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.


  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

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Information & Authors


Published In

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


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


  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



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