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Article by Luís M.A. Bettencourt, The Origins of Scaling in Cities Published in the June 21st, 2013 issue of Science |
Bettencourt's work as described in the article consists of three parts: 1. Statistical derivations of relationships of cities' characteristics as a function of city scale, 2. Simple modeling to derive such relationships, 3. More sophisticated modeling of such relationships.
The first part is not without its element of theoretical analysis. For example the choice of the variable to represent scale is very implortant. Bettencourt chose population.
There is also the matter of choosing exact definitions for the empirical characteristics unnder consideration and then putting together the data from disparate source into a consistent form.
The type of relationship Bettencourt was seeking for a characteristic Y is of the form Y=Y_{0}N^{α}, where N is population and Y_{0} and α are parameters. Here are two examples of scale relationship that Bettencourt found.
In these graphs the heavy black lines represent relationships with exponents of unity.
In other fields an exponent like α is the ratio of small integers. For example, in the relationship between the body strength S and body weight W of an athlete the exponent is 2/3 because weight is proportional to the cube of the scale and strength is proportional to the cross section area of muscles, which is proportional to the square of scale.
In the above graphs the lines for an exponent of 7/6 for income and 5/6 for road mileage are shown along with the regression lines. Those lines are essentially the same as the regression lines.
Bettencourt starts the low level modeling with defining F_{>k}^{ij} as the social network interaction of the k-th type of person i with person j. Such interactions depend upon the area of interaction a_{0} and the strength of the k-th type of interaction g_{k}. He then defines the volumee spanned by the interaction as the product of a_{0} and the average length l traveled by people or goods in a city. The other crucial variable is population density N/A, where A is the land area of the city. The third variable is average outcome per interaction and is denoted as g. Therefore an urban characteristic per person, y=Y/N, is given by
where G is ga_{0}l.
One benefit of a city to its residents is income. If y represents per capita income then
On the cost side of the characteristics of a city is the transportation cost T. Bettencourt takes the per capita transportation costs to be proportional to the tranverse distance L in the city. The transverse distance L is proportional to the square root of the land area of the city, A^{½}. Bettencourt then asserts that per capita income and per capita transportation cost must maintain a constant ratio for all city sizes; i.e.,
But y is equal to GN/A and therefore
where a=(G/ε)^{2/3}. The scale relationship of city land area is sublinear.
But generally for any characteristic Y, Y=GN²/A.Therefore substituting A=aN^{2/3} into the expression for Y gives
and hence Y has a superlinear scale relationship with population.
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