Davis二倍数规律与Zipf三参数模型的等价性证明——关于城市规模分布法则的一个理论探讨

A Proof of Davis' 2\+n law as a Special Equivalent of the Three-parameter Zipf Model

本文从城市规模分布的Ｄａｖｉｓ 二倍数规律（２ｎ 规律：ａｉ＝ ａｉ＋ ｎ·２ｎ，ｆｉ＝ ｆｉ＋ ｎ·２－ ｎ）中推导出具有一般意义的三参数Ｚｉｐｆ模型：Ｐ（ｒ）＝ Ｃ（ｒ－ α）－ ｄｚ，揭示了２ｎ 规律隐含的分形几何性质， 论证了２ｎ 法则为Ｚｉｐｆ维数ｄｚ＝ １ 时的特殊情形， 并将２ｎ 规律推广到具有普遍意义的δｎ 规律， 给出了Ｚｉｐｆ维数及分维与邻级倍数δ的数值关系：ｄｚ＝ １／Ｄ＝ ｌｎ２／ｌｎδ。最后从三个方面对文中的理论成果进行了实证分析。

The Zipf's model with three parameters, P(r)=C(r-α) - d z , is deduced from Davis' 2 n law: a i=a i+n ·2 n, f i=f i+n ·2 -n , by means of a series of mathematical transformation, where d z proves to have some nature of fractal dimension (D) because d z=1/D. The 2 n rule is generalized to δ n rule and δ represents an arbitrary number which is greater than one, namely δ >1. The relationships between δ and the fractal dimensions of city size distributions can be expressed as D=lnδ/ln2 : when δ =2, we have d z =1, so the 2 n rule is only a special case of the three parameter Zipf's model. The result of the demonstration of Davis' law as an equivalent of the generalized Zipf's law is illustrated and verified by some examples including the data in which 2 n rule of urban systems is discovered.