Research in Computational Topology pp 33-56 | Cite as

# A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs

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## Abstract

Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative description of metric graphs using topological summaries. In particular, we provide a complete characterization of the one-dimensional intrinsic Čech persistence diagrams for finite metric graphs using persistent homology. Together with complementary results by Adamaszek et al., which imply the results on intrinsic Čech persistence diagrams in all dimensions for a single cycle, our results constitute the important steps toward characterizing intrinsic Čech persistence diagrams for arbitrary finite metric graphs across all dimensions.

## Keywords

Persistence Diagrams Persistent Homology Adamaszek Simplicial Homology Graph Motif## Notes

### Acknowledgements

We are grateful for the Women in Computational Topology (WinCompTop) workshop for initiating our research collaboration. In particular, participant travel support was made possible through an NSF grant (NSF-DMS-1619908), and some additional travel support and social outings throughout the workshop were made possible through a gift from Microsoft Research. The Institute for Mathematics and its Applications generously offered the use of their facilities, their experienced staff to facilitate conference logistics, and refreshments. We appreciate their continued support of the applied algebraic topology community with regard to the WinCompTop Workshop, the special thematic program *Scientific and Engineering Applications of Algebraic Topology* held during the 2013–2014 academic year, and the Applied Algebraic Topology Network WebEx talks. Finally, we would like to thank the AWM ADVANCE grant for travel support for organizers and speakers to attend the WinCompTop special session at the AWM Research Symposium in April 2017.

EP was partially supported by the Asymmetric Resilient Cybersecurity Initiative at Pacific Northwest National Laboratory, part of the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy. During the completion of this project, RS was partially supported by the Simons Collaboration Grant, BW was partially supported by NSF-IIS-1513616, and YW was partially supported by NSF-CCF-1526513 and NSF-CCF-1618247.

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