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(r)∼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→∞. For finite size lattices we find an optimal α(L) that depends strongly on L. The convergence to α=2 as L→∞ shows interesting power-law dependence on the anisotropy strength.