動態複雜系統的隨機成長過程中之不隨機

Order from Random Growth Process in the Evolving Complex Systems

冪次法則（Power law）是自我組織複雜系統的共同特性，而區域科學中的普瑞夫法則（Zipf's law）則是冪次法則的特殊型式。本文以模擬方式顯示，符合吉伯特定理（Gibrat's law）的同質隨機成長假設可導致冪次法則的分配。然而，吉伯特定理不必然衍生出普瑞夫法則的極限分配。在成長率的標準差呈漸緩遞減時，都市成長才可能趨近普瑞夫法則的極限分配。都市成長率標準差的遞減率決定都市大小分配收斂的速度和斜率。都市成長率的標準差與區域內都市間的潛在連繫和都市間交互作用的敏感度有關。

Power taw has been shown to be a common feature of many self-organized complex systems, and Zipf's law in regional science is the most famous of all these distributions. This paper shows that the assumption of homogeneity of the random growth process as assumed in Gibrat's law will generate city size distribution as power law. However, Gibrat's law does not necessarily generate Zipf's limiting pattern. City distribution could possibily converge to a Zipf's pattern limiting distribution only with a diminishing decreasing standard deviation of the random growth rate. Moreover, the value of the diminishing rate of the standard deviation of city growth rate determines the speed of the convergence and the value of the converged slope. The homogeneous random evolving process is the essential underlying feature, which generates the common power law property of many complex systems. Nevertheless, the variation of the changing rate of increased potential connections and the sensitivity of interactions among cities are the major reasons for the differences of the slopes among self-organized systems.