Improved mixing rates of directed cycles by added connection

TL;DR

This paper demonstrates that adding random long-distance edges to a directed cycle graph significantly improves the mixing rate of the associated Markov chain, especially when non-reversibility is introduced.

Contribution

It provides a theoretical analysis showing a square factor improvement in mixing rates when augmenting directed cycles with random long-distance edges and non-reversibility.

Findings

Mixing rate improved by a square factor with added edges

Non-reversibility enhances mixing efficiency

Theoretical proof of mixing rate enhancement

Abstract

We investigate the mixing rate of a Markov chain where a combination of long distance edges and non-reversibility is introduced: as a first step, we focus here on the following graphs: starting from the cycle graph, we select random nodes and add all edges connecting them. We prove a square factor improvement of the mixing rate compared to the reversible version of the Markov chain.