A Geometric Fractal Growth Model for Scale Free Networks

We introduce a deterministic model for scale-free networks, whose degree distribution follows a power-law with an exponent γ. At each time step, each vertex generates its offsprings, whose number is proportional to the degree of that vertex with proportionality constant m − 1 (m > 1). We consider the two cases: first, each offspring is connected to its parent vertex only, forming a tree structure, and secondly, it is connected to both its parent and grandparent vertices, forming a loop structure. We find that both models exhibit power-law behaviors in their degree distributions with the exponent γ = 1 + ln(2m − 1)/ lnm. Thus, by tuning m, the degree exponent can be adjusted in the range, 2 < γ < 3. We also solve analytically a mean shortest-path distance d between two vertices for the tree structure, showing the small-world behavior, that is, d ∼ lnN/ ln k̄, where N is system size, and k̄ is the mean degree. Finally, we consider the case that the number of offsprings is the same for all vertices, and find that the degree distribution exhibits an exponential-decay behavior. PACS numbers: 89.70.+c, 89.75.-k., 05.10.-a Typeset using REVTEX 1