一个将无标度特性与Zipf定律联系起来的合理证明
A sensible proof connecting the scale-free feature with the Zipf-law
AI总结：
本文通过严格数学推导，证明了任意阶数为$N$的无标度网络模型中基于顶点排名的无标度特性与Zipf定律之间的等价关系，消除了Zipf定律缺乏理论基础的空白，为采用复杂网络方法研究Zipf定律提供了理论依据。
摘要：
自然界和社会中大多数各种大型复杂系统可以很好地描述为复杂网络（图），以便更好地理解它们背后的演化机制和动力学功能。其中一部分遵循无标度行为，即度大于或等于$k$的顶点数与整个网络阶数的比值服从表达式$P_{cum}(k)\sim k^{1-γ}$（$2<γ<3$）。同时，满足$f_{r}\sim r^{-α}$（$α$接近于1）的Zipf定律在许多复杂系统中也很普遍，例如文本中的词频和城市规模。很容易注意到，上述两者具有相同类型的表象，即已知的幂律。与通过连续统理论解析证明的无标度特性相比，后者在大多数情况下在许多科学界，特别是在社会科学中，仍然被认为是一种经验原则。因此，有必要指出两者之间的内在联系或区分彼此的差异。在此，对于任意给定阶数为$N$的无标度网络模型，我们报告了基于顶点排名的无标度特性与Zipf定律之间的等价关系。通过严格的数学推导，我们消除了Zipf定律缺乏理论基础的空白。因此，人们可以确信，采用已经用于研究复杂网络的方法来研究Zipf定律是合理的
英文摘要：
Most of various large-size complex systems in nature and society can be well described as complex networks (graphs) to better understand the evolutional mechanisms and dynamical functions behind themselves. Of some part follow scale-free behavior, that is, the ratio of the number of vertices with degree more than or equal to $k$ and order of the whole network obeys the expression $P_{cum}(k)\sim k^{1-γ}$ ($2<γ<3$). Meanwhile, the Zipf-law, which satisfies this $f_{r}\sim r^{-α}$ ($α$ close to unity), is also prevalent in many complex systems, such as word frequencies in text and city sizes. It can be easily noticed that the both above have same type of appearance, namely the known power-law. Compared to the scale-free feature proofed analytically by continuum theory, by far the latter in most cases still is thought of as an empirical principle in lots of science communities, particularly in social science. For this reason there is a need for either pointing out the inner connection between the two or distinguishing difference of one another. Here, for any arbitrary given scale-free network model of order $N$, we report an equivalent relation between scale-free feature and the Zipf-law based on the vertex rank. By rigorous mathematical derivations, we eliminate the gap, lack of theoretical fundament of the Zipf-law. Therefore one can be convinced that it is reasonable to adopt methods already used to study complex networks to do the Zipf-law
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