Compact Redistricting Plans Have Many Spanning Trees

TL;DR

This paper analytically confirms that sampling methods based on spanning trees favor more compact redistricting plans, establishing a relationship between boundary length and sampling probability, thus supporting their practical use.

Contribution

It provides a theoretical proof of the inverse exponential relationship between boundary length and sampling probability in redistricting algorithms based on spanning trees.

Findings

Longer boundaries decrease sampling probability exponentially.

Supports the use of spanning tree-based algorithms for compact redistricting.

Provides theoretical justification for observed empirical behavior.

Abstract

In the design and analysis of political redistricting maps, it is often useful to be able to sample from the space of all partitions of the graph of census blocks into connected subgraphs of equal population. There are influential Markov chain Monte Carlo methods for doing so that are based on sampling and splitting random spanning trees. Empirical evidence suggests that the distributions such algorithms sample from place higher weight on more "compact" redistricting plans, which is a practically useful and desirable property. In this paper, we confirm these observations analytically, establishing an inverse exponential relationship between the total length of the boundaries separating districts and the probability that such a map will be sampled. This result provides theoretical underpinnings for algorithms that are already making a significant real-world impact.