Scale-Free Networks on Lattices

We suggest a method for embedding scale-free networks, with degree distribution P(k)~k^-λ, 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 λ>2 can be successfully embedded up to a (Euclidean) distance ξ 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 df=d), while the dimension of the shortest path between any two sites is smaller than 1: dₘᵢₙ=(λ-2)/(λ-1-1/d), contrary to all other known examples of fractals and disordered lattices.