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Computational Tools in Weighted Persistent Homology

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Abstract

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p2 weighted persistent homology provided with some information on the mod p weighted persistent homology.

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References

  1. Adcock, A., Carlsson, E., and Carlsson, G., The ring of algebraic functions on persistence bar codes, Homology, Homotopy and Applications, 18(1), 2016, 381–402.

    Article  MathSciNet  Google Scholar 

  2. Albert, R. and Barabási, A.-L., Statistical mechanics of complex networks, Reviews of Modern Physics, 74(1), 2002, 47–49.

    Article  MathSciNet  Google Scholar 

  3. Basum, S. and Parida, L., Spectral sequences, exact couples and persistent homology of filtrations, Expositions Mathematicae, 35(1), 2017, 119–132.

    Article  MathSciNet  Google Scholar 

  4. Bell, G., Lawson, A., Martin, J., et al., Weighted persistent homology, Involve, 12(5), 2019, 823–837.

    Article  MathSciNet  Google Scholar 

  5. Bendich, P., Marron, J. S., Miller, E., et al., Persistent homology analysis of brain artery trees, The Annals of Applied Statistics, 10(1), 2016, 198–218.

    Article  MathSciNet  Google Scholar 

  6. Boccaletti, S., Latora, V., Moreno, Y., et al., Complex networks: Structure and dynamics, Physics reports, 424(4), 2006, 175–308.

    Article  MathSciNet  Google Scholar 

  7. Boissonnat, J.-D. and Maria, C., Computing persistent homology with various coefficient fields in a single pass, Lecture Notes in Comput. Sci., 8737, Springer-Verlag, Heidelberg, 2014, 185–196.

    MATH  Google Scholar 

  8. Browder, W., Torsion in H-spaces, Annals of Mathematics, 74(2), 1961, 24–51.

    Article  MathSciNet  Google Scholar 

  9. Bubenik, P., Statistical topological data analysis using persistence landscapes, The Journal of Machine Learning Research, 16(1), 2015, 77–102.

    MathSciNet  MATH  Google Scholar 

  10. Bubenik, P., and Kim, P. T., A statistical approach to persistent homology, Homology Homotopy and Applications, 9(2), 2007, 337–362.

    Article  MathSciNet  Google Scholar 

  11. Bubenik, P., and Scott, J. A., Categorification of persistent homology, Discrete & Computational Geometry, 51(3), 2014, 600–627.

    Article  MathSciNet  Google Scholar 

  12. Buchet, M., Chazal, F., Oudot, S. Y. and Sheehy, D. R., Efficient and robust persistent homology for measures, Computational Geometry, 58, 2016, 70–96.

    Article  MathSciNet  Google Scholar 

  13. Carlsson, G., Ishkhanov, T., De Silva, V. and Zomorodian, A., On the local behavior of spaces of natural images, International Journal of Computer Vision, 76(1), 2008, 1–12.

    Article  MathSciNet  Google Scholar 

  14. Chow, T. Y., You could have invented spectral sequences, Notices of the AMS, 53, 2006, 15–19.

    MathSciNet  MATH  Google Scholar 

  15. Dawson, R. J. MacG., Homology of weighted simplicial complexes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 31(3), 1990, 229–243.

    MathSciNet  MATH  Google Scholar 

  16. DeWoskin, D., Climent, J., Cruz-White, I., et al., Applications of computational homology to the analysis of treatment response in breast cancer patients, Topology and Its Applications, 157(1), 2010, 157–164.

    Article  MathSciNet  Google Scholar 

  17. Dłotko, P., and Wagner, H., Simplification of complexes of persistent homology computations, Homology Homotopy and Applications, 16(1), 2014, 49–63.

    Article  MathSciNet  Google Scholar 

  18. Doran, B., Giansiracusa, N. and David, J., A simplicial approach to effective divisors in M0, n, International Mathematics Research Notices, 2017(2), 2016, 529–565.

