Hitting Time Distributions of Random Walks on Finite Graphs

TL;DR

This paper studies the full distributions and variances of hitting times for random walks on finite graphs, providing formulas and methods to better understand their probabilistic behavior beyond expected values.

Contribution

It introduces new formulas and recurrence relations for the full distributions and variances of hitting times using spectral and Markov chain techniques.

Findings

Derived formulas for hitting time distributions

Provided recurrence relations for variance calculations

Enhanced understanding of hitting time variability

Abstract

We investigate the hitting times of random walks on graphs, where a hitting time is defined as the number of steps required for a random walker to move from one node to another. While much of the existing literature focuses on calculating or bounding expected hitting times, this approach is insufficient, as hitting time distributions often exhibit high variance. To address this gap, we analyze both the full distributions and variances of hitting times. Using general Markov chain techniques, as well as Fourier and spectral methods, we derive formulas and recurrence relations for computing these distributions. This is a preliminary work, and some results are being refined. A journal version will be available soon.