Kleinberg Navigation in Fractal Small World Networks

TL;DR

This paper investigates how navigation efficiency in fractal small world networks depends on the distribution of long-range links, confirming the optimal exponent matches the fractal dimension and identifying finite-size effects.

Contribution

It extends Kleinberg's navigation model to fractal lattices, demonstrating the optimal link distribution exponent equals the fractal dimension and analyzing finite-size corrections.

Findings

Optimal navigation occurs when alpha equals the fractal dimension d_f.

Finite-size corrections to the exponent alpha are proportional to 1/(ln N)^2.

Numerical simulations support theoretical predictions.

Abstract

We study the Kleinberg problem of navigation in Small World networks when the underlying lattice is a fractal consisting of N>>1 nodes. Our extensive numerical simulations confirm the prediction that most efficient navigation is attained when the length r of long-range links is taken from the distribution P(r)~r^{-alpha}, where alpha=d_f, the fractal dimension of the underlying lattice. We find finite-size corrections to the exponent alpha, proportional to 1/(ln N)^2.