Kleinberg navigation on anisotropic lattices

Details Kleinberg navigation on anisotropic lattices Kleinberg navigation on anisotropic lattices

2008-05-06

arxiv.org/abs/0805.0807v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

6 pages, 4 figures, data included with source

Scientific paper

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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