Asymptotic Behavior of the Kleinberg Model

We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability similar to r(-alpha)) by means of an exact master equation for the process. We show that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as T similar to ln(2)L when alpha=d, and otherwise as T similar to L-x, with x=(d-alpha)/(d+1-alpha) for alpha < d, x=alpha-d for d < d+1, and x=1 for alpha > d+1. These values of x exceed the rigorous lower bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of alpha's.