The (homological) persistence of gerrymandering

Moon Duchin; Tom Needham; Thomas Weighill

doi:10.3934/fods.2021007

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doi: 10.3934/fods.2021007

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The (homological) persistence of gerrymandering

Moon Duchin ^1, , Tom Needham ^2,, and Thomas Weighill ^3,

1.	Department of Mathematics, Tufts University, Medford, MA 02155, USA
2.	Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
3.	Department of Mathematics and Statistics, UNC Greensboro, Greensboro, NC 27402, USA

^* Corresponding author: Tom Needham

Received August 2020 Revised February 2021 Early access March 2021

Fund Project: The first and third author were supported by NSF Convergence Accelerator Grant No. OIA-1937095

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. We begin by combining geographic and electoral data from a districting plan to produce a persistence diagram. Then, to see beyond a particular plan and understand the possibilities afforded by the choices made in redistricting, we build methods to visualize and analyze large ensembles of alternative plans. Our detailed case studies use zero-dimensional homology (persistent components) of filtered graphs constructed from voting data to analyze redistricting in Pennsylvania and North Carolina. We find that, across large ensembles of partitions, the features cluster in the persistence diagrams in a way that corresponds strongly to geographic location, so that we can construct an average diagram for an ensemble, with each point identified with a geographical region. Using this localization lets us produce zonings of each state at Congressional, state Senate, and state House scales, show the regional non-uniformity of election shifts, and identify attributes of partitions that tend to correspond to partisan advantage.

The methods here are set up to be broadly applicable to the use of TDA on large ensembles of data. Many studies will benefit from interpretable summaries of large sets of samples or simulations, and the work here on localization and zoning will readily generalize to other partition problems, which are abundant in scientific applications. For the mathematically and politically rich problem of redistricting in particular, TDA provides a powerful and elegant summarization tool whose findings will be useful for practitioners.

Keywords: Topological data analysis, Fréchet means, Markov chain ensembles, political redistricting.

Mathematics Subject Classification: Primary: 62R40, 55N31; Secondary: 68T09.

Citation: Moon Duchin, Tom Needham, Thomas Weighill. The (homological) persistence of gerrymandering. Foundations of Data Science, doi: 10.3934/fods.2021007

References:

