Multifractals of central place systems: Models, dimension spectrums, and empirical analysis

Central place systems have been demonstrated to possess self-similarity in both the theoretical and empirical perspectives. A central place model can be treated as a monofractal with a single scaling process. However, a real system of human settlements is a complex network with multi-scaling processes. The simple fractal central place models are not enough to interpret the spatial patterns and evolutive processes of urban systems. It is necessary to construct multi-scaling fractal models of urban places. Based on the postulates of intermittent space filling and unequal probability of urban growth, two typical multifractal models of central places are proposed in this paper. One model is put forward to reflect the process of spatial concentration (convergence), and the generalized correlation dimension varies from 0.7306 to 1.3181; the other model is presented to describe the process of spatial deconcentration (divergence), the generalized correlation dimension ranges from 1.6523 to 1.7118. An empirical analysis was made by the cities and towns of Central Plains, China, and an analogy is drawn between the real system of urban places and the theoretical models. A finding is that urban systems take on multifractal form, and can be modeled with multi-scaling fractals. This is a preliminary attempt to develop the theory of fractal central places, and the results are helpful for understanding the similarities and differences between the dynamical process of spatial concentration and that of spatial deconcentration.