Fig. 1: Temporal evolution of the relative population structure of cities in the same environment for different initial conditions. | Nature Communications

Fig. 1: Temporal evolution of the relative population structure of cities in the same environment for different initial conditions.

From: Demography and the emergence of universal patterns in urban systems

Fig. 1

Lines shows the trajectory of each city in terms of the fraction its population, Ni to the total NT. a shows an initial situation where all cities start out with similar sizes, whereas in b they are initiated following Zipf's law (shown in log-scale). In both cases, the relative city size distribution eventually becomes stationary at long times (vertical red line) with the same population structure. This is given by the eigenvector e0, corresponding to the leading eigenvalue of the environment. The insets show how the system converges in both cases to this common structure set by the environment, because the Kullback–Leibler divergence \({D}_{{\rm{KL}}}(P| {P}_{{{\bf{e}}}_{0}})\) approaches zero at late times.

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