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Figure 3 | EPJ Data Science

Figure 3

From: Testing Heaps’ law for cities using administrative and gridded population data sets

Figure 3

Consequences of Heaps’ law. (a), The number of cities with more than 5000 inhabitants in the Unites States is proportional to the state’s population, \(\text{corr}(C,N) = 0.95\). The correlations with area (0.04) and population density (−0.08, see inset) are negligible, as illustrated by the following pairs of states with similar area or density and very different number of cities: Alaska (\(A=1.5\)M km2, \(C(5k)=22\)) vs Texas (\(A=0.7\)M km2, \(C(5k)=392\)), and Rhode Island (\(\rho =393\) km−2, \(C(5k)=35\)) vs New Jersey (\(\rho =467\) km−2, \(C(5k)=316\)). (b), Combining the result from panel (a) with Zipf’s law it is possible to estimate the number of cities with more than X inhabitants in a country with population N as \(C(N,X) \sim N/X\); Heaps’ law. As a consequence, the scattered cloud of points resulting when plotting \(C(N,X)\) against N for various X’s in the range \(5 \cdot 10^{3} - 5 \cdot 10^{6}\) (inset) collapses on a straight line when \(C(N,X)\) is plotted against the ratio \(N/X\). (c), Historical records of the number of incorporated places (C, red triangles) and the state population (N, blue circles) in Iowa from 1850 to 2000 (source: State library of Iowa, state data center). The similar growth rates of C and N entail the validity of Heaps’ law \(C \sim N\) during the 150-year period (inset). (d), The average distance to the closest city in the United States scales as the inverse of the square root of the state’s population density (here all cities with more than 5000 inhabitants are considered). The asymmetric error bars denote the standard deviations above and below the average. (e), Illustration of the relationships between total population, number of cities, and their average distance in Iowa and Connecticut. In agreement with Heaps’ law, \(C \sim N^{\alpha }\), Iowa and Connecticut have similar populations and a similar number of cities with more than 5000 inhabitants, despite Connecticut having one-twelfth the area of Iowa. In agreement with Equation (2), cities in Connecticut are closer than cities in Iowa because of the higher population density in Connecticut. By rescaling distances such that Connecticut’s area becomes equal to Iowa’s area, the two states would have the same population density and consequently the same average distance between cities

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