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The Traveling Mailman: Topological Optimization Methods for User-Centric Redistricting

Nelson A. Colón Vargas

TL;DR

The paper tackles redistricting by using the USPS postal network as a proxy for community connectivity and combining Topological Data Analysis with Markov Chain Monte Carlo to evaluate district plans. It initializes districts with SRKMeans and then uses stochastic rebalancing followed by MCMC to generate ensembles of admissible plans, measuring community preservation via edge cuts in a postal-network graph and Polsby–Popper compactness. Gaussian Mixture Model analysis shows the official Iowa plan is an outlier, with about 89% of generated plans having fewer cut edges than the official plan. The framework demonstrates improved community coherence over the official plan under relaxed population constraints, while highlighting data, computation, and scalability limitations and outlining paths for extension to other states and finer geographic units.

Abstract

This study introduces a new districting approach using the US Postal Service network to measure community connectivity. We combine Topological Data Analysis with Markov Chain Monte Carlo methods to assess district boundaries' impact on community integrity. Using Iowa as a case study, we generate and refine districting plans using KMeans clustering and stochastic rebalancing. Our method produces plans with fewer cut edges and more compact shapes than the official Iowa plan under relaxed conditions. The low likelihood of finding plans as disruptive as the official one suggests potential inefficiencies in existing boundaries. Gaussian Mixture Model analysis reveals three distinct distributions in the districting landscape. This framework offers a more accurate reflection of community interactions for fairer political representation.

The Traveling Mailman: Topological Optimization Methods for User-Centric Redistricting

TL;DR

The paper tackles redistricting by using the USPS postal network as a proxy for community connectivity and combining Topological Data Analysis with Markov Chain Monte Carlo to evaluate district plans. It initializes districts with SRKMeans and then uses stochastic rebalancing followed by MCMC to generate ensembles of admissible plans, measuring community preservation via edge cuts in a postal-network graph and Polsby–Popper compactness. Gaussian Mixture Model analysis shows the official Iowa plan is an outlier, with about 89% of generated plans having fewer cut edges than the official plan. The framework demonstrates improved community coherence over the official plan under relaxed population constraints, while highlighting data, computation, and scalability limitations and outlining paths for extension to other states and finer geographic units.

Abstract

This study introduces a new districting approach using the US Postal Service network to measure community connectivity. We combine Topological Data Analysis with Markov Chain Monte Carlo methods to assess district boundaries' impact on community integrity. Using Iowa as a case study, we generate and refine districting plans using KMeans clustering and stochastic rebalancing. Our method produces plans with fewer cut edges and more compact shapes than the official Iowa plan under relaxed conditions. The low likelihood of finding plans as disruptive as the official one suggests potential inefficiencies in existing boundaries. Gaussian Mixture Model analysis reveals three distinct distributions in the districting landscape. This framework offers a more accurate reflection of community interactions for fairer political representation.
Paper Structure (26 sections, 2 theorems, 10 equations, 17 figures, 5 tables, 7 algorithms)

This paper contains 26 sections, 2 theorems, 10 equations, 17 figures, 5 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Framework and Definitions
  3. Districting
  4. Topological Data Analysis
  5. KMeans Clustering
  6. Markov Chain Monte Carlo
  7. Rationale for Using MCMC
  8. Methodology
  9. Choice of $p$ for this Study
  10. Stochastic Rebalancing of KMeans Generated Districts
  11. KMeans
  12. Adjusting Regions to Meet Population Constraints
  13. Monte Carlo Markov Chain Simulation
  14. Gaussian Mixture Model Analysis
  15. Results
  16. ...and 11 more sections

Key Result

Theorem 1

Given a set of $n$ data points in $d$ dimensions, the KMeans algorithm iteratively improves cluster assignments and mean positions, converging to a local minimum of the within-cluster sum of squares. Although the global minimum is not guaranteed, the convergence to a local minimum ensures that the a

Figures (17)

  • Figure 1: Iowa Districts (2021)
  • Figure 2: Iowa Post Offices
  • Figure 3: Iowa Districts with Adjacency Graph
  • Figure 4: Iowa Districts with TDA Postal Network $(\epsilon = 14mi)$
  • Figure 5: Iowa's Districting Map.
  • ...and 12 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3: Polsby-Popper Score polsby1991compactness
  • Remark 1
  • Definition 4
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 5: Topological Data Analysis
  • Definition 6: Persistent Homology
  • ...and 10 more