We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for \emph{simple} graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

翻译：我们研究了对经典简单交换马尔可夫链的一种彩色推广，该链用于对具有固定度序列的图集合进行采样。此处我们考虑顶点带有颜色的图空间，其中固定度序列和由顶点着色产生的另一统计量，并证明该集合可以通过简单的保色交换或移动来连接。这些移动构成了定义不可约马尔可夫链的基础，这对于检验分区块网络数据的统计模型拟合度是必要的。我们的方法进一步推广了20世纪90年代著名的代数结果：即相应的移动可用于构造第二个超单形推广的正则三角剖分。另一方面，与单色情形相反，我们证明对于简单图，连接该空间所需移动的1-范数随颜色数量的增加而增大。