Kemeny's constant for non-backtracking random walks

TL;DR

This paper extends Kemeny's constant to non-backtracking random walks on graphs, compares it with the simple random walk case, and finds that non-backtracking walks generally have smaller expected hitting times across various graph families.

Contribution

It introduces a new definition of Kemeny's constant for non-backtracking walks and derives closed-form expressions for regular and biregular graphs, highlighting differences from simple random walks.

Findings

Non-backtracking Kemeny's constant is generally smaller than for simple random walks.

Closed-form formulas are provided for regular and biregular graphs.

Non-backtracking walks often have faster mixing times.

Abstract

Kemeny's constant for a connected graph $G$ is the expected time for a random walk to reach a randomly-chosen vertex $u$, regardless of the choice of the initial vertex. We extend the definition of Kemeny's constant to non-backtracking random walks and compare it to Kemeny's constant for simple random walks. We explore the relationship between these two parameters for several families of graphs and provide closed-form expressions for regular and biregular graphs. In nearly all cases, the non-backtracking variant yields the smaller Kemeny's constant.