Improved Mixing Rates of Directed Cycles with Additional Sparse Interconnections

TL;DR

This paper studies how adding sparse, random interconnections to directed cycles can significantly improve the mixing rates of associated Markov chains, especially when combined with asymmetry along the cycle.

Contribution

It introduces a novel analysis of spectral gaps for perturbed directed cycles, demonstrating improved mixing times with sparse interconnections and asymmetry.

Findings

Spectral gap can be bounded inversely by the longest arc length.

Sparse random perturbations lead to faster mixing compared to reversible chains.

High-probability bounds on mixing speedup are established.

Abstract

We analyze the absolute spectral gap of Markov chains on graphs obtained from a cycle of $n$ vertices and perturbed only at approximately $n^{1/\rho}$ random locations with an appropriate, possibly sparse, interconnection structure. Together with a strong asymmetry along the cycle, the gap of the resulting chain can be bounded inversely proportionally by the longest arc length (up to logarithmic factors) with high probability, providing a significant mixing speedup compared to the reversible version.