# 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 study of power-laws is closely linked to the study of critical phenomena in physics, in which emergent properties and scale invariance are central and organizing concepts. These concepts resurface in the study of complex systems, and are of particular importance in the urban scaling framework.
The phenomenon of scaling in biology is often referred to as allometric scaling. 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. 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, in cities, urban activities 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, Luís M. A. 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 for a more thorough review of the models). The ultimate explanation of scaling laws observed in cities is still debated.

## Key Concepts in Urban Scaling
### 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 ($\beta <1$).
The key focus of Urban Scaling as a field (in contrast with other fields [see "Economics" and "Sociology" sections below]) is the emphasis on studying the origin and explanation of particular values of the scaling exponents. While other fields have recognized a relationship between size and urban metrics, it is mainly researchers in the field of Urban Scaling who have been interested in the fact that, from all the possible relationships two variables can be related, and all the coefficients that can mediate the strength of the relationship, urban metrics and population size are related through power-laws and the exponents can be slightly below 1 or slightly above 1.

### Cross-sectional versus Longitudinal Scaling

The urban scaling framework mostly focuses on cross-sectional relationships. That is, it describes the power-law relationship between urban metrics across many cities for a particular point in time.
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.
Assume $c=1,\ldots ,m$ cities in an urban system, and assume their populations grow exponentially with a fixed and constant rate $g$, $N_{c}(t)=N_{c}(0)e^{gt}$. Assume cities generate some type of output $Y$, which also grows exponentially, but with another rate $h$, such that $Y_{c}(t)=Y_{c}(0)e^{ht}$. Here, $N_{c}(0)$ and $Y_{c}(0)$ represent the population and the output at time $t=0$, respectively. Together, these two assumptions imply that
${\left({\frac {Y_{c}(t)}{Y_{c}(0)}}\right)}^{1/h}={\left({\frac {N_{c}(t)}{N_{c}(0)}}\right)}^{1/g}.$
In turn, this yields the following implicit power-law relationship between output and population:
$Y_{c}(t)={a_{c}}~{N_{c}(t)}^{h/g},$
where $a_{c}=\left({\frac {Y_{c}(0)}{N_{c}(0)^{h/g}}}\right)$.
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 $c$. Furthermore, if the ratio between the initial output and population size is a constant independent of the city, $k\equiv a_{c}$, then the same power-law will describe the cross-sectional data, since the proportionality factor and the exponent in the power-law will not depend on the subscript $c$. This is a very simple example in which urban scaling would arise both in time and in space, $Y(N,t)=k{N(t)}^{\beta },$ with a scaling exponent $\beta =h/g$ equal to the ratio of the growth rates. The relationship between temporal and cross-sectional scaling can be made more general.
The question of whether there is a relationship between temporal scaling and cross-sectional scaling is addressed by noting that the outcome variable is a function of population size and time (with perhaps some random noise), $Y=f(N,t)$. 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.
Here, the following three relationships and definitions are assumed:

#### Total derivative of a multivariate function with respect to time (denoted by upper dot)

${\dot {f}}(x_{1},x_{2},\ldots ,x_{n})={\frac {\partial f}{\partial x_{1}}}{\dot {x}}_{1}+{\frac {\partial f}{\partial x_{2}}}{\dot {x}}_{2}+\ldots +{\frac {\partial f}{\partial x_{n}}}{\dot {x}}_{n}+{\frac {\partial f}{\partial t}}$

#### Longitudinal (temporal) urban scaling exponent

$\beta _{L}={\frac {\partial \ln(Y){\big /}\partial t}{\partial \ln(N){\big /}\partial t}}$

#### Cross-sectional (spatial) urban scaling exponent

$\beta _{C}={\frac {\partial \ln(Y){\big /}\partial N}{\partial \ln(N){\big /}\partial N}}.$
Note that the longitudinal scaling exponent $\beta _{L}$ is the ratio of two partial derivatives with respect to time (i.e., holding size constant for $Y$), while the cross-sectional scaling exponent is the ratio of two partial derivatives with respect to size (i.e., holding time constant).
For clarity and convenience, let $y\equiv \ln(Y)$ and $n\equiv \ln(N)$, and drop the city-specific subscript $c$. Using the above conventions, the total derivative of $y(n,t)$ with respect to time is
${\dot {y}}={\frac {\partial y}{\partial n}}{\dot {n}}+{\frac {\partial y}{\partial t}}$ .
Since $n$ is a function of time only, then ${\dot {n}}=\partial n/\partial t$. Hence, dividing on both sides by ${\dot {n}}$, we conclude that
${\frac {\dot {y}}{\dot {n}}}=\beta _{C}+\beta _{L}$ .
Based on this, one can interpret ${\frac {\dot {y}}{\dot {n}}}$ to be a "total" urban scaling exponent, and thus
$\beta _{T}=\beta _{C}+\beta _{L}$ .
However, since $N$ is a function of time only ($N=N(t)$), both $Y$ and $N$ change simultaneously over time. This interdependence makes it impossible to hold $N$ constant while observing changes in $Y$, which is necessary to directly estimate the longitudinal exponent $\beta _{L}$ from empirical data. Consequently, only the total scaling exponent $\beta _{T}$ and the cross-sectional exponent $\beta _{C}$ can be empirically estimated, while the longitudinal exponent $\beta _{L}$ remains unobservable in practice due to the confounding effect of $N$'s dependency on time.

## Pioneering Work
### Santa Fe Institute's cities group

* Luis Bettencourt, Geoffrey West, 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 ($\beta \approx 1.15$), while infrastructure shows sublinear scaling ($\beta \approx 0.85$).
* They started the research field of urban scaling with the explicit goal of understanding the power-law relationship between aggregate urban metrics and population size.

### Economics

Some early studies in economics can be seen to have contributed to early stages of the urban scaling literature (unintendedly) by their analyses of how economic outcomes change with population size. One such study is Sveikauskas' 1975 "The productivity of cities", in which he reports a positive association between the average productivity or workers and city population size.
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. Hence, a big body of literature has been focused on understanding the so-called "urban wage premium", which is the fact that nominal wages tend to be larger in larger cities.

### Sociology

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." In his work, Durkheim emphasized the collective social effects of population. He proposed that an increase in population leads to more social interactions, resulting in competition, specialization, and eventually conflict, which then necessitates the development of social norms and integration. 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.
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. He pointed out that most city dwellers have limited interactions within their neighborhoods and rarely venture into other parts of the city, contradicting the notion that social interactions scale uniformly with population size. Fischer’s criticism emphasized the need for a deeper understanding of social systems, beyond mere quantitative models.

## Implications of Urban Scaling
### 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
* "Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies", by Geoffrey B. West