Analysis of a parallel MCMC algorithm for graph coloring with nearly uniform balancing

We propose the analysis of a scalable parallel MCMC algorithm for graph coloring aimed at balancing the color class sizes, provided that a suitable number of colors is made available. Firstly, it is shown that the Markov chain converges to the target distribution by repeatedly sampling from suitable proposed distributions over the neighboring colors of each node, independently and hence in parallel manner. We prove that the number of conflicts in the improper colorings genereted thoughout the iterations of the algorithm rapidly converges in probability to 0. As for the balancing, given to the complexity of the distributions involved, we propose a qualitative analysis about the balancing level achieved. Based on a collection of multinoulli distributions arising from the color occurrences within every node neighborhood, we provide some evidence about the character of the final color balancing, which results to be nearly uniform over the color classes. Some numerical simulations on big social graphs confirm the fast convergence and the balancing trend, which is validated through a statistical hypothesis test eventually.