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.

我们提出了一种用于图形着色的可扩展并行 MCMC 算法的分析，旨在平衡颜色类大小，前提是提供了合适数量的颜色。首先，它表明马尔可夫链通过从每个节点的相邻颜色上的合适的建议分布中重复采样，独立地并因此以并行方式收敛到目标分布。我们证明了在算法的迭代过程中产生的不正确着色中的冲突数量在概率上迅速收敛到 0。对于平衡，考虑到所涉及的分布的复杂性，我们提出了对所达到的平衡水平的定性分析。基于由每个节点邻域内的颜色出现引起的多重分布的集合，我们提供了一些关于最终色彩平衡特性的证据，结果在颜色类别上几乎是均匀的。对大型社交图的一些数值模拟证实了快速收敛和平衡趋势，最终通过统计假设检验得到验证。