Scale-Free Networks on Lattices

We suggest a method for embedding scale-free networks, with degree distribution [Formula presented], 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 [Formula presented] can be successfully embedded up to a (Euclidean) distance [Formula presented] 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 [Formula presented]), while the dimension of the shortest path between any two sites is smaller than 1: [Formula presented], contrary to all other known examples of fractals and disordered lattices.