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arXiv · 0805.0807

Kleinberg navigation on anisotropic lattices

J. Mauricio Campuzano·James P. Bagrow·Daniel ben-Avraham
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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 $r$ of long-range links is taken from the distribution $P({\bf r})\sim r^{-\alpha}$, when the exponent $\alpha$ 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, $L\to\infty$. For finite size lattices we find an optimal $\alpha(L)$ that depends strongly on $L$. The convergence to $\alpha=2$ as $L\to\infty$ shows interesting power-law dependence on the anisotropy strength.

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J. Mauricio Campuzano, James P. Bagrow, Daniel ben-Avraham. 2008-05-06. Kleinberg navigation on anisotropic lattices. https://doi.org/10.1155/2008/346543

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