Persistent Path Homology of Directed Networks

TL;DR

This paper introduces persistent path homology, a novel extension of homology for directed networks, capturing their asymmetric structure across multiple resolutions, and demonstrates its effectiveness on various datasets.

Contribution

It extends the homology of digraphs to a persistent framework, enabling analysis of directed networks' structure at multiple scales.

Findings

Effectively captures asymmetric features of directed networks.

Provides multi-resolution insights into network structure.

Shows promising results on simulated and real-world data.

Abstract

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a construction of homology of digraphs due to Grigor'yan, Lin, Muranov and Yau, and extend this construction to the persistent framework. The result, which we call persistent path homology, can provide information about the digraph structure of a directed network at varying resolutions. Moreover, this method encodes a rich level of detail about the asymmetric structure of the input directed network. We test our method on both simulated and real-world directed networks and conjecture some of its characteristics.