Properties of Model Networks Generated by Modified Small-World Scale-Free Fractal Tree Algorithm

We modified the small-world scale-free fractal tree algorithm by introducing the length distribution of branches that grow from the tree structure. The new algorithm provides a convenient tool for studying the self-organisation of small-world fractal networks, taking advantage of the simplicity of the model, where only the parameter \(\gamma \) , the power-law exponent describing the degree distribution, can produce various small-world graphs with different values of the fractal dimension. The numerical result suggests the \(\gamma \) -dependency of the fractal box dimension, \(d_\textrm{b} \simeq 1 + (\gamma -2)^{-0.5}\) , and demonstrates that, when \(d_\textrm{b}\) becomes less than 2 in response to \(\gamma \) being greater than 3, it becomes difficult to maintain fractal properties while maintaining small-world properties. These results are in contrast to those for non-small-world fractal networks, where fractality can be maintained regardless of the values of \(\gamma \) and \(d_\textrm{b}\) . Understanding the relationship between \(\gamma \) and \(d_\textrm{b}\) in various types of fractal networks is important for interpreting empirical data from real networks.