Random Walks on Small World Networks

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≥ 0 is a parameter of the model. Kleinberg [33{ studied a close variant of this network model and proved that the (decentralised) routing time is O((log n)2) when r=2 and nΩ (1) when r≠ 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 ϴ (log n) for r< 2, O((log n)4) for r=2, and nΩ (1) for r> 2.