# Mixing time of PageRank surfers on sparse random digraphs

## TL;DR

This paper analyzes how the PageRank random walk converges to equilibrium on large sparse random directed graphs, revealing a trichotomy in behavior depending on the refresh probability relative to the graph's mixing time.

## Contribution

It identifies a universal three-regime behavior of PageRank convergence on sparse random digraphs, depending on the refresh probability and the graph's mixing time.

## Key findings

- When refresh probability is very small, convergence shows cutoff behavior.
- When refresh probability is large, convergence is exponential with rate equal to the refresh probability.
- Intermediate refresh probabilities lead to a mixed convergence behavior.

## Abstract

We consider the generalised PageRank walk on a digraph $G$, with refresh probability $\alpha$ and resampling distribution $\lambda$. We analyse convergence to stationarity when $G$ is a large sparse random digraph with given degree sequences, in the limit of vanishing $\alpha$. We identify three scenarios: when $\alpha$ is much smaller than the inverse of the mixing time of $G$ the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behaviour; when $\alpha$ is much larger than the inverse of the mixing time of $G$ on the contrary one has pure exponential decay with rate $\alpha$; when $\alpha$ is comparable to the inverse of the mixing time of $G$ there is a mixed behaviour interpolating between cutoff and exponential decay. This trichotomy is shown to hold uniformly in the starting point and uniformly in the resampling distribution $\lambda$.