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 simple graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

翻译：我们研究了著名的简单交换马尔可夫链的一种带颜色推广，该链用于采样具有固定度序列的图集。在此，我们考虑顶点带颜色的图空间，其中固定了度序列以及由顶点颜色产生的另一统计量，并证明了该集合可通过简单的颜色保持交换或移动实现连通性。这些移动构成了定义不可约马尔可夫链的基础，该链对于检验分块网络数据的统计模型拟合度至关重要。我们的方法进一步推广了1990年代的著名代数结果：即相应的移动可用于构造第二个超单纯形的一种推广的正则三角剖分。另一方面，与单色情况相比，我们证明对于简单图而言，实现空间连通性所需的移动的1-范数会随颜色数量的增加而增大。