Scale-Free Networks on Lattices

Abstract

We suggest a method for embedding scale-free networks, with degree distribution $P(k)\sim k^{-\lambda }$, in regular Euclidean lattices accounting for geographical properties. 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 a (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 1: $d_{\mathrm{m}\mathrm{i}\mathrm{n}}=(\lambda -2)/(\lambda -1-1/d)$, contrary to all other known examples of fractals and disordered lattices.