城市规模分布的三参数Zipf模型 ——Davis二倍数规律的理论推广及其分形性质的实证分析

将关于城市等级-规模分布的Davis二倍数（2n）规律推广为任意倍数（δn）规律：ai=ai n·2n,fi=fi n·δ-n，然后从中导出具有一般意义的三参数Zipf模型：P（r）=C（r－a）-dz．揭示了参数dz的分维性质并给出了它与分维D以及邻级倍数δ的数值关系：dz=1／D-In2／Inδ。从而证明Davis的2n规律乃是δ=2即dz=1的特殊情形；最后用Davis的原始数据对推导结果进行了实证分析．

Three-parameter Zipf's model on city-size distribution:A generalization of Davis'2n-law and demonstrations of its fractal nature

The Davis' number-doubling rule of city hierarchies,which was originally given as,αi＝αi+n.2n，fi＝fi+n.2 -n,is generalized to an universal form,αi＝αi+n.2n，fi ＝fi+n.δ-n(δ1)， from the new expression a three-parameter Z ipf's model is deduced out as follows,P(r)=C(r-α)-dz,where C= a0(f0/lnδ)lnδ／ln2， α＝f0/ln(1/δ),and d z=ln2/lnδ. The last parameter,dz,proved to have a certain nature of fractal dimension(D),i.e.,dz=1/D,so the fractal dimens ion can be expressed as D=lnδ/ln2. This implies that the so-called 2n-law is a special form of 3-parameter Zipf law,in which the Zipf's dimension (dz) as well as the fractal dimension (D) is equal to one,nam ely,dz=(1/D)=1. The results developed by the authors are demonstrated and v erified by the original data presented by K.Davis(1978).