Statistical Methods for Evaluating the Geometrical Hierarchy of a Network

Abstract

The objective o f this paper is to present statistical methods for evaluating the geometrical hierarchy o f a network in comparison with a random hierarchy network. First, the random hierarchy network is explicitly formulated b y 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 o f 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 o f these indices is laborious, an efficient computational method is developed, and its application to an actual example is described.

Hierarchy Depth in Directed Networks

“…The concept of hierarchy has been used for systems described with networks for a significant time and for different purposes. It has been considered in the context of elections [ 1 ], investigation of road network geometry [ 2 ], and neural networks [ 3 ]. More recently, the concept of hierarchy has been used for social systems [ 4 , 5 , 6 , 7 , 8 ], artificial neural networks [ 9 ], financial markets [ 10 ], communication networks [ 11 ], and the structure of cities [ 12 ].…”

Hierarchy depth in directed networks

“…The concept of hierarchy has been used for systems described with networks for a significant time and for different purposes. It has been considered in context of elections [1], investigation of road network geometry [2] and neural networks [3]. More recently, the concept of hierarchy has been used for social systems [4,5,6,7,8], artificial neural networks [9], financial markets [10] and communication networks [11].…”