本研究探討複雜理論與聚落空間分布型態的關係，試圖解釋何以冪次法則能主導聚落空間分布型態。冪次定律是指在一個自我組織的系統中事物發生頻率與規模之間的關係：物體的規模S和其出現之次數，呈S-a的比例關係。在1949年戚普夫(Zipf)提出等級大小法則，P（r）表示第r級都市之人口數，q表示為常數，則P（r）＝ K ＊ r-q ，而K為最大都市人口數。q為稱為Zipf force，通常均假設等於1。目前對於何以會產生等級大小法則尚無合理的解釋。
　　在本研究中，我們基於複雜理論的概念建構了一組電腦模擬模型。首先，我們的模擬顯示，在均質平面的假設下，依照報酬遞增機制，系統會出現符合等級大小法則的都市體系。
　　為了驗證電腦模型的結果，我們建構了都市系統演化的數學模型。結果顯示，當系統出現符合冪次法則分布的體系後，依照報酬遞增機制，系統將持續遵循報酬遞增法則，而且冪次法則係數的絕對值將隨時間而增加，這代表都市體系將朝向極化方向發展。
　　許多學者相信，由實證資料來看，Zipf force的數值並不隨時間改變。我們的研究顯示，在過去一百年間，美國的都市體系，呈現反極化的發展傾向，所以美國的實證結果與我們的模型並不一致。顯示在真實世界中，正回饋機制似乎不是主導都市發展的唯一因素。
　　在本研究的最後部份，我們把報酬遞增的吸引係數改變成會隨著都市規模而改變，以建立一般化的模型。依照我們的模擬，由報酬遞增機制下的隨機成長模型所產生的都市系統，幾乎都遵循冪次分布型態。但符合冪次分布型態未必符合等級大小法則，等級大小法則是冪次係數為1的特例，我們相信等級大小法則只出現於特定的條件限制下。另外本研究認為，「先固定後遞減」可能是最符合真實世界的都市體系成長歷程的推動機制。

　This research explores the relation between complex theory and the spatial distribution of human settlements, and explains why the power law would govern the distribution of city sizes. The power law is a functional relation between the scales of objects and the frequencies of occurrence in a self-organizing system: the frequency of the occurrence of objects with scale S is proportional to S-a for some exponential constant a. In 1949, Zipf proposed a rank-size rule about city sizes and city ranks: Let P(r) be the population of the r-th largest city, then P(r) is equal to K× r -q, where q is a constant, and K is the population of the largest city. The quantity q is sometimes referred to as the Zipf force, and is often assumed to be 1. There is no valid, explicit explanation yet of how Zipf’s law comes about.
　In this research, we constructed computer simulation models based on the idea of complexity theory. First, Our simulations showed, under the assumption of a uniform, flat region, how urban systems emerged that fit Zipf’s law.
　In order to validate the conjecture, we then constructed a mathematical model of the dynamics of urban system evolution. We showed that, assuming that the urban systems do obey a power law, and that they continue to evolve based on the principle of increasing returns, the ensuing systems would still obey the power law, and the exponential constant in the model increases in absolute values with time. This means that urban systems become more centralized with time.
　Many scholars believe, based on empirical data, that the Zipf force is constant over time. In the past 100 years, the Zipf force decreased stably for the system of cities in United States, showing a tendency of decentralization. Therefore, the empirical case of U.S. is not in accordance with our model, and shows that in the real world, positive-feedback does not seem to be the only factor governing urban developments.
　Finally, we examined more generalized models in which the attraction coefficient of the increasing returns equation varies in relation to scale. According to our simulations, the urban systems obtained from a random growth model subject to increasing returns almost always followed a power law distribution, no matter what the value of the attraction coefficient was. However, such a power law is not necessarily the rank-size rule, i. e., the special case of a power law where a=1. Therefore, although the rank-size rule is widely accepted and applied as a general law governing the distribution of cities, we believe that it can only occur under restrictive conditions for more specialized configurations. We compared the results obtained from different forms of attraction coefficient functions, and concluded that the outcome maps most closely the real-world situations when the form of the function is "stationary then decreasing".

一、中文部份

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