The Walk-Length Filtration for Persistent Homology on Weighted Directed Graphs

TL;DR

This paper introduces the walk-length filtration for directed graphs to compute persistent homology, providing stability results, an algorithm, and applications to cycle and hippocampal networks.

Contribution

It proposes a novel walk-length filtration for directed graphs, analyzes its stability, and offers an algorithm for practical computation.

Findings

Persistence is stable under a generalized L1-distance.

The walk-length filtration effectively captures graph structure.

Comparison with Dowker filtration shows distinct behaviors.

Abstract

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.