[1]	T. Abrishami, N. Guillen, P. Rule, Z. Schutzman, J. Solomon, T. Weighill and S. Wu, Geometry of graph partitions via optimal transport, SIAM J. Sci. Comput., 42 (2020), A3340-A3366. doi: 10.1137/19M1295258.
[2]	H. Adams, T. Emerson, M. Kirby, R. Neville and C. Peterson, et al., Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), 35pp.
[3]	P. K. Agarwal, K. Fox, A. Nath, A. Sidiropoulos and Y. Wang, Computing the Gromov-Hausdorff distance for metric trees, ACM Trans. Algorithms, 14 (2018), 20pp. doi: 10.1145/3185466.
[4]	P. Bajardi, M. Delfino, A. Panisson, G. Petri and M. Tizzoni, Unveiling patterns of international communities in a global city using mobile phone data, EPJ Data Science, 4 (2015). doi: 10.1140/epjds/s13688-015-0041-5.
[5]	A. Banman and L. Ziegelmeier, Mind the gap: {A} study in global development through persistent homology, in Research in Computational Topology, Assoc. Women Math. Ser., 13, Springer, Cham, 2018,125-144. doi: 10.1007/978-3-319-89593-2_8.
[6]	P. Bendich, J. S. Marron, E. Miller, A. Pieloch and S. Skwerer, Persistent homology analysis of brain artery trees, Ann. Appl. Stat., 10 (2016), 198-218. doi: 10.1214/15-AOAS886.
[7]	R. Brüel-Gabrielsson, B. J. Nelson, A. Dwaraknath, P. Skraba, L. J. Guibas and G. Carlsson, A topology layer for machine learning, preprint, arXiv: 1905.12200.
[8]	P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77-102.
[9]	P. Bubenik, J. Scott and D. Stanley, An algebraic Wasserstein distance for generalized persistence modules, preprint, arXiv: 1809.09654.
[10]	S. Cannon, M. Duchin, D. Randall and P. Rule, A reversible recombination chain for graph partitions, work in progress.
[11]	G. Carlsson, Topological pattern recognition for point cloud data, Acta Numer., 23 (2014), 289-368. doi: 10.1017/S0962492914000051.
[12]	G. Carlsson, V. de Silva and D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009,247-256. doi: 10.1145/1542362.1542408.
[13]	G. Carlsson and F. Mémoli, Characterization, stability and convergence of hierarchical clustering methods, J. Mach. Learn. Res., 11 (2010), 1425-1470.
[14]	M. Carrière, F. Chazal, Y. Ike, T. Lacombe, M. Royer and Y. Umeda, PersLay: A neural network layer for persistence diagrams and new graph topological signatures, preprint, arXiv: 1904.09378.
[15]	F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas and S. Y. Oudot, Proximity of persistence modules and their diagrams, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009,237-246. doi: 10.1145/1542362.1542407.
[16]	F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Mémoli and S. Y. Oudot, Gromov-Hausdorff stable signatures for shapes using persistence, in Computer Graphics Forum, 28, Wiley Online Library, 2009, 1393-1403. doi: 10.1111/j.1467-8659.2009.01516.x.
[17]	F. Chazal, B. T. Fasy, F. Lecci, A. Rinaldo and L. Wasserman, Stochastic convergence of persistence landscapes and silhouettes, in Computational Geometry (SoCG'14), ACM, New York, 2014,474-483. doi: 10.1145/2582112.2582128.
[18]	F. Chazal, M. Glisse, C. Labruère and B. Michel, Convergence rates for persistence diagram estimation in topological data analysis, J. Mach. Learn. Res., 16 (2015), 3603-3635.
[19]	F. Chazal and B. Michel, An introduction to topological data analysis: Fundamental and practical aspects for data scientists, preprint, arXiv: 1710.04019.
[20]	M. Chikina, A. Frieze and W. Pegden, Assessing significance in a Markov chain without mixing, Proc. Natl. Acad. Sci. USA, 114 (2017), 2860-2864. doi: 10.1073/pnas.1617540114.
[21]	S. Chowdhury, B. Dai and F. Mémoli, The importance of forgetting: Limiting memory improves recovery of topological characteristics from neural data, PLoS One, 13 (2018). doi: 10.1371/journal.pone.0202561.
[22]	D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120. doi: 10.1007/s00454-006-1276-5.
[23]	L. Crawford, A. Monod, A. X. Chen, S. Mukherjee and R. Rabadán, Predicting clinical outcomes in glioblastoma: An application of topological and functional data analysis, J. Amer. Statist. Assoc., 115 (2020), 1139-1150. doi: 10.1080/01621459.2019.1671198.
[24]	J. Curry, The fiber of the persistence map for functions on the interval, J. Appl. Comput. Topol., 2 (2018), 301-321. doi: 10.1007/s41468-019-00024-z.
[25]	Y. Dabaghian, F. Mémoli, L. Frank and G. Carlsson, A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Computational Biology, 8 (2012). doi: 10.1371/journal.pcbi.1002581.
[26]	D. DeFord and M. Duchin, Redistricting reform in Virginia: Districting criteria in context, MGGG, 2019. Available from: https://mggg.org/va-criteria.pdf.
[27]	D. DeFord, M. Duchin and J. Solomon, Comparison of districting plans for the Virginia house of delegates, MGGG, 2018. Available from: https://mggg.org/VA-report.pdf.
