An Illusion of Spatial Hierarchy: Spatial Hierarchy in a Random Configuration

In this paper we show the properties of a spatial hierarchy found in a random configuration in which points are randomly placed on a plane and ranks are randomly assigned to these points. First we show a method of detecting a spatial hierarchy in a configuration of ranked points on a plane (not necessarily the random configuration). Second, using this method we obtain the spatial hierarchy of the random configuration mentioned above, and examine its properties theoretically as well as numerically with the Monte Carlo simulation. From this examination, we find that the spatial hierarchy of the random configuration shares more or less similar properties with Christaller's spatial hierarchy. Stated explicitly, the shape of dominant regions is hexagonal on average; the areas of the same rank centers are fairly homogeneous; the K value defined by Christaller is almost constant. The constant K of the random configuration is close to seven, implying that the spatial hierarchy of the random configuration is close to Christaller's hierarchy of the administrative principle. These results explain to a certain extent why spatial hierarchies are often observed in the real world. At the same time, these results give us a warning. Even if we find a spatial hierarchy in the real world, we should question whether or not the observed hierarchy is a seeming hierarchy like the spatial hierarchy in the random configuration.