The Marked Edge Walk: A Novel MCMC Algorithm for Sampling of Graph Partitions

TL;DR

This paper introduces the marked edge walk (MEW), a new MCMC algorithm that efficiently samples graph partitions under flexible distributions, overcoming limitations of previous methods tied to spanning trees.

Contribution

The paper presents MEW, a novel MCMC algorithm that operates on spanning trees with marked edges, enabling sampling from a broader class of distributions for graph partitioning.

Findings

MEW converges under target distributions unrelated to spanning trees.

Empirical results demonstrate the algorithm's effectiveness on real-world graphs.

MEW offers a more flexible approach for ensemble generation in graph partitioning.

Abstract

Novel Markov Chain Monte Carlo (MCMC) methods have enabled the generation of large ensembles of redistricting plans through graph partitioning. However, existing algorithms such as Reversible Recombination (RevReCom) and Metropolized Forest Recombination (MFR) are constrained to sampling from distributions related to spanning trees. We introduce the marked edge walk (MEW), a novel MCMC algorithm for sampling from the space of graph partitions under a tunable distribution. The walk operates on the space of spanning trees with marked edges, allowing for calculable transition probabilities for use in the Metropolis-Hastings algorithm. Empirical results on real-world dual graphs show convergence under target distributions unrelated to spanning trees. For this reason, MEW represents an advancement in flexible ensemble generation.