A Study on the Curves of Scaling Behavior of Fractal Cities

Abstract: 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.