Distance Distributions of Human Settlements for ten countries

City-size distributions have been studied from different perspectives in past decades. However, spatial properties of city distributions have not been widely explored. Here, we study distance distribution between pairs of cities by considering their sizes for data from ten countries. We<br>find that distance distributions present different shapes, from quasi-symmetrical to right-skewed, which exhibit changes as the lower threshold of the city size is increased. The Gini index G and the normalized Shannon entropy E_S were calculated to evaluate the inequality and heterogeneity<br>(diversity) in the distance distributions. Our results show that quasi-symmetrical distributions are characterized by a low inequality and intermediate value of entropy, while right-skewed distributions lead to higher inequality indexes with different levels of diversity. We introduce the big-club (set of<br>the largest cities) coefficient to evaluate the tendency of big settlements to be located close to (or far away from) each other, finding that most of the countries have a high big-club value. Finally, we consider a simple model based on node sizes to generate three configurations with different spatial<br>correlations (random, correlated and anti-correlated), which roughly reproduce similar values of G and E_S to those found in actual distance distributions.

近数十年来，学界已从多重视角对城市规模分布（city-size distributions）展开研究。然而，城市分布的空间属性尚未得到广泛探索。本研究针对十个国家的数据集，结合城市规模，分析城市两两之间的距离分布。
我们发现，距离分布呈现出从准对称到右偏的多种形态，且随城市规模下限的提升而发生变化。我们计算了基尼系数（Gini index）G与归一化香农熵（normalized Shannon entropy）E_S，用以评估距离分布中的不平等性与异质性（多样性）。研究结果表明，准对称分布的特征为低不平等性与中等熵值；而右偏分布则对应更高的不平等指数，且多样性水平各不相同。
我们引入了大俱乐部（big-club，即规模最大的城市集合）系数，用以衡量大型聚居地彼此邻近（或远离）的分布趋势，结果发现多数国家的大俱乐部系数较高。
最后，我们提出了一种基于节点规模的简单模型，生成了具有不同空间相关性（随机、相关与反相关）的三种配置，其得到的基尼系数G与归一化香农熵E_S与实际距离分布中的数值大致相近。