[28]	D. DeFord, M. Duchin and J. Solomon, Recombination: A family of Markov chains for redistricting., preprint, arXiv: 1911.05725.
[29]	M. Duchin, Gerrymandering metrics: How to measure? What's the baseline?, Bull. Amer. Acad. Arts Sci., 71 (2018), 54-58.
[30]	M. Duchin, Outlier analysis for Pennsylvania congressional redistricting, report, 2018. Available from: https://mggg.org/uploads/md-report.pdf.
[31]	M. Duchin and O. Walch, Political Geometry, in press, Birkhäuser.
[32]	H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/mbk/069.
[33]	K. Emmett, B. Schweinhart and P. Rabadan, Multiscale topology of chromatin folding, in Proceedings of the 9th EAI International Conference on Bio-Inspired Information and Communications Technologies, ACM, 2016,177-180. doi: 10.4108/eai.3-12-2015.2262453.
[34]	M. Feng and M. A. Porter, Persistent homology of geospatial data: A case study with voting, SIAM Rev., 63 (2021), 67--99. doi: 10.1137/19M1241519.
[35]	M. Feng and M. A. Porter, Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence, Phys. Rev. Research, 2 (2020). doi: 10.1103/PhysRevResearch.2.033426.
[36]	P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math., 6 (1972), 71-103. doi: 10.1007/BF01298413.
[37]	B. Grofman and G. King, The future of partisan symmetry as a judicial test for partisan gerrymandering after LULAC v. Perry, Election Law J., 6 (2007), 2-35. doi: 10.1089/elj.2006.6002.
[38]	L. Guth, A. Nieh and T. Weighill, Three applications of entropy to gerrymandering, preprint, arXiv: 2010.14972.
[39]	G. Herschlag, H. S. Kang, J. Luo, C. Vaughn Graves, S. Bangia, R. Ravier and J. C. Mattingly, Quantifying gerrymandering in North Carolina, Stat. Public Policy, 7 (2020), 30-38. doi: 10.1080/2330443X.2020.1796400.
[40]	Y. Hiraoka, T. Shirai and K. D. Trinh, Limit theorems for persistence diagrams, Ann. Appl. Probab., 28 (2018), 2740-2780. doi: 10.1214/17-AAP1371.
[41]	C. D. Hofer, R. Kwitt and M. Niethammer, Learning representations of persistence barcodes, J. Mach. Learn. Res., 20 (2019), 45pp.
[42]	M. Lesnick, The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math., 15 (2015), 613-650. doi: 10.1007/s10208-015-9255-y.
[43]	J. Levitt, A citizen's guide to redistricting, SSRN, 2008,139pp. doi: 10.2139/ssrn.1647221.
[44]	A. Lytchak, Open map theorem for metric spaces, St. Petersburg Math. J., 17 (2006), 477-491. doi: 10.1090/S1061-0022-06-00916-2.
[45]	C. Maria, J.-D. Boissonnat, M. Glisse and M. Yvinec, The Gudhi library: Simplicial complexes and persistent homology, in International Congress on Mathematical Software, Lecture Notes in Computer Science, 8592, Springer, Berlin, Heidelberg, 2014,167-174. doi: 10.1007/978-3-662-44199-2_28.
[46]	V. Maroulas, F. Nasrin and C. Oballe, A Bayesian framework for persistent homology, SIAM J. Math. Data Sci., 2 (2020), 48-74. doi: 10.1137/19M1268719.
[47]	J. Mattingly, Report on Redistricting: Drawing the Line., Common Cause v. Rucho, 318 F.Supp.3d 777, Ex.3002, 871 (2018).
[48]	M. D. McDonald and R. E. Best, Unfair partisan gerrymanders in politics and law: A diagnostic applied to six cases, Election Law J., 14 (2015), 312-330. doi: 10.1089/elj.2015.0358.
[49]	MGGG, MAUP: The geospatial toolkit for redistricting data., Available from: https://github.com/mggg/maup.
[50]	MGGG, MGGG-states: Open collection of precincts shapefiles for U.S. states., Available from: https://github.com/mggg-states.
[51]	Y. Mileyko, S. Mukherjee and J. Harer, Probability measures on the space of persistence diagrams, Inverse Problems, 27 (2011), 22pp. doi: 10.1088/0266-5611/27/12/124007.
[52]	D. Morozov, K. Beketayev and G. Weber, interleaving distance between merge trees., Discrete Comput. Geom., 49 (2013), 22-45.
[53]	M. Nicolau, A. J. Levine and G. Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, PNAS, 108 (2011), 7265-7270. doi: 10.1073/pnas.1102826108.
[54]	W. Pegden, Pennsylvania's Congressional Districting is an Outlier: Expert Report, League of Women Voters, et. al v. the Commonwealth of Pennsylvania, et. al. 159 MM, 2017.
[55]	J. Rodden and T. Weighill, Political geography: A case study of districting in Pennsylvania., preprint, arXiv: 2010.14608.
[56]	N. O. Stephanopoulos and E. M. McGhee, Partisan gerrymandering and the efficiency gap., The University of Chicago Law Review, (2015), 831-900.
[57]	B. J. Stolz, H. A. Harrington and M. A. Porter, The topological "shape" of Brexit, SSRN, (2016), 9pp. doi: 10.2139/ssrn.2843662.
[58]	K. Turner, Y. Mileyko, S. Mukherjee and J. Harer, Fréchet means for distributions of persistence diagrams, Discrete Comput. Geom., 52 (2014), 44-70. doi: 10.1007/s00454-014-9604-7.
[59]	Voting Rights Data Institute, GerryChain, GitHub Repository, 2018.
[60]	A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274. doi: 10.1007/s00454-004-1146-y.

