# PageRank in Undirected Random Graphs

## TL;DR

This paper analyzes the behavior of PageRank in large undirected random graphs with expansion properties, showing it approximates a mixture of restart and degree distributions, with extensions to community-structured graphs.

## Contribution

It provides a theoretical approximation of PageRank in undirected random graphs and extends the analysis to stochastic block models with community effects.

## Key findings

- PageRank approximates a mixture of restart and degree distributions in large graphs.
- Extension of the approximation to stochastic block models reveals community-dependent correction terms.
- Results hold asymptotically as graph size tends to infinity.

## Abstract

PageRank has numerous applications in information retrieval, reputation systems, machine learning, and graph partitioning. In this paper, we study PageRank in undirected random graphs with an expansion property. The Chung-Lu random graph is an example of such a graph. We show that in the limit, as the size of the graph goes to infinity, PageR- ank can be approximated by a mixture of the restart distribution and the vertex degree distribution. We also extend the result to Stochastic Block Model (SBM) graphs, where we show that there is a correction term that depends on the community partitioning.