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Table 1 Exponents of Zipf’s Law and Heaps’ Law. Column 1 displays \(\beta = 1+ \alpha \); the PDF exponent of Zipf’s Law with corresponding standard deviation displayed in column 2 for each of the four continents. Values were calculated by fitting a line to the PDF of city sizes X for each continent, starting from a minimum population of \(X = 10^{5}\) (see Fig. 2(a)). Column 3 displays γ; the exponent of Heaps’ Law with corresponding standard deviation displayed in column 4 for each of the four continents. Values of γ and the error were obtained by fitting a line to the logarithm of Heaps’ Law; \(\log (C) = \gamma \log (N)\), for each continent (see Figs. 2(b)–(c) and (e)–(f)). Exponents β and γ were fit to the data using non-linear least squares

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

Continent β \(\sigma _{\beta }\) γ \(\sigma _{\gamma }\)
Africa 1.77 0.05 0.78 0.09
Asia 1.94 0.04 0.85 0.05
America 1.96 0.04 0.97 0.07
Europe 2.02 0.05 1.01 0.06