Natural spatial pattern—When mutual socio-geo distances between cities follow Benford’s law

Abstract

Benford’s law states that the first digits of numbers in any natural dataset appear with defined frequencies. Pioneering, we use Benford distribution to analyse the geo-location of cities and their population in the majority of countries. We use distances in three dimensions: 1D between the population values, 2D between the cities, based on geo-coordinates of location, 3D between cities’ location and population, which jointly reflects separation and mass of urban locations. We get four main findings. Firstly, we empirically show that mutual 3D socio-geo distances between cities and populations in most countries conform with Benford’s law, and thus the urban geo-locations have natural spatial distribution. Secondly, we show empirically that the population of cities within countries follows the composition of gamma (1,1) distributions and that 1D distance between populations also conforms to Benford’s law. Thirdly, we pioneer in replicating spatial natural distribution–we discover in simulation that a mixture of three pure point-patterns: clustered, ordered and random in proportions 15:3:2 makes the 2D spatial distribution Benford-like. Complex 3D Benford-like patterns can be built upon 2D (spatial) Benford distribution and gamma (1,1) distribution of cities’ sizes. This finding enables generating 2D and 3D Benford distributions, which may replicate well the urban settlement. Fourth, we use historical settlement analysis to claim that the geo-location of cities and inhabitants worldwide followed the evolutionary process, resulting in natural Benford-like spatial distribution and to justify our statistical findings. Those results are very novel. This study develops new spatial distribution to simulate natural locations. It shows that evolutionary settlement patterns resulted in the natural location of cities, and historical distortions in urbanisation, even if persistent till now, are being evolutionary corrected.