A local graph rewiring algorithm for sampling spanning trees

TL;DR

This paper presents a Markov Chain Monte Carlo algorithm that efficiently samples spanning trees of complete graphs using local rewiring, with proven ergodicity and results matching theoretical predictions.

Contribution

It introduces a novel local rewiring MCMC algorithm for sampling spanning trees, analyzing its properties and validating its effectiveness against theoretical values.

Findings

Algorithm is ergodic and generates independent configurations efficiently.

Autocorrelation time of graph diameter decreases with system size.

Mean graph diameter matches theoretical asymptotic values.

Abstract

We introduce a Markov Chain Monte Carlo algorithm which samples from the space of spanning trees of complete graphs using local rewiring operations only. The probability distribution of graphs of this kind is shown to depend on the symmetries of these graphs, which are reflected in the equilibrium distribution of the Markov chain. We prove that the algorithm is ergodic and proceed to estimate the probability distribution for small graph ensembles with exactly known probabilities. The autocorrelation time of the graph diameter demonstrates that the algorithm generates independent configurations efficiently as the system size increases. Finally, the mean graph diameter is estimated for spanning trees of sizes ranging over three orders of magnitude. The mean graph diameter results agree with theoretical asymptotic values.