Studies on the fractal structure of point-axis systems with spatial complexity

Abstract

Studies are preliminarily made on spatial structure of point-axis systems in the paper, which reveal that a point-axis system as a unique giant component (UGC) is in fact a fractal system. The theoretical starting point of the point-axis model in the aspect of spatial patterns is the triangular lattice that is the same as that of central place theory, but the configuration of developed point-axis systems is of irregularity based on random process to a certain extent. On the other hand, the point-axis model is as advanced as a new optimization design which signifies that an ideal point-axis system must have some kinds of optimum structure, especially the spatial structure with some kinds of ‘order’, the order may be what is called self-similarity that is always emerging ‘at the edge of chaos’ which is mathematically related to the concept of ‘self-organized criticality’. In reality a point-axis system usually appears concretely as an urban system depending on one or two ‘axes’such as seaboard, great rivers, railways and so on, and systems of cities and towns proved to be fractal systems. This implies that point-axis systems may have fractal structure. From the viewpoint of a general dynamic system that can be used to describe point-axis systems, an allometric growth equation is deduced out as x i ∝ x jα , from which, the relationship between measures and yardsticks of point-axis systems, M ( r )∝ r D , can be derived by means of the theorem of ergodicity. According to the measure-scale relationship, two kinds of fractal dimensions can be given to characterize point-axis systems. One is the point-distribution dimension that is defined by the formula N ( r )∝ r D , where r is a yardstick, N ( r ) is corresponding number of points in a point-axis system and D is fractal dimension. The other is the line-distribution dimension that can be defined by the formula L ( r )∝ r D , where L ( r ) is length of communication lines joining points together corresponding the yardstick r . In addition, the line-distribution dimension can also be defined by number of branches of the network of communication lines linking one point with another. The four growth stages of point-axis systems are reinterpreted using ideas from fractals, and theoretical results are applied to the system of cities and towns in North Henan including three urban subsystems. It is demonstrated and illustrated that the urban system as a UGC has fractal dimensionality, which only goes to prove that the hypothesis of self-similar point-axis system is correct empirically. A least squares computation of the quantities gives the values of the fractal dimension D =2.387 with a determination coefficient of 0.996 for Zhengzhou subsystem, D =1.747 with a determination coefficient of 0.997 for Kaifeng subsystem, and D =1.659 with a determination coefficient of 0.997 for Luoyang subsystem. As for the system on the whole, D =1.715 with a goodness of fit of 0.999. The results bring to light a great deal of temporal-spatial information on the development and evolution patterns of the point-axis system in Central Plains.