Table of Contents
Research Letters in Physics
Volume 2008, Article ID 346543, 4 pages
http://dx.doi.org/10.1155/2008/346543
Research Letter

Kleinberg Navigation on Anisotropic Lattices

Department of Physics, Clarkson University, Potsdam, NY 13699-5820, USA

Received 21 July 2008; Accepted 29 September 2008

Academic Editor: Peter McClintock

Copyright © 2008 J. M. Campuzano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the Kleinberg problem of navigation in small-world networks when the underlying lattice is stretched along a preferred direction. Extensive simulations confirm that maximally efficient navigation is attained when the length π‘Ÿ of long-range links is taken from the distribution 𝑃(𝐫)βˆΌπ‘Ÿβˆ’π›Ό, when the exponent 𝛼 is equal to 2, the dimension of the underlying lattice, regardless of the amount of anisotropy, but only in the limit of infinite lattice size, πΏβ†’βˆž. For finite size lattices we find an optimal 𝛼(𝐿) that depends strongly on 𝐿. The convergence to 𝛼=2 as πΏβ†’βˆž shows interesting power-law dependence on the anisotropy strength.