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

Part of the Association for Women in Mathematics Series book series (AWMS, volume 13)


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.


Persistence Diagrams Persistent Homology Adamaszek Simplicial Homology Graph Motif 
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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.


  1. 1.
    M. Aanjaneya, F. Chazal, D. Chen, M. Glisse, L. Guibas, D. Morozov, Metric graph reconstruction from noisy data. Int. J. Comput. Geom. Appl. 22(04), 305–325 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Adamaszek, H. Adams, The Vietoris-Rips complexes of a circle (2015). arXiv 1503.03669Google Scholar
  3. 3.
    M. Adamaszek, H. Adams, F. Frick, C. Peterson, C. Previte-Johnson, Nerve complexes of circular arcs. Discret. Comput. Geom. 56(2), 251–273 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Biswal, F.Z. Yetkin, V.M. Haughton, J.S. Hyde, Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. Magn. Reson. Med. 34, 537–541 (1995)CrossRefGoogle Scholar
  5. 5.
    G. Carlsson, Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Chazal, V. de Silva, S. Oudot, Persistence stability for geometric complexes. Geom. Dedicata. 173, 193–214 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9(1), 79–103 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. de Silva, R. Ghrist, Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7, 339–358 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    T.K. Dey, D. Shi, Y. Wang, Comparing graphs via persistence distortion, in Proceedings 31st International Symposium on Computational Geometry, ed. by L. Arge, J. Pach. Leibniz International Proceedings in Informatics (LIPIcs), vol. 34, pp. 491–506 (Dagstuhl, Wadern, 2015). Schloss Dagstuhl–Leibniz-Zentrum fuer InformatikGoogle Scholar
  10. 10.
    H. Edelsbrunner, J. Harer, Persistent homology - a survey. Contemp. Math. 453, 257–282 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Ghrist, Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45, 61–75 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)zbMATHGoogle Scholar
  13. 13.
    P. Kuchment, Quantum graphs I. Some basic structures. Waves Random Media 14(1), S107–S128 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    W.S. Massey, A Basic Course in Algebraic Topology. Graduate Texts in Mathematics (Springer, New York, 1991)Google Scholar
  15. 15.
    J.R. Munkres, Elements of Algebraic Topology. Advanced Book Classics (Perseus Books, New York, 1984)zbMATHGoogle Scholar
  16. 16.
    C. Previte, The \(\mathbb {D}\) -neighborhood complex of a graph. PhD thesis, Colorado State University, 2014Google Scholar
  17. 17.
    J. Rotman, An Introduction to Homological Algebra. Universitext (Springer, New York, 2009)CrossRefGoogle Scholar
  18. 18.
    D. Taylan, Matching trees for simplicial complexes and homotopy type of devoid complexes of graphs. Order 33, 459–476 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  1. 1.Union CollegeSchenectadyUSA
  2. 2.University of IowaIowa CityUSA
  3. 3.Pacific Northwest National LaboratorySeattleUSA
  4. 4.NC State UniversityRaleighUSA
  5. 5.University of UtahSalt Lake CityUSA
  6. 6.The Ohio State UniversityColumbusUSA
  7. 7.Department of Mathematics, Statistics, & Computer ScienceMacalester CollegeSaint PaulUSA

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