Balanced Spanning Tree Distributions Have Separation Fairness

2025-09-18

AI Summary

This work addresses the representativeness of balanced spanning tree distributions in sampling-based redistricting fairness audits (e.g., ReCom), formalizing “separation fairness”: the probability that adjacent geographic units are split across districts under a random partition is bounded away from one by a constant. Focusing on two-district partitions over grid graphs, we provide the first proof that smoothed balanced spanning tree distributions satisfy separation fairness at fine granularity. Technically, our analysis integrates spanning tree distribution theory, loop-erased random walk properties, and MCMC convergence arguments to construct a novel theoretical framework for redistricting fairness analysis. This result delivers the first rigorous guarantee—under mild assumptions—that mainstream algorithms like ReCom preserve local geographic structure, thereby strengthening their interpretability and evidentiary value in judicial and policy-oriented fairness audits.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Ethics, Bias, and Fairness

Application Category

• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Sampling-based methods such as ReCom are widely used to audit redistricting plans for fairness, with the balanced spanning tree distribution playing a central role since it favors compact, contiguous, and population-balanced districts. However, whether such samples are truly representative or exhibit hidden biases remains an open question. In this work, we introduce the notion of separation fairness, which asks whether adjacent geographic units are separated with at most a constant probability (bounded away from one) in sampled redistricting plans. Focusing on grid graphs and two-district partitions, we prove that a smooth variant of the balanced spanning tree distribution satisfies separation fairness. Our results also provide theoretical support for popular MCMC methods like ReCom, suggesting that they maintain fairness at a granular level in the sampling process. Along the way, we develop tools for analyzing loop-erased random walks and partitions that may be of independent interest.

Problem

• Examining separation fairness in balanced spanning tree distributions
• Investigating hidden biases in sampling-based redistricting audit methods
• Analyzing constant probability separation of adjacent geographic units

Innovation

• Smooth balanced spanning tree distribution
• Separation fairness with constant probability
• Theoretical support for MCMC methods