# Random Walks on Dynamic Graphs: Mixing Times, HittingTimes, and Return Probabilities

## TL;DR

This paper develops intuitive bounds for random walk metrics on dynamic graphs, showing their similarities to static graphs under certain conditions and exploring differences when connectivity varies.

## Contribution

It introduces a framework based on local expansion properties to analyze random walks on dynamic graphs, extending static graph results to the dynamic setting.

## Key findings

- Mixing and hitting times are O(n^2) for dynamic d-regular connected graphs.
- Refined bounds depend on the isoperimetric dimension, matching static graph results.
- Dynamic graphs with changing connectivity can exhibit strong discrepancies from static graph behavior.

## Abstract

We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allows us to capture the progress the random walk makes through $t$-step probabilities. We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that their behaviour is in a certain sense similar to static graphs. For example, we show that the mixing and hitting times of any sequence of $d$-regular connected graphs is $O(n^2)$, generalising a well-known result for static graphs. We also provide refined bounds depending on the isoperimetric dimension of the graph, matching again known results for static graphs. Finally, we investigate properties of random walks on dynamic graphs that are not always connected: we relate their convergence to stationarity to the spectral properties of an average of transition matrices and provide some examples that demonstrate strong discrepancies between static and dynamic graphs.