Poisson approximation for non-backtracking random walks

Abstract

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. The authors of [2] studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length n on a high-girth n-vertex regular expander, is typically (1 + o(1)) log n log log n , as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes.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! + o(1). Furthermore, if g = Ω(log log n), then N t /n is typically 1+o (1) et! uniformly on all t ≤ (1 − o(1)) log n log log n and 0 for all t ≥ (1 + o(1)) log n log log n . In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.

Balanced allocation on graphs: A random walk approach

“…A nonbacktracking random walk (NBRW) W of length l > 0 starting from a node is a random walk in l steps so that in each step the walker picks a neighbor uniformly at random and moves to that neighbor with an additional property that the walker never traverses an edge twice in a row. Further information about NBRWs can be found in [1] and [2]. Here we consider two families of -regular graphs, namely high-degree graphs, where = (log n) and low-degree graphs where, = O(log n).…”

“…It seems that the previous works could only handle any three of the four properties. In a different context, Alon and Lubetzky [2] showed that if a particle starts a NBRW of length n on an n-vertex regular expander graph with high-girth then the number of visits to nodes has a Poisson distribution. In particular, they showed that the maximum number of visits to a node is at most (1+o(1))⋅ log n log log n .…”

“…1), o(log n)] be a given integer and, for some h = O(log log n), G be a -regular n-vertex graph with girth at least 10h . Also, let k and be the parameters defined in (2). Assume that after allocating at most n 1 ⩽ n balls on G by A , U n 1 , ,h ∩  holds.…”

A non-backtracking Pólya’s theorem

“…Proof. Since the non-backtracking condition puts no restriction on the first step, it is clear thatà (1) = A. Forà (2) , note that A 2 counts all walks of length 2, so we must simply subtract off those that backtrack. The only walks of length 2 that backtrack are those that move from a vetex to a neghbor, then return immediately.…”

“…The convergence and mixing rate of non-backtracking random walks is studied in [1,4,8] and [9]. The distribution of the number of visits of a random walk to a vertex is studied in [2], and [3] studies non-backtracking random walks on the universal cover of a graph. In [10], non-backtracking random walks are used to study spectral clustering algorithms.…”