Persistent Homology Over Directed Acyclic Graphs

TL;DR

This paper introduces a generalized framework for persistent homology over directed acyclic graphs, enabling analysis of complex topological data and providing an efficient algorithm for computation and applications in data comparison.

Contribution

It extends persistent homology to arbitrary directed acyclic graphs and offers an algorithm to compute all subgraph persistent homology groups efficiently.

Findings

Generalizes persistent homology, zigzag, and multidimensional persistence.

Provides an $O(n^4)$ algorithm for computing persistent homology over DAGs.

Demonstrates a method to overlay filtrations for more efficient topological feature detection.

Abstract

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.