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

2025-06-27

AI Summary

Standard persistent homology frameworks struggle to accommodate weighted directed graphs. To address this, we propose Walk-Length Filtration (WLF): a filtration of simplicial complexes defined by the length of directed walks, incrementally incorporating cliques induced by short directed paths. WLF constitutes the first systematic filtration framework tailored specifically for directed graphs; it introduces a generalized L₁ metric and establishes its stability theory. Theoretically and empirically, WLF demonstrates superior sensitivity to local directional structures compared to Dowker persistence. Validation on synthetic hippocampal and ring networks confirms that WLF achieves higher discriminability and computational stability in identifying topological features such as periodicity and feedback loops.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Convert directed graphs to undirected simplicial complexes
• Propose walk-length filtration for persistent homology
• Analyze stability under generalized L1-style distance

Innovation

• Walk-length filtration for directed graphs
• Generalized L1-style stable distance
• Algorithm for computation provided