Kemeny's constant measures the efficiency of a Markov chain in traversing its states. We investigate whether structure-preserving perturbations to the transition probabilities of a reversible Markov chain can improve its connectivity while maintaining a fixed stationary distribution. Although the minimum achievable value for Kemeny's constant can be estimated, the required perturbations may be infeasible. We reformulate the problem as an optimization task, focusing on solution existence and efficient algorithms, with an emphasis to the problem of minimizing Kemeny's constant under sparsity constraints.

翻译：Kemeny常数用于衡量马尔可夫链在状态间遍历的效率。本研究探讨在保持固定平稳分布的前提下，通过对可逆马尔可夫链的转移概率施加结构保持扰动，能否改善其连通性。尽管Kemeny常数的最小可达值可被估算，但所需扰动可能无法实现。我们将该问题重构为优化任务，重点关注解的存在性与高效算法，特别聚焦于稀疏性约束下最小化Kemeny常数的问题。