Non-Backtracking Random Walks and a Weighted Ihara’s Theorem

Abstract

We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara's Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara's Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.

Non-backtracking PageRank

“…We now show with a small example that without the hypothesis of regularity the two steady states are no longer related in general when α ∈ (0, 1). For α = 0, 1 the conclusion still follows from an easy computation and from [17]. Let us consider the graph represented in Fig.…”

“…Proof When α = 1 the result follows from [17]. If α = 0, then from the description of P and P it follows that x = (1−α) n 1, andŷ = (1−α) n υ.…”

“…Let us now briefly discuss the case where the two measures behave similarly. It has been proved in [17] that for an undirected graph with nodes all having at least degree two (so that there are no dangling edges), the steady state of P = D −1 W coincides with that of P = D −1 B, once the latter has been projected in the node space via L T : so standard PageRank and the new non-backtracking PageRank-like measures are the same in the absence of teleportation. In other words, in this case where every edge is reciprocated and no teleportation is allowed, this result tells us that the long-term frequency of visits to a particular node during a random walk does not change when non-backtracking is imposed-the nodes benefit from non-backtracking precisely in proportion to their original PageRank score.…”

Correlation Bounds for Distant Parts of Factor of IID Processes

“…For regular graphs its spectrum is closely related to that of the adjacency operator, but the nonbacktracking operator often proves to be a more efficient tool, for example in understanding the spectral gap and expansion properties of random regular graphs [8,15,28]. In [1] and more recently in [7,21], the mixing time and the cutoff phenomenon were examined for nonbacktracking random walks.…”

A non-backtracking Pólya’s theorem

“…Extensive research has been done recently in the study of non-backtracking random walks. 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.…”