The spatial organization of the population density in cities

Although the average population density of a city is an extremely simple indicator, it is often used as a determinant factor for describing various aspects of urban phenomena. On the other hand, a plethora of different measures that aim at characterizing the urban form have been introduced in the literature, often with the risk of redundancy. Here, we argue that two measures are enough to capture a wealth of different forms of the population density. First, fluctuations of the local density can be very important and we should distinguish almost homogeneous cities from highly heterogeneous ones. This is easily characterized by an indicator such as the Gini coefficient $G$, or equivalently by the relative standard deviation or the entropy. The second important dimension is the spatial organization of the heterogeneities in population density. We propose a dispersion index $\eta$ that characterizes the degree of localization of highly populated areas. As far as population density is concerned, we argue that these two dimensions are enough to characterize the spatial organization of cities. We discuss this approach using a dataset of about 4, 500 cities belonging to the 10 largest urban areas in France, for which we have high resolution data, at the level of a square grid of 200 × 200 meters. Representing cities in the plane ($G$, $\eta$) allows us to construct families of cities. We find that, on average, compactness increases with heterogeneity. More precisely, we find four large categories of cities (with population larger than 10, 000 inhabitants): (i) first, homogeneous and dispersed cities where the density fluctuations are small, (ii) very heterogeneous cities with a compact organization of large densities areas. The last two groups comprise heterogeneous cities with (iii) a monocentric organization or (iv) a more delocalized, polycentric structure. We believe that integrating these two parameters in econometric analysis could improve our understanding of the impact of urban form on various socio-economical aspects.