Distance Distributions of Human Settlements for ten countries

City-size distributions have been studied from different perspectives in past decades. However,<br>spatial properties of city distributions have not been widely explored. Here, we study distance<br>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,<br>which exhibit changes as the lower threshold of the city size is increased. The Gini index G and<br>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<br>characterized by a low inequality and intermediate value of entropy, while right-skewed distributions<br>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<br>far away from) each other, finding that most of the countries have a high big-club value. Finally, we<br>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<br>and E_S to those found in actual distance distributions.