Preferential Migration and Random Mobility in Population Size Distribution of Municipalities

Abstract

Some approaches in statistical physics have been applied to analyzing economic and social events. Recently, Kobayashi, Kuninaka et al. reviewed the statistical properties of several social and biological phenomena as ‘‘complex systems’’ from the viewpoint of stochastic processes. In their review, the population size distribution of municipalities in Japan was treated as one example. They stated that the log-normal distribution is basic for complex systems, which are generally described as a multiplicative random growth given by the equation xðt þ 1Þ xðtÞ 1⁄4 ðtÞxðtÞ. Here, xðtÞ is the population size at time t for example, and ðtÞ is the growth rate, which is a random variable. This property is well known as Gibrat’s law. For the population size distribution, the following feature has been recognized: the major part of the distribution obeys the log-normality, but the tail part corresponding to a large size exhibits the power law. Sasaki, Kuninaka et al. classified Japanese municipalities into three types (i.e., village, town, and city) and found that villages and cities are fitted with the log-normal and power-law distributions, respectively. Their conclusion was that the difference in these distributions is due to the existence of a lower population threshold in cities, but not in villages. The power-law distribution is reproducible by an additional effect to the above multiplicative random growth. Particularly, the following models have been applied to the Japanese case: (i) Random growth model: This model treats multiplicative random growth with an additive random noise, namely, the Kesten process, which possesses a powerlaw property. (ii) Migration model: There are two approaches focusing on population migration. (ii-a) Tomita and Hayashi adopted an urn model where preferential attachment in complex networks was applied. (ii-b) Sasaki and coworkers proposed another model and took population thresholds into account. Depending on the existence of such thresholds, the size distributions of villages, towns, and cities have been separately reproduced. It is noted that the emergence of the power law due to a threshold has already been reported in fragmentation process. In this short note, we review model (ii-b) with modification by focusing on the preferential migration effect, which is a different viewpoint from those in Refs. 9 and 15. Now, N sites representing each municipality are prepared. The time evolution of xjðtÞ, the size of the j-th site, is described by xjðt þ 1Þ xjðtÞ 1⁄4 XN