show all references

References:

[1]	T. Abrishami, N. Guillen, P. Rule, Z. Schutzman, J. Solomon, T. Weighill and S. Wu, Geometry of graph partitions via optimal transport, SIAM J. Sci. Comput., 42 (2020), A3340-A3366. doi: 10.1137/19M1295258.
[2]	H. Adams, T. Emerson, M. Kirby, R. Neville and C. Peterson, et al., Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), 35pp.
[3]	P. K. Agarwal, K. Fox, A. Nath, A. Sidiropoulos and Y. Wang, Computing the Gromov-Hausdorff distance for metric trees, ACM Trans. Algorithms, 14 (2018), 20pp. doi: 10.1145/3185466.
[4]	P. Bajardi, M. Delfino, A. Panisson, G. Petri and M. Tizzoni, Unveiling patterns of international communities in a global city using mobile phone data, EPJ Data Science, 4 (2015). doi: 10.1140/epjds/s13688-015-0041-5.
[5]	A. Banman and L. Ziegelmeier, Mind the gap: {A} study in global development through persistent homology, in Research in Computational Topology, Assoc. Women Math. Ser., 13, Springer, Cham, 2018,125-144. doi: 10.1007/978-3-319-89593-2_8.
[6]	P. Bendich, J. S. Marron, E. Miller, A. Pieloch and S. Skwerer, Persistent homology analysis of brain artery trees, Ann. Appl. Stat., 10 (2016), 198-218. doi: 10.1214/15-AOAS886.
[7]	R. Brüel-Gabrielsson, B. J. Nelson, A. Dwaraknath, P. Skraba, L. J. Guibas and G. Carlsson, A topology layer for machine learning, preprint, arXiv: 1905.12200.
[8]	P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77-102.
[9]	P. Bubenik, J. Scott and D. Stanley, An algebraic Wasserstein distance for generalized persistence modules, preprint, arXiv: 1809.09654.
[10]	S. Cannon, M. Duchin, D. Randall and P. Rule, A reversible recombination chain for graph partitions, work in progress.
[11]	G. Carlsson, Topological pattern recognition for point cloud data, Acta Numer., 23 (2014), 289-368. doi: 10.1017/S0962492914000051.
[12]	G. Carlsson, V. de Silva and D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009,247-256. doi: 10.1145/1542362.1542408.
[13]	G. Carlsson and F. Mémoli, Characterization, stability and convergence of hierarchical clustering methods, J. Mach. Learn. Res., 11 (2010), 1425-1470.
[14]	M. Carrière, F. Chazal, Y. Ike, T. Lacombe, M. Royer and Y. Umeda, PersLay: A neural network layer for persistence diagrams and new graph topological signatures, preprint, arXiv: 1904.09378.
[15]	F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas and S. Y. Oudot, Proximity of persistence modules and their diagrams, in Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009,237-246. doi: 10.1145/1542362.1542407.
[16]	F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Mémoli and S. Y. Oudot, Gromov-Hausdorff stable signatures for shapes using persistence, in Computer Graphics Forum, 28, Wiley Online Library, 2009, 1393-1403. doi: 10.1111/j.1467-8659.2009.01516.x.
[17]	F. Chazal, B. T. Fasy, F. Lecci, A. Rinaldo and L. Wasserman, Stochastic convergence of persistence landscapes and silhouettes, in Computational Geometry (SoCG'14), ACM, New York, 2014,474-483. doi: 10.1145/2582112.2582128.
[18]	F. Chazal, M. Glisse, C. Labruère and B. Michel, Convergence rates for persistence diagram estimation in topological data analysis, J. Mach. Learn. Res., 16 (2015), 3603-3635.
[19]	F. Chazal and B. Michel, An introduction to topological data analysis: Fundamental and practical aspects for data scientists, preprint, arXiv: 1710.04019.
[20]	M. Chikina, A. Frieze and W. Pegden, Assessing significance in a Markov chain without mixing, Proc. Natl. Acad. Sci. USA, 114 (2017), 2860-2864. doi: 10.1073/pnas.1617540114.
