The ongoing rapid urbanization phenomena make the understanding of the evolution of urban environments of utmost importance to improve the well-being and steer societies towards better futures. Many studies have focused on the emerging properties of cities, leading to the discovery of scaling laws mirroring, for instance, the dependence of socio-economic indicators on city sizes. Though scaling laws allow for the definition of city-size independent socio-economic indicators, only a few efforts have been devoted to the modeling of the dynamical evolution of cities as mirrored through socio-economic variables and their mutual influence. In this work, we propose a Maximum Entropy (ME), non-linear, generative model of cities. We write in particular a Hamiltonian function in terms of a few macro-economic variables, whose coupling parameters we infer from real data corresponding to French towns. We first discover that non-linear dependencies among different indicators are needed for a complete statistical description of the non-Gaussian correlations among them. Furthermore, though the dynamics of individual cities are far from being stationary, we show that the coupling parameters corresponding to different years turn out to be quite robust. The quasi time-invariance of the Hamiltonian model allows proposing an analytic model for the evolution in time of the macro-economic variables, based on the Langevin equation. Despite no temporal information about the evolution of cities has been used to derive this model, its forecast accuracy of the temporal evolution of the system is compatible to that of a model inferred using explicitly such information.
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