冪法則及其在網路科學的意義

Abstract

新興的網路科學領域，大致可分為社會網路、網際網路及全球資訊網、生物網路、輸送網路等四大範疇。在這些典型網路結構裡，「不受度量規模影響 (scale free)」的性質特別值得我們留意，這導致「冪法則分布」的特性，其數學表徵之冪函數隨之出現。其生成原因至少包括源自統計物理學的「自我組織臨界性」及源自全球資訊網研究的「優先連結成長」等兩項因素。因此，「碎形幾何」概念的介入，以及厚尾巴導致的「黑天鵝事件」的出現，即屬必然。 我們在第二章，綜合回顧網路科學的背景，包括網路的性質及其類型，常態分布、冪法則分布的比較，以及由冪法則分布帶來的黑天鵝事件概念。同時，也包括針對社會網路在上世紀20年代所作的經典社會科學實驗及其在當今網路社會所扮演的角色。第三章則針對冪法則分布作更進一步的探索。包括冪法則的源起及其歷史回顧，具體實例，生成原因，不受度量規模影響的碎形特點及其數學呈現的唯一性。第四章則針對冪法則分布的厚尾所帶來黑天鵝事件作深入的探討，包括古典的及當代的黑天鵝事件。至於金融風暴所帶來的黑天鵝事件則在第五章作探討。 除了理論上的探討之外，我們也針對中央研究院語言學研究所提供的中文詞頻數據，以及華人姓氏分布數據等兩組數據為基礎深入探討，也發現了預期中冪法則的蹤影。

The newly emerging discipline of network science essentially consists of four category including social networks, internet and World Wild Web, biological networks and transportation networks. Scale-free distribution is one of the most important characteristic among them. It will then leads to power laws, as well as power functions. Stimulated by this observation, this research will focus on studying of the power laws. Self-organized criticality, from statistical physics, and growing by preferential attachment, based on the study of World Wide Web, are among the most important factors leading to scale-free distributions. It follows that the fractal geometry and blacks swan events associated with the fat tails of power law come out. A comprehensive review on network science, including properties and types of networks, normal distributions, power law distributions followed by black swan events, is given in Chapter 2. Some classical social science experiments done around 1950s are included too for their significant roles played in nowadays networks. A further study of power laws is given in Chapter 3, including its origins, historical backgrounds, prototyped examples, its unique mathematical presentation. Those black swan events, historical and contemporary, associated with the fat tail of power law distributions are surveyed in Chapter 4. Those financial crisis brought by black swan events will be studied in Chapter 5. In addition to the foregoing theoretical study, we will conduct experimental study on power law in Chapter 6 based on Chinese word frequency data, organized by the Institute of Chinese language, Academia Sinica, and on the statistical distribution data of Chinese surnames in detail. The power law will be confirmed in both studies as we expected.