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.
We consider a dynamic random graph on n vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction α n of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as n → ∞, when α n is chosen such that lim n→∞ α n (log n) 2 = β ∈ [0, ∞]. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order 1/ √ α n when β = ∞. In the present paper we investigate what happens when β ∈ [0, ∞). It turns out that the mixing time is of order log n, with the scaled mixing time exhibiting a one-sided cutoff when β ∈ (0, ∞) and a two-sided cutoff when β = 0.The occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes from a competition between the time scales of mixing on the static graph, as identified by , and the regeneration time of first stepping across a rewired edge.Mathematics Subject Classification 2010. 60K37, 82C27.
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