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

2025-10-20

AI Summary

Existing graph partitioning MCMC algorithms—such as Reversible and Metropolized Forest Recombination—are restricted to sampling from distributions induced by spanning trees, lacking expressivity for arbitrary target distributions. This work introduces Marked-Edge Walking (MEW), a novel MCMC-based algorithm that first constructs a spanning-tree space augmented with marked edges, thereby decoupling graph partitioning from the choice of target distribution. Within this space, MEW defines computationally tractable and provably convergent transition probabilities via the Metropolis–Hastings framework, enabling flexible and efficient sampling of graph partitions under arbitrary target distributions. Theoretical analysis establishes irreducibility and aperiodicity—ensuring ergodicity—and guarantees convergence to the desired stationary distribution. Experiments on real-world dual graphs demonstrate that MEW significantly outperforms baseline methods in both convergence speed and sample quality.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Sampling graph partitions under tunable distributions using MCMC
• Overcoming constraints of spanning tree-based sampling algorithms
• Enabling flexible ensemble generation for redistricting plan analysis

Innovation

• MEW algorithm samples graph partitions via MCMC
• Uses marked edges on spanning trees for transitions
• Enables flexible ensemble generation beyond tree constraints