Fig. 4: The dynamics of the rank-size distribution for cities in the US between 1790 and 1990. | Nature Communications

Fig. 4: The dynamics of the rank-size distribution for cities in the US between 1790 and 1990.

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

Fig. 4

Population structure distributions for the largest 100 cities in the US53 are shown as dashed lines, while the average over all years is shown in red. The inset depicts the DKL(PPz) to Zipf’s law of the population structure for each available year (blue) and of the cumulative average over all structure vectors up the specific year (red). We see that the temporal average steadily approaches Zipf’s law as more years are added. After an initial phase, the average is always closer to Zipf’s law than the distribution in any single year. Note that in recent decades, the population structure vector began consistently deviating from Zipf’s law, leading also to a small divergence of the cumulative temporal averages. This effect is to a large extent the consequence of mid-sized and large metropolitan areas growing faster over this period than the largest cities in the US urban system.

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