[21]	S. Chowdhury, B. Dai and F. Mémoli, The importance of forgetting: Limiting memory improves recovery of topological characteristics from neural data, PLoS One, 13 (2018). doi: 10.1371/journal.pone.0202561.
[22]	D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120. doi: 10.1007/s00454-006-1276-5.
[23]	L. Crawford, A. Monod, A. X. Chen, S. Mukherjee and R. Rabadán, Predicting clinical outcomes in glioblastoma: An application of topological and functional data analysis, J. Amer. Statist. Assoc., 115 (2020), 1139-1150. doi: 10.1080/01621459.2019.1671198.
[24]	J. Curry, The fiber of the persistence map for functions on the interval, J. Appl. Comput. Topol., 2 (2018), 301-321. doi: 10.1007/s41468-019-00024-z.
[25]	Y. Dabaghian, F. Mémoli, L. Frank and G. Carlsson, A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Computational Biology, 8 (2012). doi: 10.1371/journal.pcbi.1002581.
[26]	D. DeFord and M. Duchin, Redistricting reform in Virginia: Districting criteria in context, MGGG, 2019. Available from: https://mggg.org/va-criteria.pdf.
[27]	D. DeFord, M. Duchin and J. Solomon, Comparison of districting plans for the Virginia house of delegates, MGGG, 2018. Available from: https://mggg.org/VA-report.pdf.
[28]	D. DeFord, M. Duchin and J. Solomon, Recombination: A family of Markov chains for redistricting., preprint, arXiv: 1911.05725.
[29]	M. Duchin, Gerrymandering metrics: How to measure? What's the baseline?, Bull. Amer. Acad. Arts Sci., 71 (2018), 54-58.
[30]	M. Duchin, Outlier analysis for Pennsylvania congressional redistricting, report, 2018. Available from: https://mggg.org/uploads/md-report.pdf.
[31]	M. Duchin and O. Walch, Political Geometry, in press, Birkhäuser.
[32]	H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/mbk/069.
[33]	K. Emmett, B. Schweinhart and P. Rabadan, Multiscale topology of chromatin folding, in Proceedings of the 9th EAI International Conference on Bio-Inspired Information and Communications Technologies, ACM, 2016,177-180. doi: 10.4108/eai.3-12-2015.2262453.
[34]	M. Feng and M. A. Porter, Persistent homology of geospatial data: A case study with voting, SIAM Rev., 63 (2021), 67--99. doi: 10.1137/19M1241519.
[35]	M. Feng and M. A. Porter, Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence, Phys. Rev. Research, 2 (2020). doi: 10.1103/PhysRevResearch.2.033426.
[36]	P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math., 6 (1972), 71-103. doi: 10.1007/BF01298413.
[37]	B. Grofman and G. King, The future of partisan symmetry as a judicial test for partisan gerrymandering after LULAC v. Perry, Election Law J., 6 (2007), 2-35. doi: 10.1089/elj.2006.6002.
[38]	L. Guth, A. Nieh and T. Weighill, Three applications of entropy to gerrymandering, preprint, arXiv: 2010.14972.
[39]	G. Herschlag, H. S. Kang, J. Luo, C. Vaughn Graves, S. Bangia, R. Ravier and J. C. Mattingly, Quantifying gerrymandering in North Carolina, Stat. Public Policy, 7 (2020), 30-38. doi: 10.1080/2330443X.2020.1796400.
[40]	Y. Hiraoka, T. Shirai and K. D. Trinh, Limit theorems for persistence diagrams, Ann. Appl. Probab., 28 (2018), 2740-2780. doi: 10.1214/17-AAP1371.
[41]	C. D. Hofer, R. Kwitt and M. Niethammer, Learning representations of persistence barcodes, J. Mach. Learn. Res., 20 (2019), 45pp.
[42]	M. Lesnick, The theory of the interleaving distance on multidimensional persistence modules, Found. Comput. Math., 15 (2015), 613-650. doi: 10.1007/s10208-015-9255-y.
[43]	J. Levitt, A citizen's guide to redistricting, SSRN, 2008,139pp. doi: 10.2139/ssrn.1647221.
[44]	A. Lytchak, Open map theorem for metric spaces, St. Petersburg Math. J., 17 (2006), 477-491. doi: 10.1090/S1061-0022-06-00916-2.