    Article  Google Scholar 

  19. Edelsbrunner, H., and Morozov, D., Persistent homology: Theory and practice, European Congress of Mathematics, 31–50, Eur. Math. Soc., Zürich, 2013

  20. González, J. L., Gunther, E. and Zhang, O., Balanced complexes and effective divisors on M0,n, Comm. Algebra, 48(6), 2020, 2662–2680.

    Article  MathSciNet  Google Scholar 

  21. Hilton, P. J. and Stammbach, U., A Course in Homological Algebra, Graduate Texts in Mathematics, 4, Springer-Verlag, New York, 1997.

    Book  Google Scholar 

  22. Lubkin, S., Cohomology of Completions, North-Holland Mathematics Studies, 42, Notas de Mathematica, 71, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1980.

    Google Scholar 

  23. MacLane, S., Homology, Springer Science & Business Media, Berlin, 2012.

    MATH  Google Scholar 

  24. May, P. J. and Ponto, K., More concise algebraic topology, Localization, completion, and model categories, University of Chicago Press, Chicago, 2012.

    MATH  Google Scholar 

  25. McCleary, J., A user’s guide to spectral sequences, Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001.

    MATH  Google Scholar 

  26. Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005.

    Google Scholar 

  27. Munkres, J. R., Elements of Algebraic Topology, 2, Addison-Wesley, Menlo Park, 1984.

    MATH  Google Scholar 

  28. Neisendorfer, J. A., Algebraic methods in unstable homotopy theory, New Mathematical Monographs, 12, Cambridge University Press, Cambridge, 2010.

    Book  Google Scholar 

  29. Petri, G., Scolamiero, M., Donato, I. and Vaccarino, F., Topological strata of weighted complex networks, PloS one, 8(6), 2013, e66506.

    Article  Google Scholar 

  30. Ren, S. Q., Wu, C. Y. and Wu, J., Weighted persistent homology, Rocky Mountain Journal of Mathematics, 48(8), 2018, 2661–2687.

    Article  MathSciNet  Google Scholar 

  31. Romero, A., Heras, J., Rubio, J. and Sergeraert, F., Defining and computing persistent Z-homology in the general case, arXiv:1403.7086, 2014.

  32. Stanley, R. P., Combinatorics and Commutative Algebra, Progress in Mathematics, 41, Birkhauser Boston, Inc., Boston, 1996.

    MATH  Google Scholar 

  33. Strogatz, S. H., Exploring complex networks, Nature, 410(6825), 2001, 268–276.

    Article  Google Scholar 

  34. Zomorodian, A., The tidy set: A minimal simplicial set for computing homology of clique complexes, Computational Geometry, 43, 257–266, 2010.

    Article  MathSciNet  Google Scholar 

  35. Zomorodian, A. and Carlsson, G., Computing persistent homology, Discrete & Computational Geometry, 33(2), 2005, 249–274.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Shiquan Ren

  2. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Chengyuan Wu

  3. Institute of High Performance Computing, A*STAR, Singapore, 138632, Singapore

    Chengyuan Wu

  4. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Jie Wu

  5. Center for Topology and Geometry based Technology, Hebei Normal University, Shijiazhuang, 050024, China

    Jie Wu

Authors
  1. Shiquan Ren
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  2. Chengyuan Wu
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  3. Jie Wu
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Corresponding author

Correspondence to Chengyuan Wu.

Additional information

This work was supported by the Singapore Ministry of Education Research Grant (AcRF Tier 1 WBS No.R-146-000-222-112), the Postdoctoral International Exchange Program of China 2019 Project from the Office of China Postdoctoral Council, China Postdoctoral Science Foundation, the President’s Graduate Fellowship of National University of Singapore, the Natural Science Foundation of China (Nos. 11971144, 12001310), High-Level Scientific Research Foundation of Hebei Province and China Postdoctoral Science Foundation (No. 2019–2021).

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Ren, S., Wu, C. & Wu, J. Computational Tools in Weighted Persistent Homology. Chin. Ann. Math. Ser. B 42, 237–258 (2021). https://doi.org/10.1007/s11401-021-0255-8

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  • DOI: https://doi.org/10.1007/s11401-021-0255-8

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