Hitting times for second-order random walks

TL;DR

This paper extends classical results on mean hitting and return times from standard to second-order random walks on graphs, providing formulas and lemmas that facilitate their computation and analysis.

Contribution

It introduces formulas for mean hitting and return times of second-order random walks and develops second-order versions of Kac's and the random target lemmas.

Findings

Derived simple formulas for hitting and return times.

Introduced the 'pullback' process for second-order walks.

Extended classical lemmas to second-order case.

Abstract

A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is not allowed to revisit a node in one step. Second-order random walks can model physical diffusion phenomena in a more realistic way than traditional random walks and have been very successfully used in various network mining and machine learning settings. However, numerous questions are still open for this type of stochastic processes. In this work we extend well-known results concerning mean hitting and return times of standard random walks to the second-order case. In particular, we provide simple formulas that allow us to compute these numbers by solving suitable systems of linear equations. Moreover, by introducing the "pullback" first-order stochastic process of a second-order random walk, we provide second-order versions of the renowned Kac's and random target lemmas.