Mixing times of random walks on dynamic configuration models

Abstract

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on n vertices, is known to be of order log n. In this paper we investigate what happens when the random graph becomes dynamic, namely, at each unit of time a fraction α n of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every ε ∈ (0, 1) the ε-mixing time of random walk without backtracking grows like 2 log(1/ε)/ log(1/(1 − α n )) as n → ∞, provided that lim n→∞ α n (log n) 2 = ∞. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.

Random walks on dynamic configuration models:A trichotomy

“…The following hold whp in η and x: The proof of Theorem 1.4 is organised as follows. Theorem 1.4(1) was already proved in [1]. In Section 2 we show that Theorems 1.4(2)-(3) follow from a key proposition (Proposition 2.1 below), which will be proved in Sections 3-4.…”

“…Condition 1.2 was used in [1] to deal with the regime of 'fast graph dynamics'. Conditions (R1) and (R2) are minimal requirements to guarantee that the graph is locally tree-like.…”

“…In this section, we prove Proposition 2.1. We use the notation introduced in [1] and recall some of the definitions that are needed along the way. For a fixed sequence of half-edges x [0,t] with x 0 = x and a fixed set of times T ⊆ [t], we use the short-hand notation…”

“…In an earlier paper [1] we consider a discrete-time dynamic version of the configuration model, where at each unit of time a fraction α n of the edges is sampled and rewired uniformly at random. Our dynamics preserves the degrees of the vertices.…”

“…In Section 1.2 we define the model. This is a verbatim repetition of what was written in [1,Section 1.2], in which we introduce notation and set the stage. In Section 1.3 we state our main theorem, which is a trichotomy for the cases β = ∞, β ∈ (0, ∞) and β = 0.…”