Mixing time trichotomy in regenerating dynamic digraphs

TL;DR

This paper analyzes how random walks on regenerating dynamic directed graphs reach stationarity, revealing three distinct regimes based on the interplay between regeneration frequency and graph size, with implications for understanding mixing times.

Contribution

It introduces a trichotomy in the mixing time behavior of random walks on regenerating dynamic digraphs, depending on the asymptotic relation between regeneration rate and graph size.

Findings

Three regimes of convergence based on $\alpha \log n$

Stationarity achieved through different mechanisms in each regime

Provided control over approximations of the stationary distribution

Abstract

We study convergence to stationarity for random walks on dynamic random digraphs with given degree sequences. The digraphs undergo full regeneration at independent geometrically distributed random time intervals with parameter $\alpha$. Relaxation to stationarity is the result of a competition between regeneration and mixing on the static digraph. When the number of vertices $n$ tends to infinity and the parameter $\alpha$ tends to zero, we find three scenarios according to whether $\alpha\log n$ converges to zero, infinity or to some finite positive value: when the limit is zero, relaxation to stationarity occurs in two separate stages, the first due to mixing on the static digraph, and the second due to regeneration; when the limit is infinite, there is not enough time for the static digraph to mix and the relaxation to stationarity is dictated by the regeneration only; finally, when the limit is a finite positive value we find a mixed behaviour interpolating between the two extremes. A crucial ingredient of our analysis is the control of suitable approximations for the unknown stationary distribution.