The (homological) persistence of gerrymandering

TL;DR

This paper introduces a novel application of persistent homology from topological data analysis to study electoral redistricting, enabling visualization and analysis of gerrymandering signals through persistence diagrams.

Contribution

It develops a new method combining geographic and election data with persistent homology to analyze redistricting plans and detect gerrymandering signals, with theoretical stability results and practical case studies.

Findings

Persistence diagrams are stable under data perturbations.

The method successfully identifies gerrymandering signals in case studies.

The approach provides a new topological perspective on redistricting analysis.

Abstract

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a persistence diagram. We are then able to visualize and analyze large ensembles of computer-generated districting plans of the type commonly used in modern redistricting research (and court challenges). We set out three applications: zoning a state at each scale of districting, comparing elections, and seeking signals of gerrymandering. Our case studies focus on redistricting in Pennsylvania and North Carolina, two states whose legal challenges to enacted plans have raised considerable public interest in the last few years. To address the question of robustness of the persistence diagrams to perturbations in vote data and in district boundaries, we translate the classical stability theorem of Cohen--Steiner et al. into our setting and find that it can be phrased in a manner that is easy to interpret. We accompany the theoretical bound with an empirical demonstration to illustrate diagram stability in practice.