  The curves of scaling behavior is a significant concept in fractal dimension
analysis of complex systems. However, the underlying rationale of this kind of
curves for fractal cities is not yet clear. This paper is devoted to
researching a set of basic problems of the scaling behavior curves in urban
studies by using mathematical reasoning and empirical analysis. The main
findings are as follows. First, the formula of scaling behavior curves is
derived from a fractal model based on hierarchical structure of urban systems.
Second, the relationships between the formula of scaling behavior curves and
similarity dimension and scale-dependent fractal dimension are revealed. Third,
according to the fractal dimension measurement methods, the scaling behavior
curves are divided into two different types. Fourth, empirically, 1-dimensional
spatial autocorrelation function of scaling behavior curves can be employed to
reveal the basic property of scaling behavior curves. In comparison, scaling
behavior curves are more sensitive to power law relations than the fractal
distribution functions. The scaling curves can be utilized to evaluate fractal
development extent of urban systems and identify scaling ranges of fractal
cities. In positive studies, the curves can be used to help distinguish the
boundaries between urban areas and rural areas and self-affine fractal
structure behind the dynamical process of spatial correlation.
