  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.
