4.1 Theoretical framework
　　　　Concerning the distribution of city population, the “rank-size rule” was first mentioned by Auberbach in 1913 and analyzed in depth by Zipf(1949). This rank-size rule is expressed as;

　　　　This formula signifies that the second city’s population is the half of the top city’s population, the third city’s population is the 1/3 and likewise afterwards, as in the acoustic harmonic distribution. Graphically speaking, when the city population is plotted logarithmically to its rank, a linear distribution with a negative one (-1) slope is obtained. It has been shown that this rank-size rule applies to various cases of modern cities.
　　　　However, in the historical urban context, it is known that the rank-size distribution is close to a linear approximation but the slope is variable, either sharper or milder. For example, Berry(1961) showed that the slope had changed according to the stages of colonization in Latin America. A sharp slope was observed in the beginning of colonization due to the comparatively large size of the town which houses the colonizer’s office and later the slope became milder as the second, third or more cities grew in size. Skinner (1977) found that in 18th century China, regional city distribution was normal (slope was close to negative one) but China as a whole had a gentler slope. The gentler slope of rank-size distribution in the pre-modern times is also mentioned by Russell in his various studies on medieval cities²) or de Vries(1984) on the European cities from 1500 to 1800.
　　　　Thus the above mentioned formula (1) can be rewritten to generally express various historical settings as;

　　　　According to this formula, the n^th city population is expressed with the top city population(P₁) and the slope of the distribution (-a). For example, using this formula, the slope of the city population distribution in the world in 2000 is -0.5791 and the linearity is as strong as R=-0.9802, it can be said that the generalized rank-size rule applies (Figure 5).

Figure 5 City population rank-size distribution (World : 2000)

Note : Urban agglomeration in UN(2004b) is used as city population

　　　　Rank-size rule has been used to analyze the city population, i.e. the agglomerations of larger size. Here we will see if this rule also applies to smaller sized cities or even to villages.
　　　　In Japan, the lowest administrative division is called Shi-Cho-Son (City-Town-Village) and the whole territory is divided either by a Shi, a Cho or a Son. Although recently, the merger of these administrative units is ongoing and the size of one unit is getting larger, originally this distinction was introduced to designate the city as Shi, the town as Cho and the village as Son. Thus observing the population of Shi-Cho-Son, one can get the rank-size distribution of all agglomerations regardless of the appellation. The rank-size distribution of total 3,223 Shi-Cho-Son population is linear (R=-0.9945), until around the rank 2,500th (Figure 6).

Figure 6 Shi-Cho-Son population rank-size distribution (Japan : 2000)

Source : Population Census of Japan(2000)

　　　　This shows that the Shi-Cho-Son population distribution is continuous throughout the different unit of agglomeration such as the city, town or village, until the community size becomes very small to a population of 5,000.
　　　　The same kind of agglomeration population data can be obtained from the French data. The rank-size distribution of the French commune population also shows strong continuity and linearity (R=-0.9927) until about the rank 30,000th where the population of the commune is around 150 persons.
　　　　For both Japanese and French data, the very small-sized community population, ranked more than 2,500th in Japan and 30,000th in France shows no continuous trend. However, the share of the population of these small communities against total population is 1.8% in Japan and 1.0% in France and it can be considered negligible.
　　　　From these examples of Japan and France, it can be assumed that the generalized rank-size rule is applied not only to large cities but also to the whole communities. In this case, using the formula (2), the total population P_T can be expressed as follows;

　　　　From this formula (3), the total population(P_T) is expressed using the top city population(P₁), the slope of rank-size distribution(-a) and the number of communities(N). When -a and N is stable, the change of P_T is reflected to P₁ change, which is followed by the lower rank(2^nd, 3^rd...) cities’ population change. As a result, with the total population change, the rank-size distribution shifts horizontally, which can be illustrated using the data of China (Figure 7). In this case, the change of larger cities’ population is parallel to the change of total population and the city population ratio is stable.

Figure 7 Vertical shift of rank-size distribution according to the total population PT (in 1,000) (China)

　　　　However, even before the border year, there was a minor change of city population ratio and this can be attributed to the other parameters; -a and N. From the formula (3), it can be said that the steeper the slope –a and smaller the number of the community N, the smaller the total population (P_T) gets and thus larger the city population ratio gets. For example, in China up to 1300, there is a strong correlation (R=-0.9392, Figure 8) between the slope –a and city population ratio. This correlation does not apply for the period from 1400 to 1850 but from a partial evidence of increased number of towns and cities (Liu 1987), the city population ratio might be influenced by the number of communities N in this period.

Figure 8 Correlation of Slope -a and city population ratio (Up to 1300)
Note : For the excluded year of 900 and 1000, please refer to the 4th paragraph of section ‎2.2.

　　　　Among the 3 factors determining the total population, the P₁ might be the most important one but the other two, -a and N are closely related to the social structure and would be useful for the more detailed estimate. However the manner and degree in which these variables affect the city population ratio remains to be investigated in the future.

2. Russell had modified the standard rank-size distribution formula to “reduce the numerator”. The formula proposed is ;

Using this formula, the hypothetical slope of rank-size distribution for the top 10 cities is calculated to be -0.8903.