GEOGRAPHICAL RESEARCH ›› 1998, Vol. 17 ›› Issue (1): 82-89.

### FRACTAL DIMENSIONS OF HIERARCHICAL STRUCTURE OF URBAN SYSTEMS AND THE METHODS OF THEIR DETERMINATION

Liu Jisheng1, Chen Yanguang2

1. 1. Department of Geography, Northeast Normal University, Changchun 130024;
2. Department of Geography, Xinyang Normal College, Xinyang 464000
• Received:1997-05-05 Revised:1997-12-22 Online:1998-03-15 Published:1998-03-15

Abstract: Three parameters were presented in the paper to characterize the hierarchical structure, especially size distribution of cities, of an urban system, including Zipf's dimension, fractal structure factor (FSF), and difference degree.1.Zipf's dimension.Zipf's law is very familiar to urban geographers, it is equivalent to Pareto's distribution and can be expressed mathematically as P(k)=P1K-q, where q is sometimes called Zipf's dimension, which is actually the reciprocal of fractal dimension, namely, q=1/D.2.FSF.The parameter is put forwards by two Chinese geo scientists and has been introduced to the studies of urban geography by the authors of the paper.FSF can be defined as I(r)=I0+DI lnr, where I(r) is information capacity of size distribution of cities corresponding to a certain scale (r, when r=1, I(r)=I0), and DI is what is called FSF.3.Difference degree.The parameter is given by the authors of the paper, it can be defined as follows: C=1-I/Imax, where I=- PilnPi, Imax=lnN, and PiP(k)/ P(k),P(k) is the population of the kth city of an urban system (k=1, 2, …, N).Difference degree C can be linked with fractal dimension D by means of Zipf's formula under some conditions.The geographical meanings of the three parameters were illuminated, the methods of using them were illustrated, in particular, the degenerational forms of Zipf's model were discussed so as to connect fractal studies with non-fractal studies of urban systems, and finally, a preliminary comparison was made between them.

PACS:

• K928.5