Spanning Trees and Redistricting: New Methods for Sampling and Validation

TL;DR

This work advances neutral baseline sampling for redistricting by formalizing a spanning-tree distribution $\pi(P) \propto \prod_i N_{\sf ST}(P_i)$ over graph partitions and introducing RevReCom, a reversible recombination Markov chain that targets $\pi$ under exact and approximate population balance. It provides rigorous detailed-balance proofs for exact balance and practical extensions to $\epsilon$-balanced partitions, along with a high-performance Rust implementation and parallelization strategy. Through extensive benchmarking on a $7\times7$ grid and on real-world-like state graphs (PA, VA), the authors demonstrate that RevReCom, Forest ReCom, and SMC yield convergent, comparable null models, with RevReCom scaling effectively to larger district counts. The results establish spanning-tree-based ensembles as a robust, policy-relevant tool for evaluating gerrymandering, while offering open-source software and diagnostics for convergence and cross-validation in legal and legislative contexts.

Abstract

Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now signed on to decisions that find that computational methods can provide key evidence. Today, the leading approaches for benchmarking districting plans are based on the use of spanning trees for sampling graph partitions. We present a new *reversible recombination* algorithm and rigorously prove its fundamental properties. Furthermore, we argue for a canonical sampling distribution called the *spanning tree distribution* that is well adapted to redistricting and provides a principled foundation for comparing and validating methods. Together with a highly efficient (and open-source) implementation that can generate and handle large datasets, this work provides the most powerful null model to date for the gerrymandering problem, meeting an urgent democratic challenge with sound scientific methodology.

Figures (10)

• Figure 1: Illustration of the ensemble method in Pennsylvania. TOP LEFT: An ensemble of Pennsylvania districting plans produced with RevReCom is shown in the histograms (in blue). To make it, we overlay the plans with voting from two elections to determine how many of the 18 districts have more D votes than R votes; we call these Democratic seats. These "blindly drawn" plans usually gives fewer Democratic seats than the proportional outcome of nearly nine---an empirical finding based on the detailed geography of votes. TOP RIGHT: Four Congressional plans can then be compared to the ensemble, with colored dots on the histograms marking their performance. The (Republican) legislature's plans both secure more (Republican) partisan advantage than the bulk of the ensemble. BOTTOM: To investigate the consistency of tree-based sampling methods, we repeat this process and plot the mean of each trial. We include ten independent ensembles made with RevReCom and with four other samplers and plot the means with light gray bars, which appear darker when they overlap. (The means from the histograms above are shown here in teal.) The methods with asymptotic distributional guarantees ( RevReCom , Forest ReCom , and SMC) require more computation and give slightly more variable results, while the heuristic ReCom variants (B,C) give fast and stable results, but come with only an approximate description for their target distribution. See https://github.com/mggg/RRC-Replication/tree/main/figure_and_table_generation/table_outputs for details.
• Figure 2: Weighting partitions with spanning trees. Two configurations or districting plans $P$ and $Q$ are shown here, for the $7\times 7\to 7$ districting problem. There are 28 cut edges in plan $P$ and 42 cut edges in plan $Q$. In this example, each district $Q_i$ in $Q$ has just one spanning tree, while each district $P_i$ in $P$ has 15 spanning trees. This means a sample from the spanning tree distribution $\pi$ is exactly $15^7$ times as likely to choose the "plump" plan $P$ as it is to choose the "spindly" plan $Q$---a factor of over 170 million.
• Figure 3: Convergence in distribution. For the $7 \times 7$ grid divided into $7$ equal-sized districts, these plots show the distribution of cut edges in an ensemble of districting plans created after 10,000, 1 million, and 100 million RevReCom proposal steps (red) compared to the distribution of cut edges in all districting plans, weighted by $\pi$ (green). The million-step sample was collected in under ten seconds on a laptop computer, and the time growth is linear.
• Figure 4: Convergence diagnostics for the $7\times 7$ grid. In this figure, we compare different methods at their largest practical sample sizes. TOP: A trace plot of Wasserstein distance between cut edge distributions. Long runs are compared against each other, and against the $\pi$-weighted ground truth, showing excellent accuracy for the Markov chain methods. For RevReCom , the time to 3 million accepted steps on a laptop is about ten minutes. BOTTOM LEFT: The ReCom --A,B,C,D variants stabilize quickly, though they do not exactly converge to $\pi$. BOTTOM RIGHT: SMC is not a Markov chain, so outputs are shown here with dots representing whole runs. Performance improves with batch size. See also Figure \ref{['fig:50x50-2']}.
• Figure 5: Virginia example. TOP: The $L^1$ Wasserstein trace plot for Democratic vote share by district (from the 2016 Presidential contest) across three different pairwise comparisons of RevReCom and Forest ReCom runs. BOTTOM: The corresponding box-and-whiskers plot showing the partisan shares by district. The left-most column shows the range of shares in the least Democratic district in each plan; the next column in the second-least Democratic; and so on. In this race, Clinton received 52.8% of the major-party share, and this plot shows that in a $\pi$-typical plan, 6/11 districts would have a Democratic advantage, 4/11 would have a Republican advantage, and the last would be very competitive, if people voted as they did in this (Clinton vs. Trump) vote pattern.