Distance distributions of human settlements

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 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 (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 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 correlations (random, correlated and anti-correlated), which roughly reproduce similar values of G and E_S to those found in actual distance distributions.