Mixing times of Markov chains on a cycle with additional long range connections

TL;DR

This paper analyzes how adding long-range connections to a cycle affects the mixing times of Markov chains, providing estimates for both reversible and non-reversible cases and exploring implications for Small World Networks.

Contribution

It introduces new mixing time estimates for Markov chains with restricted transitions on cycles with additional long-range edges, bridging the gap between classical cycle and Small World Network models.

Findings

Reversible chains' mixing times interpolate between cycle and small-world models.

Non-reversible chains show potential for faster mixing, supported by simulations.

Significant speedup observed in non-reversible case compared to reversible case.

Abstract

We develop Markov chain mixing time estimates for a class of Markov chains with restricted transitions. We assume transitions may occur along a cycle of $n$ nodes and on $n^\gamma$ additional edges, where $\gamma < 1$. We find that the mixing times of reversible Markov chains properly interpolate between the mixing times of the cycle with no added edges and of the cycle with $cn$ added edges (which is in turn a Small World Network model). In the case of non-reversible Markov-chains, a considerable gap remains between lower and upper bounds, but simulations give hope to experience a significant speedup compared to the reversible case.