Statistical Methods for Evaluating the Geometrical Hierarchy of a Network

The objective of this paper is to present statistical methods for evaluating the geometrical hierarchy of a network in comparison with a random hierarchy network. First, the random hierarchy network is explicitly formulated by four stochastic processes in which line segments or lines, which represent roads, are randomly placed and their placement is independent of their ranks. Second, under these stochastic processes, this paper derives the expected value and variance of i) the number of intersections between roads of different ranks; and ii) the distance from an arbitrary point to the nearest highest-ranked road through two kinds of routes. Using these expected values, the intersection index and the detour index are proposed, and it is shown that these indicators are useful for evaluating the geometrical hierarchy of a network. As calculation of these indices is laborious, an efficient computational method is developed, and its application to an actual example is described.     

Statistical Analysis of the Distribution of Points on a Network

This paper shows four statistical methods that examine the distribution of points on a network (such as the distribution of retail stores along streets). The first statistical method is an extension of the nearest-neighbor distance method (the Clark-Evans statistic) defined on a plane to the method defined on a network. The second statistical method examines the effect of categorical attribute values of links (say, types of streets) on the distribution of activity points on a network. The third statistical method examines the effect of infrastructural elements (such as railway stations) on the distribution of activity points on a network. The fourth statistical method examines the compound effect of multiple kinds of infrastructural elements (say, railway stations and big parks) on the distribution of activity points on a network. These methods are discussed with empirical examples.