Fig. 2
Population density distribution by the model and validation with empirical data. a The relationship between C and β for different sizes (P(t) = 104, 5 × 104, 105), which can be well fitted by a power law with an exponential cutoff (\(\beta \sim 0.00350C^{ - 0.697}e^{ - 46.280C}, R^2 = 0.996\), see Supplementary Fig. 1 and Supplementary Note 1 for more details). β is estimated by the non-linear least squares (NLS) fitting of the simulation. Error bars mean±sd, obtained from ten different realisations. b Simulated AP density on a log-log plot along the distance for various system sizes with C = 0.002 and P(t) = 104, 5 × 104, 105. The absolute density increases as the system grows larger (see Supplementary Fig. 2 and Supplementary Note 1 for the re-scaled population density). c The empirical AP density along the distance to the central areas of London (β = 0.296) and Beijing (Inset, β = 0.0888) (see Table S1 for further details on the central areas of cities)
