Repeats a similar analysis for industries of different complexity. Note that, in the case of extremely heavy-tailed distributions within cities (for example, power-law distributions with a scaling parameter α < 2), tail differences by city size—and therefore superlinear scaling—may occur completely at random due to a dependency between sample size and the expected value of such distributions 19 , 57 . The α parameters for the six indicators considered here range from 2.1 to 4.4. Confirming the robustness of our results, randomizing units across cities 57 does not produce scaling exponents that are significantly different from 0 (average z score and P value across 1,000 randomized populations per indicator and pooled across indicators are 0.795 and 0.381, respectively), and neither are they close to the observed scaling exponents (average z score and P value across 1,000 randomized populations per indicator and pooled across indicators, in paired Z -tests, are 6.788 and P < 0.001, respectively) for any of the indicators. We refrain from using α to capture the skewness of within-city distributions 59 because the non-parametric measure d proved more robust in smaller cities, where the number of observations for some indicators drops below 50.
