We pose the central problem of defining a measure of complexity, specifically for spatial systems in general, city systems in particular. The measures we adopt are based on Shannon’s (in Bell Syst Tech J 27:379–423, 623–656, 1948) definition of information. We introduce this measure and argue that increasing information is equivalent to increasing complexity, and we show that for spatial distributions, this involves a trade-off between the density of the distribution and the number of events that characterize it; as cities get bigger and are characterized by more events—more places or locations, information increases, all other things being equal. But sometimes the distribution changes at a faster rate than the number of events and thus information can decrease even if a city grows. We develop these ideas using various information measures. We first demonstrate their applicability to various distributions of population in London over the last 100 years, then to a wider region of London which is divided into bands of zones at increasing distances from the core, and finally to the evolution of the street system that characterizes the built-up area of London from 1786 to the present day. We conclude by arguing that we need to relate these measures to other measures of complexity, to choose a wider array of examples, and to extend the analysis to two-dimensional spatial systems.

中文翻译：

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熵、复杂性和空间信息。

我们提出了定义复杂性度量的中心问题，特别是对于一般的空间系统，特别是城市系统。我们采用的措施基于香农 (Bell Syst Tech J 27:379–423, 623–656, 1948) 的信息定义。我们引入了这个度量并认为增加信息等同于增加复杂性，我们表明对于空间分布，这涉及分布密度和表征它的事件数量之间的权衡；随着城市越来越大，事件越来越多——更多的地方或地点，信息增加，所有其他条件都相同。但有时分布的变化速度比事件数量快，因此即使城市增长，信息也会减少。我们使用各种信息措施来发展这些想法。我们首先证明它们适用于过去 100 年来伦敦的各种人口分布，然后适用于更广泛的伦敦地区，该地区被划分为距核心越来越远的区域带，最后适用于以街道系统为特征的演变伦敦从 1786 年至今的建成区。最后，我们认为我们需要将这些度量与其他复杂性度量联系起来，选择更广泛的示例，并将分析扩展到二维空间系统。最后是从 1786 年至今伦敦建成区的街道系统的演变。最后，我们认为我们需要将这些度量与其他复杂性度量联系起来，选择更广泛的示例，并将分析扩展到二维空间系统。最后是从 1786 年至今伦敦建成区的街道系统的演变。最后，我们认为我们需要将这些度量与其他复杂性度量联系起来，选择更广泛的示例，并将分析扩展到二维空间系统。