Asymptotic behavior of the Kleinberg model

TL;DR

This paper analyzes the asymptotic scaling of Kleinberg navigation in a lattice with long-range links, revealing how delivery time depends on the exponent a and the dimension d, and extends understanding of search efficiency.

Contribution

It provides an exact master equation analysis of Kleinberg navigation, deriving precise asymptotic scaling laws for delivery time based on the parameters a and d.

Findings

Delivery time scales as (ln L)^2 when a=d.

For a<d, delivery time scales as L^{(d-a)/(d+1-a)}.

For a>d+1, delivery time scales linearly with L.

Abstract

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 proportional to r^{-a}) 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 (ln L)^2 when a=d, and otherwise as L^x, with x=(d-a)/(d+1-a) for a<d, x=a-d for d<a<d+1, and x=1 for a>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 a's.