Abstract

A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex). Software packages for computing persistent homology typically construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing simplicial complexes and the effects of making associated choices during simplicial-complex construction on the output of persistent-homology algorithms. We present two new methods for constructing simplicial complexes from two-dimensional geospatial data (such as maps). We apply these methods to a California precinct-level voting data set, and we thereby demonstrate that our new constructions can capture geometric characteristics that are missed by distance-based constructions. Our new constructions can thus yield more interpretable persistence modules and barcodes for geospatial data. In particular, they are able to distinguish short-persistence features that occur only for a narrow range of distance scales (e.g., voting patterns in densely populated cities) from short-persistence noise by incorporating information about other spatial relationships between regions.

Keywords

  1. persistent homology
  2. topological data analysis
  3. voting data
  4. geospatial data

MSC codes

  1. Primary
  2. 55N31; Secondary
  3. 55-04
  4. 55U10
  5. 62R40
  6. 91D20

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
M. Adamaszek and H. Adams, The Vietoris--Rips complexes of a circle, Pacific J. Math., 290 (2017), pp. 1--40.
2.
H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta, and L. Ziegelmeier, Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), pp. 218--252.
3.
P. Bajardi, M. Delfino, A. Panisson, G. Petri, and M. Tizzoni, Unveiling patterns of international communities in a global city using mobile phone data, European Phys. J. Data Sci., 4 (2015), art. 3.
4.
A. Banman and L. Ziegelmeier, Mind the gap: A study in global development through persistent homology, in Research in Computational Topology, E. W. Chambers, B. T. Fasy, and L. Ziegelmeier, eds., Springer International Publishing, Cham, Switzerland, 2018, pp. 125--144.
5.
R. Barnes and J. Solomon, Gerrymandering and Compactness: Implementation Flexibility and Abuse, preprint, https://arxiv.org/abs/1803.02857, 2018.
6.
U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, Phat---Persistent homology algorithms toolbox, in Mathematical Software---ICMS 2014, H. Hong and C. Yap, eds., Springer-Verlag, Heidelberg, Germany, 2014, pp. 137--143.
7.
P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, Homology and robustness of level and interlevel sets, Homology Homotopy Appl., 15 (2013), pp. 51--72.
8.
O. Bobrowski, S. Mukherjee, and J. E. Taylor, Topological consistency via kernel estimation, Bernoulli, 23 (2017), pp. 288--328.
9.
P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), pp. 77--102.
10.
P. Bubenik, M. Hull, D. Patel, and B. Whittle, Persistent homology detects curvature, Inverse Problems, 36 (2020), art. 025008.
11.
H. M. Byrne, H. A. Harrington, R. Muschel, G. Reinert, B. J. Stolz, and U. Tillmann, Topology characterises tumour vasculature, Math. Today, 55 (2019), pp. 206--210.
12.
G. Carlsson, Topological methods for data modelling, Nat. Rev. Phys., 2 (2020), pp. 697--708
13.
G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, On the local behavior of spaces of natural images, Internat. J. Comput. Vision, 76 (2008), pp. 1--12.
14.
C. Curto, What can topology tell us about the neural code?, Bull. Amer. Math. Soc., 54 (2017), pp. 63--78.
15.
M. Duchin and B. E. Tenner, Discrete Geometry for Electoral Geography, preprint, https://arxiv.org/abs/1808.05860, 2018.
16.
H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, AMS, Providence, RI, 2010.
17.
H. Edelsbrunner, D. Kirkpatrick, and R. Seidel, On the shape of a set of points in the plane, IEEE Trans. Inform. Theory, 29 (1983), pp. 551--559.
18.
K. Emmett, B. Schweinhart, and R. Rabadan, Multiscale topology of chromatin folding, in Proceedings of the 9th EAI International Conference on Bio-inspired Information and Communications Technologies (Formerly BIONETICS), BICT '15, ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering), 2016, pp. 177--180.
19.
R. Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., 45 (2008), pp. 61--75.
20.
F. Gibou, R. Fedkiw, and S. Osher, A review of level-set methods and some recent applications, J. Comput. Phys., 353 (2018), pp. 82--109.
21.
The Gudhi Project, Gudhi User and Reference Manual, Version 3.0.0, Gudhi Editorial Board, 2015, https://gudhi.inria.fr/doc/3.0.0/.
22.
C. Giusti, R. Ghrist, and D. S. Bassett, Two's company, three (or more) is a simplex, J. Comput. Neurosci., 41 (2016), pp. 1--14.
23.
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, UK, 2002.
24.
D. P. Humphreys, M. R. McGuirl, M. Miyagi, and A. J. Blumberg, Fast estimation of recombination rates using topological data analysis, Genetics, 211 (2019), pp. 1191--1204.
25.
P. S. P. Ignacio and I. K. Darcy, Tracing patterns and shapes in remittance and migration networks via persistent homology, European Phys. J. Data Sci., 8 (2019), art. 1.
26.
