Random Walks on Small World Networks
Abstract
We study the mixing time of random walks on smallworld networks modelled as follows: starting with the 2dimensional periodic grid, each pair of vertices $\{u,v\}$ with distance $d>1$ is added as a "longrange" 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 $r>2$.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.02467
 Bibcode:
 2017arXiv170702467D
 Keywords:

 Computer Science  Discrete Mathematics
 EPrint:
 To appear in Transactions of Algorithms (TALG)