A Geometric Fractal Growth Model for Scale Free Networks

Abstract

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 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 $\gamma =1+\ln (2m-1)/\ln m$. 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, $d\sim \ln N/\ln \bar{k}$, where N is system size, and $\bar{k}$ 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.

References in corpus (5)

• Statistical mechanics of complex networks
• Dynamical and correlation properties of the Internet
• Deterministic Scale-Free Networks
• Degree Distributions of Growing Networks
• Fluctuation-driven dynamics of the Internet topology

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