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 $m-1$ $\mathrm{}(m>\mathrm{}1\mathrm{}).$ 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 $\gamma =1+\ln \left(2m-1\right)/\ln m.\mathrm{}$ Thus, by tuning m, the degree exponent can be adjusted in the range, $2<\mathrm{}\gamma <3.\mathrm{}$ We also solve analytically a mean shortest-path distance d between two vertices for the tree structure, showing the small-world behavior, that is, $d\sim \ln N/\ln \overline{k},$ where N is system size, and $\overline{k}$ 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.