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

TL;DR

This paper investigates the mixing times of non-backtracking random walks on static and dynamic random graphs, establishing a link between their behaviors and identifying various cut-off phenomena under different rewiring regimes.

Contribution

It introduces a novel coupling scheme to relate static and dynamic graph mixing times and characterizes new subregimes in mesoscopic dynamics.

Findings

Link established between static and dynamic mixing times.

Identified conditions for no cut-off, one-sided, and two-sided cut-off.

Discovered new behaviors with six subregimes in mesoscopic dynamics.

Abstract

This paper considers non-backtracking random walks on random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph tends to infinity. Subject to mild general conditions, we link two mixing times: one for a static version of the random graph, the other for a class of dynamic versions of the random graph in which the edges are randomly rewired but the degrees are preserved. The link is provided by the probability that the random walk has not yet stepped along a previously rewired edge. We use this link to compute the scaling of the mixing time for three specific classes of random rewirings. Depending on the speed and the range of the rewiring relative to the current location of the random walk, the mixing time may exhibit no cut-off, one-sided cut-off or two-sided cut-off, a trichotomy that was also found in earlier work. Interestingly, for a class of dynamics that are `mesoscopic', i.e., non-local and non-global, we find new behaviour with six subregimes. Proofs are built on a new and flexible coupling scheme, in combination with sharp estimates on the degrees encountered by the random walk in the static and the dynamic version of the random graph. Some of these estimates require sharp control on possible short-cuts in the graph between the edges that are traversed by the random walk.