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Kleinberg navigation on anisotropic lattices

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 $P({\bf r})\sim r^{-α}$, 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, $L\to\infty$. For finite size lattices we find an optimal $α(L)$ that depends strongly on $L$. The convergence to $α=2$ as $L\to\infty$ shows interesting power-law dependence on the anisotropy strength.