Revealing configurational attractors in the evolution of modern Australian and US cities.

The spatial structure of modern cities exhibits highly diverse patterns and keeps evolving under numerous constraints. Two key dimensions have recently achieved prominence in characterizing this diversity: heterogeneity and spreading. However, modern settlements do not fill the entire heterogeneity--spreading space. Yet, the dynamic mechanisms leading to emergence of the observed layouts are unclear. Here, we assess the heterogeneity and spreading of population density in 25 Australian and 175 US cities. We observe that larger cities tend to form a cluster with a low degree of spreading and a high degree of heterogeneity, and relate this observation to the dynamic properties of intra-urban migration in these cities. In doing so, we introduce a model consistent with the relocation data which predicts such highly compact and heterogeneous structure for the majority of cities, in concordance with the actual layout data. In addition, we analyze the stability of the long-term dynamics of urban configurations with respect to changes in the mobility characteristics, such as social disposition and relocation impedance near their equilibrium states. As a result, we report three qualitatively distinct feasible phases of urban structures: uniform, monocentric, and polycentric. These phases are shown to be separated by either smooth or sharp transitions, observed in the space of suitably chosen configurational parameters. Finally, this analysis reveals that the set of all possible equilibrium configurations ("configurational attractors") form a narrow region in the heterogeneity--spreading space, thus explaining the emergence of clustering patterns.