As described in the introduction, city size scaling can be traced back to work that emerged in the early 1970s, in which Michel Fontaine and colleagues started looking for power law breakdowns in data collected from the French department of the Somme (Fontaine et al. 1971; Fontaine 1974). In these studies, the null expectation that people in large cities are larger than in smaller cities was tested. The results showed that power law scaling was a more likely explanation for cities than traditional exponential growth models, which at that point still had broad roots in the study of city patterns and apparently had not yet been challenged or replaced (Bettencourt et al. 2007). More than 50 years later, the Fontaine’s findings have been picked up by many researchers, who have investigated urban phenomena in various areas, including but not limited to, cities and regions in the US (Bettencourt and West 2010; Bettencourt and West 2013), World Bank cities (Leombruni et al. 2017), cities in low- and high-income countries (Bettencourt and West 2013; Bettencourt et al. 2017), historical regions (Ferrier et al. 2017) and in the Middle East (Fontaine 1991; Fontaine and Stevens 1991). For a wide overview, see Altaweel and Palmisano, 2019.
One example of the power law scaling revealed in (Bettencourt et al. 2007) is the value of the concentration parameter K. K is the exponent/slope from a simple power law equation, K = Cαp, where p stands for the site population. C is a constant that reflects the starting population. α was 0.85 in the range 350–350,000 for the Netherlands (Bettencourt et al. 2007; Bettencourt and West 2013). In this paper, we use the term urban scaling to define such relationships.
The changes in the value of urban scaling over time, which Bettencourt et al. (2020) are especially interested in, are also of importance for other areas: this equation has been used, for example, to properly describe the cities of the Roman Empire, the Byzantine Empire and the Abbasid Empire (Fontaine 1991; Fontaine 1991; Fontaine et al. d2c66b5586