Multiscale Parallel Tempering for Fast Sampling on Redistricting Plans

TL;DR

This paper tackles the challenge of auditing redistricting plans by requiring ensembles drawn from policy driven distributions that match non partisan criteria. It introduces a multiscale parallel tempering framework that operates on a hierarchy of contracted graphs and uses a novel swap mechanism to propagate mixing across scales. The method achieves fast mixing on a large CT precinct graph and demonstrates how varying compactness and other measures shape partisan outcomes, including a comparison with forest based measures. The approach broadens the class of admissible policy based distributions available for principled redistricting audits and has practical implications for more robust, situation aware comparisons of district plans.

Abstract

When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given plan, one must ensure that the non-partisan criteria are matched so that we may conclude that partisan differences come from bias rather than, for example, levels of compactness or differences in community preservation. Certain sampling algorithms allow one to explicitly state the policy-based probability distribution on plans, however, these algorithms have shown poor mixing times for large graphs (i.e. redistricting spaces) for all but a few specialized measures. In this work, we generate a multiscale parallel tempering approach that makes local moves at each scale. The local moves allow us to adopt a wide variety of policy-based measures. We examine our method in the state of Connecticut and succeed at achieving fast mixing on a policy-based distribution that has never before been sampled at this scale. Our algorithm shows promise to expand to a significantly wider class of measures that will (i) allow for more principled and situation-based comparisons and (ii) probe for the typical partisan impact that policy can have on redistricting.

Figures (13)

• Figure 1: Example hierarchy and overall method for 2 districts on a 2x4 grid graph. (For this example, each unit square has population 1.)
• Figure 2: Geographic hierarchies on a 2x4 grid graph and the precinct graph of Randolph County, NC.
• Figure 3: Part of the swap operation must project fine scale district plans, $\xi_F$, to some plan that can be represented on the coarse structure $H_C$. If a coarsened node in $H_{C}$ is mapped to different districts at the finer scale, we must resolve the fine scale assignment as we project the fine scale redistricting plan onto the coarsened space.
• Figure 4: The swap mechanism transforms redistricting plans on adjacent levels in order to allow them to swap levels in the parallel tempering scheme.
• Figure 5: We show the congressional districts and precincts from the 2021 Connecticut redistricting cycle. Data comes from the VEST project CTvest via the Redistricting Data Hub redistrictingDataHub.