Scale-Free Networks on Lattices

ROZENFELD, A.F.; COHEN, R.; BEN-AVRAHAM, D.; HAVLIN, S.

PHYSICAL REVIEW LETTERS

AMER PHYSICAL SOC

Año: 2002 vol. 89 p. 1 - 4

0031-9007

<!-- @page { size: 8.5in 11in; margin: 0.79in } P { margin-bottom: 0.08in } --> We suggest a method for embedding scale-free networks, with degree distribution  P(k)~k-l, 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.