Draft:Urban Scaling

Urban scaling is an area of research within the study of cities as complex systems. It examines how various urban indicators change systematically with city size.

The literature on urban scaling was motivated by the success of scaling theory in biology, itself motivated in turn by the success of scaling in physics. Crucial insights from scaling analysis applied to a system can emerge from finding power-law function relationships between variables of interest and the size of the system (as opposed to finding power-law probability distributions). Power-laws have an implicit self-similarity which suggests universal mechanisms at work, which in turn support the search for fundamental laws.

The phenomenon of scaling in biology is often referred to as allometric scaling. These relationships were originally mentioned 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. A theoretical explanation of allometric scaling laws in biology was provided by the Metabolic Scaling Theory.

The application of scaling in the context of cities is inspired by the idea that cities are emergent phenomena arising from the interactions of many individuals in close physical proximity. 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. If such regularities are observed, then it would support the search for a general mathematical theory of cities.

Indeed, Luis Bettencourt, Geoffrey West, and Jose Lobo's seminal work demonstrated that many urban indicators are associated with population size through a power-law relationship, in which socio-economic quantities tend to scale superlinearly, while measures of infrastructure (such as the number of gas stations) scale sublinearly with population size. They argue for a quantitative, predictive framework to understand cities as collective wholes, guiding urban policy, improving sustainability, and managing urban growth.

The literature has grown, with many theoretical explanations for these emergent power-laws. Ribeiro and Rybski summarized these in their paper "Mathematical models to explain the origin of urban scaling laws". Examples include Arbesman et al.'s 2009 model, Bettencourt's 2013 model , Gomez-Lievano et al.'s 2017 model , and Yang et al.'s 2019 model , among others (see ). The ultimate explanation of scaling laws observed in cities is still debated.

Power Laws and Scaling Exponents

 * Urban scaling often follows power-law relationships, where the form of the scaling can be expressed as
 * $$ Y = Y_0 N^\beta $$,

where $$Y$$ is the urban indicator, $$Y_0$$ is a constant, $$N $$ is the population size, and $$\beta$$ is the scaling exponent.
 * The exponent $$\beta$$ indicates whether the relationship is superlinear ($$\beta>1$$), sublinear ($$\beta = 1$$), or linear ($$\beta<1$$).

Pioneering Work and Key Studies
==== Geoffrey West, Luis Bettencourt, Jose Lobo and the Santa Fe Institute's cities group ====
 * Geoffrey West, Luis Bettencourt, Jose Lobo, and their colleagues at the Santa Fe Institute, conducted seminal work on urban scaling  . They identified consistent scaling laws across cities worldwide, showing that larger cities tend to be more innovative and productive but also face challenges such as increased crime rates and disease spread.
 * Their research demonstrated that many urban characteristics, from GDP to infrastructure, follow predictable scaling patterns. For example, they found that economic indicators typically have a superlinear scaling exponent ($$\beta\approx1.15$$), while infrastructure shows sublinear scaling ($$\beta\approx0.85$$).

Urban Planning and Policy

 * Understanding urban scaling helps policymakers and planners make more informed decisions. For example, recognizing the efficiencies of larger cities can guide infrastructure investments and resource allocation.
 * Scaling laws can also inform strategies to manage the challenges associated with urban growth, such as congestion, pollution, and social inequality.

Economic Development

 * The superlinear scaling of economic activity suggests that larger cities are engines of economic growth. Policies that support urbanization and the development of large metropolitan areas can potentially boost national and regional economies.

Sustainability and Resilience

 * Sublinear scaling of infrastructure highlights the potential for larger cities to be more sustainable by using resources more efficiently. However, this also requires careful management to avoid negative externalities like pollution and overconsumption.
 * Understanding the scaling properties of cities can also help in designing more resilient urban systems that can better withstand shocks such as natural disasters or economic downturns.

Criticisms of Urban Scaling Theory
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) 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) 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) and Altmann (2020) pointed out potential misspecifications in the statistical analysis, such as incorrect distribution assumptions and the independence of observations. These concerns highlight the need for theory to guide the choice of statistical methods. Additionally, the issue of defining city boundaries raises conceptual challenges. Arcaute et al. (2015) 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.

Another criticism of the urban scaling approach relates to the over-reliance on averages in measuring individual-level quantities such as average wages, or average number of patents produced. Complex systems, such as cities, exhibit distributions of their individual components that are often heavy-tailed. Heavy-tailed distributions are very different from normal distributions, and tend to generate extremely large values. The presence of extreme outliers can invalidate the Law of Large Numbers, making averages unreliable. Gomez-Lievano et al. (2021) 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.

Further Materials

 * Bettencourt, L. M. A., Lobo, J., Helbing, D., Kühnert, C., & West, G. B. (2007). Growth, innovation, scaling, and the pace of life in cities. Proceedings of the National Academy of Sciences, 104(17), 7301-7306.
 * Bettencourt, L. M. A. (2013). The origins of scaling in cities. Science, 340(6139), 1438-1441.
 * Bettencourt, L. M. A., & West, G. B. (2010). A unified theory of urban living. Nature, 467(7318), 912-913.
 * The surprising math of cities and corporations – TED Talk