看到的定向Flag复合物不同。我们进一步介绍了由连接预订启用的简单路径分析。作为应用程序，我们通过计算最长的简单路径来表征各种大脑网络之间的结构差异。
Directed graphs are ubiquitous models for networks, and topological spaces they generate, such as the directed flag complex, have become useful objects in applied topology. The simplices are formed from directed cliques. We extend Atkin's theory of $q$-connectivity to the case of directed simplices. This results in a preorder where simplices are related by sequences of simplices that share a $q$-face with respect to directions specified by chosen face maps. We leverage the Alexandroff equivalence between preorders and topological spaces to introduce a new class of topological spaces for directed graphs, enabling to assign new homotopy types different from those of directed flag complexes as seen by simplicial homology. We further introduce simplicial path analysis enabled by the connectivity preorders. As an application we characterise structural differences between various brain networks by computing their longest simplicial paths.