L. Kanari, P. Dłotko, M. Scolamiero, R. Levi, J. C. Shillcock, K. Hess, and H. Markram, A topological representation of branching neuronal morphologies, Neuroinform., 16 (2018), pp. 3--13.
27.
M. Kerber and R. Sharathkumar, Approximate Čech Complex in Low and High Dimensions, preprint, https://arxiv.org/abs/1307.3272, 2013.
28.
M. Kramár, A. Goullet, L. Kondic, and K. Mischaikow, Quantifying force networks in particulate systems, Phys. D, 283 (2014), pp. 37--55.
29.
R. Kwitt, S. Huber, M. Niethammer, W. Lin, and U. Bauer, Statistical topological data analysis---A kernel perspective, in Advances in Neural Information Processing Systems 28, C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, eds., Curran Associates, 2015, pp. 3070--3078.
30.
D. Lo and B. Park, Modeling the spread of the Zika virus using topological data analysis, PLoS ONE, 13 (2018), art. e0192120.
31.
C. Maria, Filtered complexes, in Gudhi User and Reference Manual, Version 3.0.0, Gudhi Editorial Board, 2015, https://gudhi.inria.fr/doc/3.0.0/group__simplex__tree.html.
32.
C. Maria, P. Dłotko, V. Rouvreau, and M. Glisse, Rips complex, in Gudhi User and Reference Manual, Version 3.0.0, Gudhi Editorial Board, 2016, https://gudhi.inria.fr/doc/3.0.0/group__rips__complex.html.
33.
M. E. J. Newman, Networks, 2nd ed., Oxford University Press, Oxford, UK, 2018.
34.
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Appl. Math. Sci. 153, Springer-Verlag, Heidelberg, Germany, 2003.
35.
S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton--Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12--49.
36.
N. Otter, M. A. Porter, U. Tillmann, P. Grindrod, and H. A. Harrington, A roadmap for the computation of persistent homology, European Phys. J. Data Sci., 6 (2017), art. 17.
37.
L. Papadopoulos, M. A. Porter, K. E. Daniels, and D. S. Bassett, Network analysis of particles and grains, J. Complex Networks, 6 (2018), pp. 485--565.
38.
QGIS Association, QGIS 2.18.17: A Free and Open Source Geographic Information System, 2016, http://www.qgis.org.
39.
J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt, A stable multi-scale kernel for topological machine learning, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015, pp. 4741--4748.
40.
J. W. Rocks, A. J. Liu, and E. Katifori, The Topological Basis of Function in Flow Networks, preprint, https://arxiv.org/abs/1901.00822, 2019.
41.
H. Ronellenfitsch, J. Lasser, D. C. Daly, and E. Katifori, Topological phenotypes constitute a new dimension in the phenotypic space of leaf venation networks, PLOS Comput. Biol., 11 (2015), art. e1004680.
42.
V. Rouvreau, Alpha complex, in Gudhi User and Reference Manual, Version 3.0.0, Gudhi Editorial Board, 2015, https://gudhi.inria.fr/doc/3.0.0/group__alpha__complex.html.
43.
V. Rouvreau, Python interface, in Gudhi User and Reference Manual, Version 3.0.0, Gudhi Editorial Board, 2016, https://gudhi.inria.fr/python/3.0.0.
44.
J. Schleuss, J. Fox, and P. Krishnakumar, California 2016 Election Precinct Maps, https://github.com/datadesk/california-2016-election-precinct-maps, 2016. (See https://www.latimes.com/projects/la-pol-ca-california-neighborhood-election-results/ for the associated newspaper article.)
45.
L. Speidel, H. A. Harrington, S. J. Chapman, and M. A. Porter, Topological data analysis of continuum percolation with disks, Phys. Rev. E, 98 (2018), art. 012318.
46.
B. J. Stolz, H. A. Harrington, and M. A. Porter, The Topological “Shape” of Brexit, preprint, https://arxiv.org/abs/1610.00752, 2016.
47.
B. J. Stolz, H. A. Harrington, and M. A. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series, Chaos, 27 (2017), art. 047410.
48.
D. Taylor, F. Klimm, H. A. Harrington, M. Kramár, K. Mischaikow, M. A. Porter, and P. J. Mucha, Topological data analysis of contagion maps for examining spreading processes on networks, Nature Commun., 6 (2015), art. 7723.
49.
L. Vietoris, Über den höheren zusammenhang kompakter räume und eine klasse von zusammenhangstreuen abbildungen, Math. Ann., 97 (1927), pp. 454--472.
50.
K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility, and folding, Internat. J. Numer. Methods Biomed. Engrg., 30 (2014), pp. 814--844.
51.
W. Zhou and H. Yan, Alpha shape and Delaunay triangulation in studies of protein-related interactions, Briefings Bioinform., 15 (2014), pp. 54--64.
52.
X. Zhu, A. Vartanian, M. Bansal, D. Nguyen, and L. Brandl, Stochastic multiresolution persistent homology kernel, in Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI '16, AAAI Press, 2016, pp. 2449--2455.
53.
A. Zomorodian, Fast construction of the Vietoris--Rips complex, Computers & Graphics, 34 (2010), pp. 263--271.
54.
A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), pp. 249--274.

Information & Authors

Information

Published In

cover image SIAM Review
SIAM Review
Pages: 67 - 99
ISSN (online): 1095-7200

History

Submitted: 29 January 2019
Accepted: 25 March 2020
Published online: 4 February 2021

Keywords

  1. persistent homology
  2. topological data analysis
  3. voting data
  4. geospatial data

MSC codes

  1. Primary
  2. 55N31; Secondary
  3. 55-04
  4. 55U10
  5. 62R40
  6. 91D20

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.