  Several fundamental properties of real complex networks, such as the
small-world effect, the scale-free degree distribution, and recently discovered
topological fractal structure, have presented the possibility of a unique
growth mechanism and allow for uncovering universal origins of collective
behaviors. However, highly clustered scale-free network, with power-law degree
distribution, or small-world network models, with exponential degree
distribution, are not self-similarity. We investigate networks growth mechanism
of the branching-deactivated geographical attachment preference that learned
from certain empirical evidence of social behaviors. It yields high clustering
and spectrums of degree distribution ranging from algebraic to exponential,
average shortest path length ranging from linear to logarithmic. We observe
that the present networks fit well with small-world graphs and scale-free
networks in both limit cases (exponential and algebraic degree distribution
respectively), obviously lacking self-similar property under a length-scale
transformation. Interestingly, we find perfect topological fractal structure
emerges by a mixture of both algebraic and exponential degree distributions in
a wide range of parameter values. The results present a reliable connection
among small-world graphs, scale-free networks and topological fractal networks,
and promise a natural way to investigate universal origins of collective
behaviors.
