Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the underlying graph between the spaces is directed and acyclic. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of arbitrary 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) time, as well as an algorithm to compute persistence for any arbitrary subgraph in the same running time. 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 calculate significant barcodes using considerably fewer points than standard persistence. ∗Department of Mathematics and Computer Science, Saint Louis University, {echambe5,letscher}@slu.edu. †Research supported in part by the National Science Foundation under Grant No. CCF 1054779.