Power-laws have an implicit self-similarity which suggests universal mechanisms at work, which in turn support the search for fundamental laws.
These concepts resurface in the study of complex systems,[4][5] and are of particular importance in the urban scaling framework.
Some of these relationships were studied by Galileo (e.g., in terms of the area width of animals' legs as a function of their mass) and then studied a century ago by Max Kleiber (see Kleiber's law) in terms of the relationship between basal metabolic rate and mass.
This is in contrast to applying scaling to countries or other social group delineations, which are more ad-hoc sociological constructions.
The expectation is that collective effects in cities should result in the form of large-scale quantitative urban regularities that ought to hold across cultures, countries and history.
[6] Indeed, Luís M. A. Bettencourt, Geoffrey West, and Jose Lobo's seminal work[7] demonstrated that many urban indicators are associated with population size through a power-law relationship, in which socio-economic quantities tend to scale superlinearly,[8] while measures of infrastructure (such as the number of gas stations) scale sublinearly with population size.
Ribeiro and Rybski summarized these in their paper "Mathematical models to explain the origin of urban scaling laws".
The framework can be extended to understand whether a given city will follow or will deviate from the power-law relationship describing the whole urban system.
cities in an urban system, and assume their populations grow exponentially with a fixed and constant rate
That is, if population size and output grow exponentially at different rates, they will be longitudinally related through a power-law for any single city
There is a certain debate in the published literature on this topic, due to a lack of explicit definitions about what scaling means in time and in space.
), while the cross-sectional scaling exponent is the ratio of two partial derivatives with respect to size (i.e., holding time constant).
Today, the field of urban economics is focused on understanding the causal underpinnings of the benefits that accrue when people come together in physical space.
The field of sociology has also investigated the relationship between socioeconomic variables and the size and density of populations.
For example, Émile Durkheim, a French sociologist, highlighted the sociological impacts of population density and growth in his 1893 dissertation, "The Division of Labour in Society."
This concept, known as "dynamic density," was later expanded by American sociologist Louis Wirth, particularly in the context of urban settings.
However, it wasn't until the 1970s that these ideas were translated into (sociological) mathematical models, sparking debates among sociologists about the complexities of urban agglomeration.
[22][23][24] Critics like Claude S. Fischer argued that mathematical models oversimplified the reality of social interactions in cities.
Fischer contended that these models assumed urbanites interact randomly, akin to marbles in a jar, which fails to capture the nuanced and localized nature of city life.
Fischer’s criticism emphasized the need for a deeper understanding of social systems, beyond mere quantitative models.
[25] Since the formulation of the urban scaling hypothesis, several researchers from the complexity field have criticized the framework and its approach.
These criticisms often target the statistical methods used, suggesting that the relationship between economic output and city size may not be a power law.
For instance, Shalizi (2011)[26] argues that other functions could fit the relationship between urban characteristics and population equally well, challenging the notion of scale invariance.
Bettencourt et al. (2013)[27] responded that while other models might fit the data, the power-law hypothesis remains robust without a better theoretical alternative.
Other critiques by Leitão et al. (2016)[28] and Altmann (2020)[29] pointed out potential misspecifications in the statistical analysis, such as incorrect distribution assumptions and the independence of observations.
Arcaute et al. (2015)[30] and subsequent studies showed that different boundary definitions yield different scaling exponents, questioning the premise of agglomeration economies.
They suggest that models should consider the intra-city composition of economic and social activities rather than relying solely on aggregate measures.
The presence of extreme outliers can invalidate the Law of Large Numbers, making averages unreliable.
Gomez-Lievano et al. (2021)[31] showed that in log-normally distributed urban quantities (such as wages), averages only make sense for sufficiently large cities.
Otherwise, artificial correlations between city size and productivity can emerge, misleadingly suggesting the appearance of urban scaling.