As the city system is dynamically changing through migration and diffusion of innovations, we should consider the city system not only statically but also dynamically. This paper attempts to analyze the changing city system of Niigata Prefecture in Japan using a Markov chain model. According to an absorbing Markov chain model, the population vector after n periods is
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where
W(0) is an initial vector,
x is a vector of births,
Q is a submatrix of transitional probability matrix, and
I is an identity matrix.
By this model, the projections of population in 1985 and in 1995 are made for 57 urban places with more than 10, 000 residents. Using data of the projected population and the population statistics in 1955, 1965 and 1975, the structural changes in city size distribution between 1955 and 1995 are analyzed in terms of the rank-size distribution and the measures of areal differentiation. The measurement of areal differentiation in the city system is
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The results of these analyses are to be summarized as follows.
(1) When fitting the rank- size rule to the city system of Niigata Prefecture at the successive decades from 1955 to 1975, the system is found to move towards unification rather than diversification. In addition, the force of unification is greater in the period between 1955 and 1965 than in the period of succeeding decade. While the structural changes in rank-size distribution between 1955 and 1965 are mainly due to the growth of upper-ranked cities, their changes between 1965 and 1975 consist of both the growth of upper-ranked cities and the decline of lower-ranked cities.
(2) The similar trends are supported by the measurement of areal differentiation using Shannon's entropic measure of uncertainty. That is to say, there are tendencies that entropy in the city system is becoming smaller year by year and the population energy is concentrated in the upper-ranked cities. These processes show that the areal differentiation of this system becomes greater.
(3) The projection of population is made by using a Markov chain model. As the result, it is predicted that Niigata City and three urban places near by will experience the increase in population and the number of population of these four urban places, whose aggregated population is about 490, 000 in 1975, will increase up to about 620, 000 in 1995.
(4) By fitting the rank-size distribution to the projected population data and examining their parameters, it is found that the force of unification will also be stronger than one of diversification in the next two decades, but the rates of its increase in 1985 and 1995 will be one-half and one-third of that in 1975, respectively.
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