Random Walks on Small World Networks

TL;DR

This paper analyzes how the mixing time of random walks on small-world networks changes with the parameter r, revealing a phase transition at r=2 with different scaling behaviors.

Contribution

It establishes the phase transition in the mixing time of random walks on small-world networks at r=2, extending Kleinberg's routing results to mixing times.

Findings

Mixing time is Θ(log n) for r<2.

Mixing time is O((log n)^4) at r=2.

Mixing time is polynomial for r>2.

Abstract

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices $\{u,v\}$ with distance $d>1$ is added as a "long-range" edge with probability proportional to $d^{-r}$, where $r\geq 0$ is a parameter of the model. Kleinberg studied a close variant of this network model and proved that the (decentralised) routing time is $O((\log n)^2)$ when $r=2$ and $n^{\Omega(1)}$ when $r\neq 2$. Here, we prove that the random walk also undergoes a phase transition at $r=2$, but in this case the phase transition is of a different form. We establish that the mixing time is $\Theta(\log n)$ for $r<2$, $O((\log n)^4)$ for $r=2$ and $n^{\Omega(1)}$ for $r>2$.