阿特拉斯模型中的齐夫定律
Zipf's Law for Atlas Models
AI总结：
本文研究了阿特拉斯模型和一阶模型如何通过保守性和完整性条件解释齐夫定律的普遍性，揭示了不同排名数据遵循齐夫定律或非齐夫帕累托分布的原因。
摘要：
具有正值的数据服从帕累托分布，当其值与排名的对数对数图近似为直线时。帕累托分布满足齐夫定律当且仅当对数对数图的斜率为-1。由于许多类型的排名数据遵循齐夫定律，因此被视为一种普遍性。我们基于阿特拉斯模型和一阶模型提出了一种数学解释，这些模型是具有仅依赖于排名参数的正连续半鞅系统。我们证明，阿特拉斯模型的稳定分布在且仅在保守性和完整性两个自然条件得到满足时才会遵循齐夫定律。由于阿特拉斯模型和一阶模型可以构造以近似时间依赖的基于排名的数据系统，我们的结果可以解释此类系统中齐夫定律的普遍性。然而，由其他方法生成的排名数据可能遵循非齐夫帕累托分布。因此，我们的结果解释了为什么词频、企业规模、家庭财富和城市规模遵循齐夫定律，而地震 magnitude、累计书籍销量、太阳耀斑强度和战争强度则不遵循齐夫定律，这些都遵循非齐夫帕累托分布。
英文摘要：
A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
