Scale-free networks on lattices

We suggest a method for embedding scale-free networks, with degree distribution P(k) similar to 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(min) = (lambda - 2)/(lambda - 1 - 1/d), contrary to all other known examples of fractals and disordered lattices.