Random walk on sparse random digraphs

Abstract

A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains.Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure. by n under the equilibrium measure (right) for the random walk on a random digraph with n = 5000 + 5000 + 5000 vertices with respective degrees (d + , d − ) = (3, 2), (3, 4) and (4, 4).

Cutoff at the “entropic time” for sparse Markov chains

“…One could easily deduce from our proofs that, with high probability, the entropy of the distribution P t (i, ·) on the timeinterval [0, t ent ] grows roughly linearly, at rate H. This in turns implies that the entropy of π is (1 − o(1)) log n with high probability. Consequently, we see that the cutoff occurs precisely when the entropy of the chain reaches the entropy of the invariant distribution, and that the mixing time is given by the entropy at stationarity divided by the average single step entropy H. Interestingly, the same interpretation can be given to the main results in the models studied in [21,7,6,9]. It is thus perhaps tempting to believe that this scenario should apply to a much larger class of Markov chains in random environments, although we do not have a precise conjecture to propose at the present time.…”

“…The present paper considerably extends the results in [9] by establishing cutoff for a large class of non-reversible sparse stochastic matrices, not necessarily arising as the transition matrix of the random walk on a (directed) graph. The time-irreversibility actually plays a crucial role in our proofs: despite the lack of an explicit underlying structure, the Markov chains that we consider turn out to exhibit a spontaneous "non-backtracking" tendency which allows us to establish a certain i.i.d.…”

“…To the best of our knowledge, the occurence of a cutoff phenomenon is new even in the special case where d 1 = · · · = d n = r for some fixed integer r ≥ 2, known as the random r−out digraph. We emphasize that the results of [9] do not apply here, since with high probability the minimum in-degree is zero and the maximum in-degree diverges (logarithmically) with n. We also note that the structure of the stationary measure π on random r−out digraphs has been investigated in details by Addario-Berry, Balle and Perarnau [1].…”

“…The above mentioned references are all concerned with the reversible case of undirected graphs, where the associated simple random walk and non-backtracking random walk have explicitly known stationary distributions. In our recent work [9], we investigated the non-reversible case of random walk on sparse directed graphs Figure 1. Distance to equilibrium along time for the n × n random matrix in Theorem 2, with Pareto(α) entry distribution, i.e.…”

“…approximation for the environment seen by a typical walker. While the overall strategy of proof of our main result is closely related to the one we introduced in [9], the level of generality allowed for in the transition probabilities requires an entirely new analysis of path weights. For instance, one of the features making the control of trajectory weights much more challenging here is the lack of nontrivial upper bounds on the probability of any particular transition (as opposed to the model studied in [9] where the minimal out-degree was assumed to be at least 2).…”