Markov chains are one of the well-known tools for modeling and analyzing stochastic systems. At the same time, they are used for constructing random walks that can achieve a given stationary distribution. This paper is concerned with determining the transition probabilities that optimize the mixing time of the reversible Markov chains towards a given equilibrium distribution. This problem is referred to as the Fastest Mixing Reversible Markov Chain (FMRMC) problem. It is shown that for a given base graph and its clique lifted graph, the FMRMC problem over the clique lifted graph is reducible to the FMRMC problem over the base graph, while the optimal mixing times on both graphs are identical. Based on this result and the solution of the semidefinite programming formulation of the FMRMC problem, the problem has been addressed over a wide variety of topologies with the same base graph. Second, the general form of the FMRMC problem is addressed on stand-alone topologies as well as subgraphs of an arbitrary graph. For subgraphs, it is shown that the optimal transition probabilities over edges of the subgraph can be determined independent of rest of the topology.

翻译：马尔可夫链是建模与分析随机系统的著名工具之一，同时也被用于构建能够实现给定平稳分布的随机游走。本文关注于确定使可逆马尔可夫链向给定平衡分布的混合时间最优化的转移概率。该问题被称为最快混合可逆马尔可夫链问题。研究表明，对于给定的基图及其团提升图，团提升图上的FMRMC问题可归约为基图上的FMRMC问题，且两图上的最优混合时间相同。基于此结果以及FMRMC问题的半定规划公式解，该问题已在具有相同基图的多种拓扑结构上得到解决。其次，FMRMC问题的一般形式在独立拓扑结构以及任意图的子图上进行了探讨。对于子图，研究表明子图边上的最优转移概率可独立于拓扑结构的其余部分进行确定。