  A fractal is in essence a hierarchy with cascade structure, which can be
described with a set of exponential functions. From these exponential
functions, a set of power laws indicative of scaling can be derived. Hierarchy
structure and spatial network proved to be associated with one another. This
paper is devoted to exploring the theory of fractal analysis of complex systems
by means of hierarchical scaling. Two research methods are utilized to make
this study, including logic analysis method and empirical analysis method. The
main results are as follows. First, a fractal system such as Cantor set is
described from the hierarchical angle of view; based on hierarchical structure,
three approaches are proposed to estimate fractal dimension. Second, the
hierarchical scaling can be generalized to describe multifractals, fractal
complementary sets, and self-similar curve such as logarithmic spiral. Third,
complex systems such as urban systems are demonstrated to be a self-similar
hierarchy. The human settlements in Germany and the population of different
languages in the world are taken as two examples to make empirical analyses.
This study may be revealing for associating fractal analysis with other types
of scaling analysis of complex systems, and spatial optimization theory may be
developed in future by combining the theories of fractals, allometry, and
hierarchy.

Physics and Society