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arXiv · 0901.4535

Asymptotic behavior of the Kleinberg model

Shai Carmi·Stephen Carter·Jie Sun·Daniel ben-Avraham
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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+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.

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BibTeXRIS

Shai Carmi, Stephen Carter, Jie Sun, Daniel ben-Avraham. 2009-01-28. Asymptotic behavior of the Kleinberg model. https://doi.org/10.1103/physrevlett.102.238702

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