Empirical Sampling of Connected Graph Partitions for Redistricting

TL;DR

This paper explores the connection between statistical physics and redistricting, analyzing Markov chain mixing times and the robustness of districting plan properties using graph theory and phase transition insights.

Contribution

It introduces a novel analysis of redistricting sampling methods through the lens of statistical physics, focusing on phase transitions and Markov chain behavior.

Findings

Self-avoiding walk phase transitions influence mixing times.

Connectivity and population constraints complicate Markov chain analysis.

Robustness of districting properties varies with score functions and graph discretization.

Abstract

The space of connected graph partitions underlies statistical models used as evidence in court cases and reform efforts that analyze political districting plans. In response to the demands of redistricting applications, researchers have developed sampling methods that traverse this space, building on techniques developed for statistical physics. In this paper, we study connections between redistricting and statistical physics, and in particular with self-avoiding walks. We exploit knowledge of phase transitions and asymptotic behavior in self avoiding walks to analyze two questions of crucial importance for Markov Chain Monte Carlo analysis of districting plans. First, we examine mixing times of a popular Glauber dynamics based Markov chain and show how the self-avoiding walk phase transitions interact with mixing time. We examine factors new to the redistricting context that complicate the picture, notably the population balance requirements, connectivity requirements, and the irregular graphs used. Second, we analyze the robustness of the qualitative properties of typical districting plans with respect to score functions and a certain lattice-like graph, called the state-dual graph, that is used as a discretization of geographic regions in most districting analysis. This helps us better understand the complex relationship between typical properties of districting plans and the score functions designed by political districting analysts. We conclude with directions for research at the interface of statistical physics, Markov chains, and political districting.