标题： 城市和地区的交叉排名：人口与收入 显示英文标题

标题： Cross Ranking of Cities and Regions: Population vs. Income

摘要： 本文通过官方数据的秩-秩分析，结合统计物理的思想，在两种技术中探讨了社区内部经济结构与其人口分布之间的关系。 数据来源于意大利城市。 为了区分“宏观”和“微观”方面，分析既在全局（国家）层面，也在更局部（地区）层面进行。 首先，发现等级规模法则并非像许多其他研究中那样是标准的幂律，而是一个双重递减的幂律。 接下来，计算了衡量成对一致性和两个排名波动之间相关性的肯德尔和斯皮尔曼等级相关系数——就像热力学中的相关函数一样，以寻找人口统计学和财富之间的等级相关性（如果有的话）。 结果不仅显示了整个（国家）集合的全球差异，而且在比较各地区的城市数量、城市的居民数量和地区的居民数量，以及比较城市的总税收收入和地区税收收入时，也显示出（地区）差异。 指出了不同的异常值并进行了合理解释。 有趣的是，发现了国家中的两类城市和国家中的两类地区。 “常识”社会、政治和经济考虑支持这些发现。 更重要的是，这些方法表明，当特定标准在数值上合理时，它们能够非常清晰地区分社区。 提出了一个具体的模型来解释这些发现，即基于统计理论（例如，瓮填充）的双重递减幂律和两相系统模型。 当在其他领域观察到类似的等级关系特征时，可以预期这些模型思想是成立的。 强调这种分析比通过皮尔逊值-值相关分析更有意义。 显示英文摘要

摘要： This paper explores the relationship between the inner economical structure of communities and their population distribution through a rank-rank analysis of official data, along statistical physics ideas within two techniques. The data is taken on Italian cities. The analysis is performed both at a global (national) and at a more local (regional) level in order to distinguish "macro" and "micro" aspects. First, the rank-size rule is found not to be a standard power law, as in many other studies, but a doubly decreasing power law. Next, the Kendall and the Spearman rank correlation coefficients which measure pair concordance and the correlation between fluctuations in two rankings, respectively, - as a correlation function does in thermodynamics, are calculated for finding rank correlation (if any) between demography and wealth. Results show non only global disparities for the whole (country) set, but also (regional) disparities, when comparing the number of cities in regions, the number of inhabitants in cities and that in regions, as well as when comparing the aggregated tax income of the cities and that of regions. Different outliers are pointed out and justified. Interestingly, two classes of cities in the country and two classes of regions in the country are found. "Common sense" social, political, and economic considerations sustain the findings. More importantly, the methods show that they allow to distinguish communities, very clearly, when specific criteria are numerically sound. A specific modeling for the findings is presented, i.e. for the doubly decreasing power law and the two phase system, based on statistics theory, e.g., urn filling. The model ideas can be expected to hold when similar rank relationship features are observed in fields. It is emphasized that the analysis makes more sense than one through a Pearson value-value correlation analysis.