Modeling the self-affine structure and optimization conditions of city systems using the idea from fractals

This paper demonstrates self-affine fractal structure of city systems by means of theoretical and empirical analyses. A Cobb–Douglas-type function (C–D function) of city systems is derived from a general urban response equation, and the partial scaling exponent of the C–D function proved to be the fractal dimension reflecting the self-affine features of city systems. As a case, the self-affine fractal model is applied to the city of Zhengzhou, China, and the result is satisfying. A fractal parameter equation indicative of structural optimization conditions is then obtained from the C–D function. The equation suggests that priority should be given to the development of the urban element with a lower fractal dimension, or a higher partial scaling exponent, for utility maximization. Moreover, the fractal dimensions of different urban elements tend to become equivalent to each other in the long term. Accordingly, it is self-similar fractals rather than self-affine fractals that represent the optimal structure of city systems under ideal conditions.