Non-Backtracking Random Walks Mix Faster

Abstract

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is.As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o (1)) log n log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

Poisson approximation for non-backtracking random walks

“…Indeed, using Θ(n) random bits, it is possible to simulate a distribution which resembles the resulting distribution of throwing n balls to n bins uniformly and independently, as opposed to the naive approach, which requires n log n random bits. Theorem 1.5 also immediately gives the result of [2] regarding the maximal number of visits that a non-backtracking random walk makes to a single vertex, with an improved threshold window, replacing the o log n log log n error term by o (log n)(log log log n) (log log n) 2 : Corollary 1.6. For any fixed d ≥ 3 and fixed λ < d the following holds: if G is an (n, d, λ) graph whose girth is larger than 10 log d−1 log n, then the maximal number of visits to a single vertex made by a non-backtracking random walk of length n on G is with high probability 1 + (1 + o(1)) log log log n log log n log n log log n , where the o(1)-term tends to 0 as n → ∞.…”

“…In many applications for random walks, it seems that using non-backtracking random walks may yield better results, and that these walks possess better random-looking properties than those of simple random walks. This motivated the authors of [2] to study the mixing-rate of a non-backtracking random walk, and show that it may be up to twice as fast as that of a simple random walk. They further show that the number of times that such a walk visits the vertices of a high-girth expander is random-looking, in the sense that its maximum is typically asymptotically the same as the maximal load of the classical balls and bins experiment.…”

“…, w k ), where w i is chosen uniformly over all neighbors of w i−1 excluding w i−2 . The mixing-rate of a non-backtracking random walk on a regular graph, in terms of its eigenvalues, was computed in [2], using some properties of Chebyshev polynomials of the second kind. It is shown in [2] that this rate is always better than that of the simple random walk, provided that d = n o (1) .…”

“…As an application, the authors of [2] analyzed the maximal number of visits that a nonbacktracking random walk of length n makes to a vertex of G, an (n, d, λ) graph of fixed degree and girth Ω(log log n). Using a careful second moment argument, they proved that this quantity is typically (1 + o(1)) log n log log n , as is the typical maximal number of balls in a single bin when throwing n balls to n bins uniformly at random (more information on the classical balls and bins experiment may be found in [7]) .…”

“…In this work, we answer the above question by combining a generalized form of Brun's sieve with some extensions of the ideas in [2]. Let N t denote the number of vertices visited precisely t times by a non-backtracking random walk of length n on a regular n-vertex expander of fixed degree and girth g. We prove that if g = ω(1), then for any fixed t, N t /n is typically 1 et!…”