Basically, the exponential relationship between the object’s size and the amount of its interactions or interaction results is addressed from biology, known as allometric scaling problem (1, 2). After some explanatory models for this allometric scaling phenomena (2, 3), social science researchers have tried to apply these biological characteristics into urbanization process and its result(4, 5, 6, 7).
The scaling results show that different urban features which are not related to each other have also exponential scalings with the size of city over various cities worldwide. In particular, the scaling exponents of urban features, which represent the levels of linearity, are shown to be different depending on the characteristics of the features. For example, quantities related to material infrastructure, such as the number of gas station, the length of electrical cables are characterized as sublinear(exponent value is smaller than 1.), while those about related to social currenice, sucah as information, innovation or wealth are characterized as superlinear(exponent value is bigger than 1.)(6).
To explain the process that this urban scaling result is emerged, Luís M. A. Bettencourt constructed a model assuming four characteristics of urban behavior: mixing population, incremental growth of infrastructure network, bounded human effort, proportional socio-economic output to local interactions(7).
The Linearity of Residuals
As shown in Fig. 1, the amount of residuals also has a relationship with the size of crimes (thus, also with the size of cities). That means, the data may have a problem of [heteroscedacity]. According to Wikipedia,
“(…) regression analysis using heteroscedastic data will still provide an unbiased estimate for the relationship between the predictor variable and the outcome, but standard errors and therefore inferences obtained from data analysis are suspect. Biased standard errors lead to biased inference, so results of hypothesis tests are possibly wrong.” (8)
If it is true, can we say more than “Bigger the city is, more the crimes are.”? Furthermore, what should we care when we apply existing regression methods on log-scaled values?
Overemphasis on Population
Assume that there are some set of quantities named A, B, and C. If A has a linear relationship with B, and B has a linear relationship with C, B is highly likely to have a linear relationship with C either. In this situation, the three linear relationships between A, B, and C themselves cannot explain any underlying causalities among them.
By the way, as far as I understand, some explanations in urban scaling implictly claim that the city size – generally measured by its population – is the main factor to cause the quantitative differences. For example, Luis Bettencourt mentioned that
“On average, as city size increases, per capita socio-economic quantities such as wages, GDP, number of patents produced and number of educational and research institutions all increase by approximately 15% more than the expected linear growth.”(5)
However, does it sound also natural that higher average wages of a specific city makes people to move into the city so the city gets bigger? Similarly, we can say that higher level of educational, informational environments or higher profits of companies made the city bigger by gravitating people nearby to move in. Furthermore, the multiple positive feedbacks between combinations of various urban features result in the growth of city.
1 – West, Geoffrey B., James H. Brown, and Brian J. Enquist. “A general model for the origin of allometric scaling laws in biology.” Science 276.5309 (1997): 122-126.
2 – West, Geoffrey B., James H. Brown, and Brian J. Enquist. “The fourth dimension of life: fractal geometry and allometric scaling of organisms.” science 284.5420 (1999): 1677-1679.
3 – West, Geoffrey B., James H. Brown, and Brian J. Enquist. “A general model for the structure and allometry of plant vascular systems.” Nature 400.6745 (1999): 664-667.
4 – Batty, Michael. “The size, scale, and shape of cities.” science 319.5864 (2008): 769-771.
5 – Bettencourt, Luis, and Geoffrey West. “A unified theory of urban living.” Nature 467.7318 (2010): 912-913.
6 – Bettencourt, Luís MA, et al. “Growth, innovation, scaling, and the pace of life in cities.” Proceedings of the national academy of sciences 104.17 (2007): 7301-7306.
7 – Bettencourt, Luís MA. “The origins of scaling in cities.” science 340.6139 (2013): 1438-1441.
8 – “Heteroscedasticity”, Wikipedia.org