# Mixing Time of Random Walk on Poisson Geometry Small World

## TL;DR

This paper studies the mixing time of random walks on a Poisson geometry small world network, showing that the diameter and mixing time grow at most polynomially with the logarithm of network size.

## Contribution

It introduces a new model of small world networks based on Poisson percolation and proves polynomial bounds on diameter and mixing time of random walks.

## Key findings

- Diameter grows at most polynomially in log n
- Random walk mixes rapidly, in polynomial time in log n
- Establishes properties of Poisson geometry small world networks

## Abstract

This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph $\mathscr G_n$. In the present paper, we first prove that the diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the random walk mixes in time at most polynomially large in $\ln n$.