[45]	C. Maria, J.-D. Boissonnat, M. Glisse and M. Yvinec, The Gudhi library: Simplicial complexes and persistent homology, in International Congress on Mathematical Software, Lecture Notes in Computer Science, 8592, Springer, Berlin, Heidelberg, 2014,167-174. doi: 10.1007/978-3-662-44199-2_28.
[46]	V. Maroulas, F. Nasrin and C. Oballe, A Bayesian framework for persistent homology, SIAM J. Math. Data Sci., 2 (2020), 48-74. doi: 10.1137/19M1268719.
[47]	J. Mattingly, Report on Redistricting: Drawing the Line., Common Cause v. Rucho, 318 F.Supp.3d 777, Ex.3002, 871 (2018).
[48]	M. D. McDonald and R. E. Best, Unfair partisan gerrymanders in politics and law: A diagnostic applied to six cases, Election Law J., 14 (2015), 312-330. doi: 10.1089/elj.2015.0358.
[49]	MGGG, MAUP: The geospatial toolkit for redistricting data., Available from: https://github.com/mggg/maup.
[50]	MGGG, MGGG-states: Open collection of precincts shapefiles for U.S. states., Available from: https://github.com/mggg-states.
[51]	Y. Mileyko, S. Mukherjee and J. Harer, Probability measures on the space of persistence diagrams, Inverse Problems, 27 (2011), 22pp. doi: 10.1088/0266-5611/27/12/124007.
[52]	D. Morozov, K. Beketayev and G. Weber, interleaving distance between merge trees., Discrete Comput. Geom., 49 (2013), 22-45.
[53]	M. Nicolau, A. J. Levine and G. Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, PNAS, 108 (2011), 7265-7270. doi: 10.1073/pnas.1102826108.
[54]	W. Pegden, Pennsylvania's Congressional Districting is an Outlier: Expert Report, League of Women Voters, et. al v. the Commonwealth of Pennsylvania, et. al. 159 MM, 2017.
[55]	J. Rodden and T. Weighill, Political geography: A case study of districting in Pennsylvania., preprint, arXiv: 2010.14608.
[56]	N. O. Stephanopoulos and E. M. McGhee, Partisan gerrymandering and the efficiency gap., The University of Chicago Law Review, (2015), 831-900.
[57]	B. J. Stolz, H. A. Harrington and M. A. Porter, The topological "shape" of Brexit, SSRN, (2016), 9pp. doi: 10.2139/ssrn.2843662.
[58]	K. Turner, Y. Mileyko, S. Mukherjee and J. Harer, Fréchet means for distributions of persistence diagrams, Discrete Comput. Geom., 52 (2014), 44-70. doi: 10.1007/s00454-014-9604-7.
[59]	Voting Rights Data Institute, GerryChain, GitHub Repository, 2018.
[60]	A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249-274. doi: 10.1007/s00454-004-1146-y.

Figure 1. This paper studies the aggregation of votes from geographic units to districts, which is intractable to study by complete enumeration of possibilities. We develop persistent homology techniques to summarize the districting problem. This figure shows a precinct map of Pennsylvania with key cities identified. The precincts are colored by voting results from the 2016 Presidential election (blue for Democratic, red for Republican). Current districts in Pennsylvania are shown with the same coloring. Note that the Congressional districts are balanced to within one-person population deviation (by 2010 Census count)

	PRES12	PRES16	SEN10	SEN12	SEN16	GOV10	ATG12	ATG16
R %	47.29	50.35	51.05	45.44	50.72	54.52	42.58	48.57

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	PRES12	PRES16	SEN10	SEN14	SEN16	GOV12	GOV16
R %	51.08	51.98	56.02	49.17	53.02	55.87	49.95

	PRES12	PRES16	SEN10	SEN14	SEN16	GOV12	GOV16
R %	51.08	51.98	56.02	49.17	53.02	55.87	49.95

American Institute of Mathematical Sciences

The (homological) persistence of gerrymandering

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