The (homological) persistence of gerrymandering

Abstract

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.

Geometric averages of partitioned datasets

“…The Democratic-leaning districts 11-13 near Charlotte and Raleigh, however, became more Democratic. This finding is consistent with the examination of (non)-uniformity of vote shifts in North Carolina using topological data analysis in [40].…”

“…Note the unusually low Democratic vote shares in District 3 in both the 2012 and 2016 plan despite the very narrow range in ensemble values. This phenomenon was observed in [40] using other methods. The Judges plan has no outliers, while the 2020 plan has one: District 9.…”

“…We demonstrate our method on Congressional redistricting in North Carolina, an area which has seen a great deal of litigation in the last decade, and which has already been extensively studied using ensemble methods in the literature [28,34,40,43]. For some context on North Carolina's political geography, we show the 2016 Presidential two-way vote shares by precinct in Figure 4.…”

“…Typically, these ensembles are analyzed (and compared against the plan being evaluated) at the level of summary statistics -for example, the number of Republican seats won under historical vote data. Two recent papers have also analyzed spatial characteristics of redistricting ensembles: via graph optimal transport [1] and topological data analysis [40]. Our method allows us to combine a geometric perspective in line with these two papers with a classical summary statistics approach.…”

Topological data analysis for geographical information science using persistent homology

“…The method is demonstrated to identify regions with voting patterns different from surrounding regions. In related work, Duchin et al (2021) propose a method for analysing the topological properties of election gerrymandering. Corcoran and Jones (2021) propose a method for analysing the connectivity of street networks.…”