Fastest Mixing Reversible Markov Chain: Clique Lifted Graphs and Subgraphs

2020-01-06

IEEE Transactions on Signal and Information Processing over Networks

AI Summary

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.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• Fastest Mixing Reversible Markov Chains
• Graph Theory
• Convergence Rate

Innovation

• Accelerated Markov Chain
• Graph Simplification
• Optimal Local Variation Independence