On the Decomposability of the Zipf's Rank-size Rule

Abstract

Sometimes, we can find the fact that when the Zipf's rank-size rule can be successfully applied to the population of cities within a country, the rule can be simultaneously applied to the population of cities within a region of the country. The purpose of this paper is to clarify the reason why we can fid such a fact.The Zipf's rank-size rule which is applied to the population of cities within a country is expressed by logP=-alogR+b(1) where P is population of cities within a country. R is the rank of the population P, and a and b are parameters.If we can find relationship which is expressed by R=KRκ+e (0_??_e_??_K-1)(2) between the rank of the population of cities within the k th region (k=1, 2, ····, K) in a country which is given by counting the rank among “the population of cities within the k th region”, Rk and that which is given by counting the rank among “the population of cities within the whole country”, R, when the whole country observed is divided into K regions, we can obtain the relationship between the population of cities within the k th region Pk and the rank of the population Rk, which is exactly or approximately expressed by logPκ=-aκlogPκ+bκ (κ=1, 2, ····, K) where e, ak and bk are parameters (If we substitute equation (2) into equation (1) and write P and b-alogK(3) by Pk and bk respectively, we can obtain equation (3), when e is regarded as sufficiently or negligibly small.). Therefore, it can be said that when the Zipf's rank-size rule can be applied to the population within a country and the condition which is written by equation (2) is satisfied, the rule can be also applied simultaneously to the population of cities within a region of the country. This characteristic which the Zipf's rank-size rule has is called here “decomposability” of the rank-size rule.In this paper, the Zipf's rank-size rule was applied to the actual population of cities within each of 9 regions in Japan. The rule was successfully applied to the population. This result suggests that the relationship between the rank of the population of cities within each of the 9 regions which is given by counting the rank among “the population of cities whthin each of the 9 regions” and that which is given by counting” the rank among the population of cities within the whole Japan” satisfied the condition written by equation (2).