The hierarchical structure of geographic or urban space can be well-characterized by the concept of living structure, a term coined by Christopher Alexander. All spaces, regardless of their size, possess certain degrees of livingness that can be mathematically quantified. While previous studies have successfully quantified the livingness of small spaces such as images or artworks, the livingness of geographic space has not yet been characterized in a recursive manner. Zipf’s law has been observed in urban systems and intra-urban structures. However, whether Zipf’s law is applicable to the hierarchical substructures of geographic space has rarely been investigated. In this study, we recursively extract the substructures of geographic space using global nighttime light imagery. We quantify the livingness of global cities considering the number of substructures (S) and their inherent hierarchy (H). We further investigate the scaling properties of the extracted substructures across scales and the relationships between livingness and population for global cities. The results demonstrate that all substructures of global cities form a living structure that conforms to Zipf’s law. The degree of livingness better captures population distribution than nighttime light intensity values for the global cities. This study contributes in three aspects: First, it considers global cities as a whole to quantify spatial livingness. Second, it applies the concept of livingness to cities to better capture the spatial structure of the population using nighttime light data. Third, it introduces a novel method to recursively extract substructures from nighttime images, offering a valuable tool to investigate urban structures across multiple spatial scales.

中文翻译：

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使用全球夜间灯光数据表征跨尺度地理空间的活力


地理或城市空间的层次结构可以通过生活结构的概念来很好地表征，这是克里斯托弗·亚历山大创造的一个术语。所有空间，无论其大小，都具有一定程度的活力，可以通过数学量化。虽然之前的研究已经成功地量化了图像或艺术品等小空间的活力，但地理空间的活力尚未以递归方式表征。齐普夫定律已在城市系统和城市内部结构中得到遵守。然而，齐普夫定律是否适用于地理空间的层次结构却很少被研究。在这项研究中，我们使用全球夜间灯光图像递归地提取地理空间的子结构。我们考虑子结构（S）的数量及其固有的层次结构（H）来量化全球城市的活力。我们进一步研究了跨尺度提取的子结构的尺度特性以及全球城市的生活与人口之间的关系。结果表明，全球城市的所有底层结构都形成了符合齐普夫定律的生命结构。全球城市的活跃度比夜间光强度值更好地反映了人口分布。这项研究有三个方面的贡献：首先，它将全球城市作为一个整体来量化空间活力。其次，它将生活的概念应用到城市中，利用夜间灯光数据更好地捕捉人口的空间结构。第三，它引入了一种从夜间图像中递归提取子结构的新方法，为研究跨多个空间尺度的城市结构提供了有价值的工具。