Computational Tools in Weighted Persistent Homology

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 p ² weighted persistent homology provided with some information on the mod p weighted persistent homology.

Keywords

Algebraic topology / Persistent homology / Weighted persistent homology / Bockstein spectral sequence

References

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

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

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

[4] Bell G, Lawson A, Martin J Weighted persistent homology. Involve, 2019, 12(5): 823-837

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

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

[7] Boissonnat J-D, Maria C. Computing persistent homology with various coefficient fields in a single pass, 2014, Heidelberg: Springer-Verlag 185-196

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

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

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

[11] Bubenik P, Scott J A. Categorification of persistent homology. Discrete & Computational Geometry, 2014, 51(3): 600-627

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

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

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

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

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

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

[18] Doran B, Giansiracusa N, David J. A simplicial approach to effective divisors in M _{0, n}. International Mathematics Research Notices, 2016, 2017(2): 529-565

[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, Zhang O. Balanced complexes and effective divisors on M_0,n. Comm. Algebra, 2020, 48(6): 2662-2680

[21] Hilton P J, Stammbach U. A Course in Homological Algebra, 1997, New York: Springer-Verlag 4

[22] Lubkin S. Cohomology of Completions, North-Holland Mathematics Studies, 1980, Amsterdam, New York, Oxford: North-Holland Publishing Company 42

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

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

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

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

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

[28] Neisendorfer J A. Algebraic methods in unstable homotopy theory, 2010, Cambridge: Cambridge University Press

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

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

[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, 1996, Boston: Birkhauser Boston, Inc.

[33] Strogatz S H. Exploring complex networks. Nature, 2001, 410(6825): 268-276

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

[35] Zomorodian A, Carlsson G. Computing persistent homology. Discrete & Computational Geometry, 2005, 33(2): 249-274