Mixing time trichotomy in regenerating dynamic digraphs

Abstract

We study the 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 α. 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 α tends to zero, we find three scenarios according to whether α 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.

Linking the mixing times of random walks on static and dynamic random graphs

“…Trichotomies were also found in subsequent work. For example, [14] derived a trichotomy for a directed version of the configuration model. Contrary to our setting, in the digraph setup the rewiring no longer preserves the in-degree and out-degree sequences and hence the analysis must be restricted to a rewiring mechanism in which all the edges are freshly resampled at each step of the random walk.…”