The science of cities seeks to understand and explain regularities observed in the world's major urban systems. Modelling the population evolution of cities is at the core of this science and of all urban studies. Quantitatively, the most fundamental problem is to understand the hierarchical organization of city population and the statistical occurrence of megacities. This was first thought to be described by a universal principle known as Zipf's law1,2; however, the validity of this model has been challenged by recent empirical studies3,4. A theoretical model must also be able to explain the relatively frequent rises and falls of cities and civilizations5, but despite many attempts6-10 these fundamental questions have not yet been satisfactorily answered. Here we introduce a stochastic equation for modelling population growth in cities, constructed from an empirical analysis of recent datasets (for Canada, France, the UK and the USA). This model reveals how rare, but large, interurban migratory shocks dominate city growth. This equation predicts a complex shape for the distribution of city populations and shows that, owing to finite-time effects, Zipf's law does not hold in general, implying a more complex organization of cities. It also predicts the existence of multiple temporal variations in the city hierarchy, in agreement with observations5. Our result underlines the importance of rare events in the evolution of complex systems11 and, at a more practical level, in urban planning.

中文翻译：

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城市增长方程

城市科学旨在理解和解释在世界主要城市系统中观察到的规律。城市人口演变建模是这门科学和所有城市研究的核心。从数量上讲，最根本的问题是了解城市人口的等级组织和特大城市的统计发生情况。这首先被认为是由称为 Zipf 定律 1,2 的普遍原理来描述的；然而，该模型的有效性受到了最近的实证研究 3、4 的挑战。理论模型还必须能够解释城市和文明相对频繁的兴衰5，但尽管进行了多次尝试6-10，但这些基本问题仍未得到令人满意的回答。在这里，我们引入了一个用于模拟城市人口增长的随机方程，根据对最近数据集（加拿大、法国、英国和美国）的实证分析构建而成。这个模型揭示了城市增长是多么罕见但巨大的城市间迁移冲击。该方程预测了城市人口分布的复杂形状，并表明，由于有限时间效应，齐夫定律一般不成立，这意味着城市的组织更为复杂。它还预测了城市层次结构中存在多个时间变化，与观察结果一致5。我们的结果强调了罕见事件在复杂系统的演变中的重要性，在更实际的层面上，在城市规划中。该方程预测了城市人口分布的复杂形状，并表明，由于有限时间效应，齐夫定律一般不成立，这意味着城市的组织更为复杂。它还预测城市层次结构中存在多种时间变化，与观察结果一致5。我们的结果强调了罕见事件在复杂系统的演变中的重要性，在更实际的层面上，在城市规划中。该方程预测了城市人口分布的复杂形状，并表明，由于有限时间效应，齐夫定律一般不成立，这意味着城市的组织更为复杂。它还预测城市层次结构中存在多种时间变化，与观察结果一致5。我们的结果强调了罕见事件在复杂系统的演变中的重要性，在更实际的层面上，在城市规划中。