Relationship between Centrality-Index and Distance, and the Construction of Equilibrium Circle

Abstract

The purpose of this paper is to verify the mutual relationship of retail trade between two central-places and to make an equilibrium circle between them.1. How to make an Equilibrium CircleConsider there are two central-places A and B, and centrality-index of A is larger than that of B. The loci of equilibrium points should make a circle of which the center is nearer to B than to A. Now how to make such a circle? One method is as follows. Assume: (I) centrality-indices of two central-places A and B are 9 and 1 respectively, (II) the distance between A and B is 8km, (III) the trade area of each central-place expands circularly.The trade area of A should be nine times larger than that of B; that is, πr2:πr'2=9:1. This results in r:r'=3:1. From this radius ratio, we are able to determine the equilibrium circle. The point P on the equilibrium circle must be at the distance of 6km from A and 2km from B on the line AB. The other point Q must be at the distance of 12km from A and 4km from B on the line AB. Furthermore, such equilibrium points must be on the circle with the diameter PQ (Fig. 2). The circle is the equilibrium line, and the inner sphere of the cirle is the trade area of B.Such an equilibrium circle can also be made by means of analytical geometry: See Godlund, S., “The function and growth of bus traffic within the sphere of urban influence, ” Lund Studies in Geography, Nr. 18, 1956. And this circle results in just same as the former.2. Relationship between Centrality-Index and DistanceAt the equilibrium point, we can suppose that the strength of two central-places are directly proportional to the centrality-indice and inversely proportional to the power n of the distances from the central-places.The power n of the distance is the function of centrality-index. If the centrality-index increases, the power n of the distance decreases, or the converse holds good. Now, the centrality-index changes in accordance with the indicator of the centrality. Therefore, the power n of the distance changes according to the indicator of the centrality. For example, if the population is chosen for its indicator, as in Reilly's Law, the power n will be square. But if the other indicator is chosen, the power n may not be square. Moreover, even if we take the same indicator, the power n will change according to regions or to times, owing to the difference of the traffic facilities or of the standards of living.3. Determination of Centrality Indicator and the Construction of the Equilbrium CircleIt is difficult to determine the most suitable centrality indicator theoretically. Then, the indicator should be determined empirically. Five indicators were chosen for this purpose. They were; (1) values of retail sales, (2) values of retail sales-values of retail food sales, (3) basic retail activity, i.e. [(2)-(population of builtup area)×(values of retail sales per capita-values of retail food sales per capita)], (4) population of the built-up area, (5) Coefficient of Localization, i.e. [(percentage of (2) against the values of Prefecture)-(Percentage of (4) against the population of Prefecture)].On the other hand, the equilibriump oints were decided by the field work in Yamaguchi and Hyogo Prefectures. And the relationship between centrality-indices and the power n of the distances at the equilibrium points were tested respectively. Table II shows the power n of the distance in each indicator.In this table we can see the power n in each indicator fluctuates respectively. Perhaps, the fluctuation comes from the following two reasons: (I) the indicator of centrality is not suitable, (II) the inevitable error occured from sampling investigation. Now, in five indicators, the fluctuation of n of (2) is lesser than that of the others; see S (standard deviation) and V (=S/x coefficient of variation) in the Table.