Document Open Access Logo

Vietoris-Rips and Cech Complexes of Metric Gluings

Authors Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, Lori Ziegelmeier

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2018.3.pdf
  • Filesize: 0.59 MB
  • 15 pages

Document Identifiers

Author Details

Michal Adamaszek
Henry Adams
Ellen Gasparovic
Maria Gommel
Emilie Purvine
Radmila Sazdanovic
Bei Wang
Yusu Wang
Lori Ziegelmeier

Cite As Get BibTex

Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. Vietoris-Rips and Cech Complexes of Metric Gluings. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SoCG.2018.3

Abstract

We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.

Subject Classification

Keywords
  • Vietoris-Rips and Cech complexes
  • metric space gluings and wedge sums
  • metric graphs
  • persistent homology

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Mridul Aanjaneya, Frederic Chazal, Daniel Chen, Marc Glisse, Leonidas Guibas, and Dmitry Morozon. Metric graph reconstruction from noisy data. International Journal of Computational Geometry and Applications, 22(04):305-325, 2012. Google Scholar
  2. Michał Adamaszek. Clique complexes and graph powers. Israel Journal of Mathematics, 196(1):295-319, 2013. Google Scholar
  3. Michał Adamaszek and Henry Adams. The Vietoris-Rips complexes of a circle. Pacific Journal of Mathematics, 290:1-40, 2017. Google Scholar
  4. Michał Adamaszek, Henry Adams, Florian Frick, Chris Peterson, and Corrine Previte-Johnson. Nerve complexes of circular arcs. Discrete & Computational Geometry, 56(2):251-273, 2016. Google Scholar
  5. Michał Adamaszek, Henry Adams, Ellen Gasparovic, Marix Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. Vietoris-Rips and Čech complexes of metric gluings. arXiv:1712.06224, 2018. Google Scholar
  6. Eric Babson and Dmitry N. Kozlov. Complexes of graph homomorphisms. Israel Journal of Mathematics, 152(1):285-312, 2006. Google Scholar
  7. Jonathan Ariel Barmak and Elias Gabriel Minian. Simple homotopy types and finite spaces. Advances in Mathematics, 218:87-104, 2008. Google Scholar
  8. Bharat Biswal, F. Zerrin Yetkin, Victor M. Haughton, and James S. Hyde. Functional connectivity in the motor cortex of resting human brain using echo-planar MRI. Magnetic Resonance in Medicine, 34:537-541, 1995. Google Scholar
  9. Anders Björner. Topological methods. Handbook of Combinatorics, 2:1819-1872, 1995. Google Scholar
  10. Karol Borsuk. On the imbedding of systems of compacta in simplicial complexes. Fundamenta Mathematicae, 35(1):217-234, 1948. Google Scholar
  11. Martin R. Bridson and André Haefliger. Metric Spaces of Non-Positive Curvature. Springer-Verlag, 1999. Google Scholar
  12. Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry, volume 33. American Mathematical Society Providence, RI, 2001. Google Scholar
  13. Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009. Google Scholar
  14. Joseph Minhow Chan, Gunnar Carlsson, and Raul Rabadan. Topology of viral evolution. Proceedings of the National Academy of Sciences, 110(46):18566-18571, 2013. Google Scholar
  15. Frédéric Chazal, David Cohen-Steiner, Leonidas J. Guibas, Facundo Mémoli, and Steve Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. Computer Graphics Forum, 28(5):1393-1403, 2009. Google Scholar
  16. Frédéric Chazal, Vin De Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. Springer, 2016. Google Scholar
  17. Frédéric Chazal, Vin de Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, pages 1-22, 2013. Google Scholar
  18. Fengming Dong, Khee-Meng Koh, and Kee L Teo. Chromatic Polynomials and Chromaticity of Graphs. World Scientific, 2005. Google Scholar
  19. Herbert Edelsbrunner and John L Harer. Computational Topology: An Introduction. American Mathematical Society, 2010. Google Scholar
  20. Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. A complete characterization of the one-dimensional intrinsic Čech persistence diagrams for metric graphs. In Research in Computational Topology, 2018. Google Scholar
  21. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. Google Scholar
  22. Jean-Claude Hausmann. On the Vietoris-Rips complexes and a cohomology theory for metric spaces. Annals of Mathematics Studies, 138:175-188, 1995. Google Scholar
  23. Dmitry N. Kozlov. Combinatorial Algebraic Topology, volume 21 of Algorithms and Computation in Mathematics. Springer-Verlag, 2008. Google Scholar
  24. Peter Kuchment. Quantum graphs I. Some basic structures. Waves in Random Media, 14(1):S107-S128, 2004. Google Scholar
  25. Janko Latschev. Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold. Archiv der Mathematik, 77(6):522-528, 2001. Google Scholar
  26. Michael Lesnick, Raul Rabadan, and Daniel Rosenbloom. Quantifying genetic innovation: Mathematical foundations for the topological study of reticulate evolution. In preparation, 2018. Google Scholar
  27. Jiří Matoušek. LC reductions yield isomorphic simplicial complexes. Contributions to Discrete Mathematics, 3(2), 2008. Google Scholar
  28. Steve Oudot and Elchana Solomon. Barcode embeddings, persistence distortion, and inverse problems for metric graphs. arXiv: 1712.03630, 2017. Google Scholar
  29. Thierry Sousbie, Christophe Pichon, and Hajime Kawahara. The persistent cosmic web and its filamentary structure - II. Illustrations. Monthly Notices of the Royal Astronomical Society, 414(1):384-403, 2011. Google Scholar
  30. Katharine Turner. Generalizations of the Rips filtration for quasi-metric spaces with persistent homology stability results. arXiv:1608.00365, 2016. Google Scholar
  31. Žiga Virk. 1-dimensional intrinsic persistence of geodesic spaces. arXiv:1709.05164, 2017. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact