  We propose a general model of unweighted and undirected networks having the
scale-free property and fractal nature. Unlike the existing models of fractal
scale-free networks (FSFNs), the present model can systematically and widely
change the network structure. In this model, an FSFN is iteratively formed by
replacing each edge in the previous generation network with a small graph
called a generator. The choice of generators enables us to control the
scale-free property, fractality, and other structural properties of
hierarchical FSFNs. We calculate theoretically various characteristic
quantities of networks, such as the exponent of the power-law degree
distribution, fractal dimension, average clustering coefficient, global
clustering coefficient, and joint probability describing the nearest-neighbor
degree correlation. As an example of analyses of phenomena occurring on FSFNs,
we also present the critical point and critical exponents of the
bond-percolation transition on infinite FSFNs, which is related to the
robustness of networks against edge removal. By comparing the percolation
critical points of FSFNs whose structural properties are the same as each other
except for the clustering nature, we clarify the effect of the clustering on
the robustness of FSFNs. As demonstrated by this example, the present model
makes it possible to elucidate how a specific structural property influences a
phenomenon occurring on FSFNs by varying systematically the structures of
FSFNs. Finally, we extend our model for deterministic FSFNs to a model of
non-deterministic ones by introducing asymmetric generators and reexamine all
characteristic quantities and the percolation problem for such
non-deterministic FSFNs.
