最快混合可逆马尔可夫链：团提升图与子图
Fastest Mixing Reversible Markov Chain: Clique Lifted Graphs and Subgraphs
AI总结：
本文研究最快混合可逆马尔可夫链问题，证明团提升图上的问题可约化为基图问题且最优混合时间相同，并进一步解决独立拓扑与子图上的一般形式问题。
摘要：
马尔可夫链是建模和分析随机系统的著名工具之一。同时，它们也被用于构造能够达到给定平稳分布的随机游走。本文研究确定转移概率，以优化可逆马尔可夫链向给定平衡分布混合的时间。该问题被称为最快混合可逆马尔可夫链（FMRMC）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问题可约化为基图上的FMRMC问题，且两个图上的最优混合时间相同。基于这一结果以及FMRMC问题的半定规划形式的解，该问题已在具有相同基图的多种拓扑上得到解决。其次，针对独立拓扑以及任意图的子图，研究了一般形式的FMRMC问题。对于子图，证明了子图边上的最优转移概率可以独立于拓扑的其余部分来确定。
英文摘要：
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.
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