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Figure 5 | EPJ Data Science

Figure 5

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

Figure 5

Zipf’s law and Heaps’ law for urban clusters. (a), Counter Cumulative Distribution Functions of areas of the clusters in all United States regions of \(128 \times 128\) km2 having urbanised area up to 5%. Regions are grouped in six groups according to their total urbanised area, \(N_{a}\), and the CCDFs of each group are computed separately for the different values of the CCA parameter m (see the legend in Fig. 4(f) for the m values). (b), The CCDFs of panel (a) collapse on the same curve when the axes are properly rescaled. The dashed grey line is a power law with exponent −1 as a guide for the eye. (c), Average number of clusters as a function of the total urbanised area, \(N_{a}\), for the \(128 \times 128\) km2 United States regions (circles). The lower and upper values of the dashed areas denote the 10th and 90th percentile of 100 realisations of the null model. For clarity, curves have been shifted by \(m^{2}\) along the x-axis. (d), Counter Cumulative Distribution Functions of populations of the clusters in all United States regions of \(128 \times 128\) km2 having urbanised area up to 5%. Regions are grouped in six groups according to their total population, \(N_{x}\), and the CCDFs of each group are computed separately for the different values of the CCA parameter m. (e), The CCDFs of panel (d) collapse on the same curve when the axes are properly rescaled. The dashed grey line is a power law with exponent −1 as a guide for the eye. (f), Average number of clusters as a function of the total population, \(N_{x}\), for the \(128 \times 128\) km2 United States regions (circles). The lower and upper values of the dashed areas denote the 10th and 90th percentile of 100 realisations of the null model. For clarity, curves have been shifted by \(m^{2}\) along the x-axis

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