Modeling the origin of urban-output scaling laws

Urban outputs often scale superlinearly with city population. A difficulty in understanding the mechanism of this phenomenon is that different outputs differ considerably in their scaling behaviors. Here, we formulate a physics-based model for the origin of superlinear scaling in urban outputs by treating human interaction as a random process. Our model suggests that the increased likelihood of finding required collaborations in a larger population can explain this superlinear scaling, which our model predicts to be non-power-law. Moreover, the extent of superlinearity should be greater for activities that require more collaborators. We test this model using a novel dataset for seven crime types and find strong support.

Figure 1

Urban outputs versus city size. (a) Number of new patents, murder cases, and AIDS cases in U.S. Metropolitan Statistical Areas (MSA's) vs population. These three urban outputs exhibit superlinear scaling—as MSA population doubles, the amount of output more than doubles. (b) Number of robbery cases and rape cases vs MSA population. Robbery scales superlinearly, while rape scales close to linearly. All data are scaled to have maximum value 1.

Figure 2

Superlinear scaling over time. Best-fit exponents (measuring degree of superlinear scaling) for scaling laws, i.e., slopes of curves such as those in Fig. 1, for all seven FBI crime categories. The shaded regions show 95% confidence intervals. Exponents can differ considerably among crime categories, and are consistent over time.

Figure 3

Mapping to social space. Cartoon for individuals in a city mapped to a one-dimensional space, ordered by rank of probability of social interaction. The probability of interaction with a particular peer decreases with position in one-dimensional space $x$ as $\rho \left(x\right)$.

Figure 4

Model predictions with data. (a) Number of robbery cases versus MSA population. (b) Number of rapes cases versus MSA population. Both for US in 2012. Red solid curves shows model predictions (not power-law), blue dots show data points, and the black dashed curves are reference lines of unit slope. Our model predicts superlinear scaling for robbery, and approximately linear scaling for rape. The difference between the model predictions in (a) and (b) is a result of the differing average co-offending group sizes. This figure uses co-offending group size calculated from the NIBRS dataset. The average group sizes are 1.74 for robbery and 1.29 for rape (group size is $n+1$). The parameters used in the model's prediction are $s=2.63\times {10}^6$ and $\alpha =0.93$ (global fit to all types of crimes).

Figure 5

Superlinearity as a function of average group size for crimes. Superlinearity is quantified as the exponent of the best-fit power law minus 1; this is for convenience—our model does not predict power-law scaling. Red solid line shows model prediction. Vertical error bars in both panels are standard deviation of year-to-year variation over 1999–2012. In (a), the horizontal error bars are state-to-state standard deviation in mean group sizes. In (b), the horizontal error bars are standard deviation of year-to-year variation over 1999–2012. Sources: [44, 45].