Fastest Mixing Reversible Markov Chain: Clique Lifted Graphs and Subgraphs
This paper addresses the design of fastest-mixing reversible Markov chains: given a target stationary distribution, how to construct a reversible chain with minimal mixing time. Focusing on two fundamental graph structures—clique-lifted graphs and arbitrary subgraphs—the work makes two key contributions: (1) It establishes, for the first time, a strict reduction of the fastest-mixing problem on clique-lifted graphs to that on their base graphs, preserving the optimal mixing time; (2) It proves that the optimal transition probabilities on a subgraph can be determined independently of the ambient graph, ensuring consistency between local optimization and global optimality. Methodologically, the approach integrates semidefinite programming (SDP), graph-theoretic techniques (clique lifting and subgraph embedding), and spectral analysis of reversible chains. Experiments demonstrate that the framework yields analytically optimal solutions across diverse topologies, combining theoretical rigor with computational scalability.