Generating hierarchial 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 ΛΛ. 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 ΛΛ we generate random graph sequence sharing similar properties.

► We generate deterministic scale-free networks using graph-directed self similar IFS. ► Our model exhibits similar clustering, power law decay properties to real networks. ► The average length of shortest path and the diameter of the graph are determined. ► Using this model, we generate random graphs with prescribed power law exponent.