Generating hierarchical scale free-graphs from fractals

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabási, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal ¿. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale-free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. We point out a connection between the power law exponent of the degree distribution and some intrinsic geometric measure theoretical properties of the underlying fractal. Using our (deterministic) fractal ¿ we generate random graph sequence sharing similar properties.