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 ~r^-α) 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~ln²L when α=d, and otherwise as T~Lˣ, with x=(d-α)/(d+1-α) for α<d, x=α-d for d<α<d+1, and $x=1$ for α>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 α's.