TITLE:
Non-Backtracking Random Walks and a Weighted Ihara’s Theorem
AUTHORS:
Mark Kempton
KEYWORDS:
Graph, Random Walk, Non-Backtracking Random Walk, Ihara Zeta Identity, Mixing Rate
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.6 No.4,
August
16,
2016
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.