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 (DAGs), 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.
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Acknowledgements
The authors would like to thank Afra Zomorodian for suggesting this problem during a visit as well as for additional comments and suggestions along the way. We would also like to thank Greg Marks and Michael May for helpful conversations during the course of this work. Finally, we would like to thank the anonymous reviewers for a number of suggestions for improvement of this paper.
Research supported in part by the National Science Foundation under Grant No. CCF 1054779 and IIS-1319573.
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Chambers, E.W., Letscher, D. (2018). Persistent Homology over Directed Acyclic Graphs. In: Chambers, E., Fasy, B., Ziegelmeier, L. (eds) Research in Computational Topology. Association for Women in Mathematics Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-89593-2_2
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