Figure 4
From: Understanding the mesoscopic scaling patterns within cities
Gravity model, Moran’s I, and super-linear scaling. (a) Fitting results between AP and gravity-based interactions for Beijing (\(\gamma = 1\)). (b) Urban scaling exponent \(\beta _{sup}\) changes with the values of \(\gamma\). The mean values of \(\beta _{sup}\) (y-axis) were calculated based on the simulation results of the ten studied cities (with ± one standard deviation). Interestingly, we find a linear relationship between \(\gamma\) and \(\beta _{sup}\) when \(\gamma\) ranges from 1 to 2 (the red line), and the cases \(\gamma =1\) and 2 effectively reproduce the theoretical estimations of \(\beta = 7/6\) and 4/3, respectively. (c) The super-linear scaling between interaction and the number of nodes (population) with \(\sigma = 1\) and \(\gamma = 1\). (d) Moran’s I and the scaling exponent \(\beta\). \(\beta\) increases monotonically as Moran’s I increases, and the theoretical value \(\beta = 7/6\) corresponds to Moran’s I = 0.66, the similar value to the empirical results of Moran’s I.
