Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected simplicial complex filtration required by the standard persistent homology framework. In this paper, we present a new filtration constructed from a directed graph, called the walk-length filtration. This filtration mirrors the behavior of small walks visiting certain collections of vertices in the directed graph. We show that, while the persistence is not stable under the usual $L_\infty$-style network distance, a generalized $L_1$-style distance is, indeed, stable. We further provide an algorithm for its computation, and investigate the behavior of this filtration in examples, including cycle networks and synthetic hippocampal networks with a focus on comparison to the often used Dowker filtration.

翻译：有向图出现在许多应用中，其中计算持久同调有助于编码输入信息的形状和结构。然而，将图信息转化为标准持久同调框架所需的无向单纯复形过滤的方法为数不多。本文提出一种从有向图构造的新过滤，称为漫步长度过滤。该过滤模拟了访问有向图中某些顶点集合的小型漫步行为。我们证明，虽然持久性在通常的$L_\infty$型网络距离下不稳定，但广义的$L_1$型距离确实是稳定的。我们进一步提供了其计算算法，并通过实例研究了该过滤的行为，包括循环网络和合成海马网络，重点是与常用的Dowker过滤进行比较。