Scaling behavior in urban development process of Tokyo City and hierarchical dynamical structure

We study a geometric structure of urban development process which pays particular attention to scaling properties in the settlement area and inhabitant population through changes in the scaling exponents. Both the degree to which the space is fulfilled and the rate at which it is filled are obtained for the residential development in Tokyo. For distances larger than the city boundary, there is a sharp cross-over to a suburban region with a quite intriguing variation with a distance from the center of the city. The population densities in this region are found to collapse into a single scaling function with the scaling exponent 0.678 in the early 1990s in which the growth of the population attenuates. We propose a cellular automata model using the simulated annealing method that succeeds in reproducing the qualitative similar structural complexity of the actual city by taking into account the transportation system, especially railroad network. Finally, a possible theoretical consideration is given in analogous with fluid dynamics. Scaling of the population density is obtained assuming that there is a dynamical hierarchical structure in the scaling region where the stationarity is fulfilled. The theoretically obtained exponent 2/3 agrees well with the observed one.