Scale-free Networks on Lattices

TL;DR

This paper presents a method to embed scale-free networks with degree distribution P(k) ~ k^-lambda into Euclidean lattices, revealing properties of their structure and path dimensions.

Contribution

It introduces a natural length-minimization embedding technique for scale-free networks on lattices, applicable for lambda>2, and analyzes their fractal and path dimensions.

Findings

Networks with lambda>2 can be embedded up to a controllable Euclidean distance.

Clusters of chemical shells are compact with fractal dimension d_f=d.

Shortest path dimension d_min is less than one, depending on lambda and lattice dimension.

Abstract

We suggest a method for embedding scale-free networks, with degree distribution P(k) k^-lambda, in regular Euclidean lattices. The embedding is driven by a natural constraint of minimization of the total length of the links in the system. We find that all networks with lambda>2 can be successfully embedded up to an (Euclidean) distance xi which can be made as large as desired upon the changing of an external parameter. Clusters of successive chemical shells are found to be compact (the fractal dimension is d_f=d), while the dimension of the shortest path between any two sites is smaller than one: d_min=(lambda-2)/(lambda-1-1/d), contrary to all other known examples of fractals and disordered lattices.