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 the exponent gamma. At each time step, each vertex generates its offspring, 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. Second, 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 gamma = 1+ln(2m-1)/ln m. Thus, by tuning m, the degree exponent can be adjusted in the range, 2 < gamma < 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 approximately ln N/ln K macro, where N is system size, and k macro is the mean degree. Finally, we consider the case that the number of offspring is the same for all vertices, and find that the degree distribution exhibits an exponential-decay behavior.