Random Walks on Small World Networks

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 $d&gt;1$ is added as a "long-range" edge with probability proportional to $d<sup>{-r}$, 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 $O((\log n)<sup>2)$ when $r=2$ and $n<sup>{\Omega(1)}$ 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 $r&lt;2$, $O((\log n)<sup>4)$ for $r=2$ and $n<sup>{\Omega(1)}$ for $r&gt;2$.