Kleinberg navigation on anisotropic lattices

TL;DR

This paper investigates how anisotropy in a lattice affects optimal navigation strategies in small-world networks, confirming that the best long-range link distribution exponent is 2 in the infinite limit, regardless of anisotropy.

Contribution

It extends Kleinberg's navigation model to anisotropic lattices, revealing the optimal exponent remains 2 in the infinite size limit and analyzing finite-size effects.

Findings

Optimal exponent α=2 for infinite lattices regardless of anisotropy.

Finite size lattices have a size-dependent optimal α(L).

Convergence to α=2 follows a power-law with anisotropy strength.

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.