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

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.