Date Published: February 22, 2018
Publisher: Public Library of Science
Author(s): Eszter Bokányi, Zoltán Szállási, Gábor Vattay, Dan Braha.
We explain the anomaly of election results between large cities and rural areas in terms of urban scaling in the 1948–2016 US elections and in the 2016 EU referendum of the UK. The scaling curves are all universal and depend on a single parameter only, and one of the parties always shows superlinear scaling and drives the process, while the sublinear exponent of the other party is merely the consequence of probability conservation. Based on the recently developed model of urban scaling, we give a microscopic model of voter behavior in which we replace diversity characterizing humans in creative aspects with social diversity and tolerance. The model can also predict new political developments such as the fragmentation of the left and the immigration paradox.
Formation of cities is the result of socio-economic advantages of concentrating human populations in space outpacing associated costs. Urban agglomeration effects are systematic changes in socio-economic performance, innovation, trade and infrastructure characteristics of all cities as functions of their size. A variety of disciplines including economics [1–3], geography [4, 5], engineering  and complex systems [7–9] explain the existence of agglomeration or scaling effects and relate macroscopic properties of a city to its scale (population size). Such relations are known across the sciences as scaling relations , and the systematic study of such relationships in cities is known as urban scaling [11–14]. Using the population N as the measure of city size, power law scaling takes the form
where Y can denote material resources such as energy or infrastructure or measures of social activity such as wealth, patents and pollution; Y0 is a normalization constant. The exponent β reflects general dynamic rules at play across the urban system. Similar scale-free, fractal-like behavior has been observed in many human social networks  including cities . Therefore, it is natural and compelling that the essential features of a quantitative, predictive theory of cities originate in the dynamics and structure of social [17, 18] and infrastructural networks , and that these underlie the observed scaling relations and the values of the exponents [20–22]. In the case of innovation , scaling has been related to the long-distance ties that are prevalent in a higher proportion when a larger population provides the potential for productive social interactions.
First, we analyze data for the votes cast for the two main political parties in urban areas in all post-World War II US presidential elections  and in the UK EU referendum  (see Sections A-B in S1 File for method details). In Fig 1A we show votes for the political options as a function of voter turnout for the 912 largest Metropolitan and Micropolitan Statistical Areas representing about 82% of the total voter population for the 2016 presidential election in the US. Fig 1B shows the votes as a function of voter turnout for the Remain and Leave opinions in the 2016 EU referendum for the urban electoral districts of the UK. The votes for Democrats and Remain in the EU scale superlinearly with exponents βD ≈ 1.14 and βrem ≈ 1.09, while votes for Republicans and Leave the EU follow sublinear scaling with βR ≈ 0.92 and βlea ≈ 0.91, with high regression coefficients R2 ≥ 0.9 indicating robust urban scaling. While the elections took place in two different political situations, nevertheless they show very similar exponents.
We applied urban scaling theory to the number of votes cast in the Metropolitan and Micropolitan Statistcal Areas in the 1948–2016 presidential elections of the US and the votes cast in the urban areas of the 2016 EU referendum in the UK. We found that out of the two voting options (Democrat/Republican, Remain/Leave), one always follows a superlinear, while the other a sublinear scaling. Using the historical dataset, we showed that instead of four parameters (two for both scaling fits), the single exponent of the superlinearly scaling party is enough to characterize all processes across the elections and the parties. We derived the other exponent from the superlinear exponent by using the conservation of voting probabilities, and showed that the city turnout distribution determines that this other exponent must be sublinear. We then analyzed the fluctuations around the scaling curve distributions and found that the distribution corresponding to the superlinear exponent is lognormal. We concluded that the two parties play different roles in urban scaling. The party with superlinear exponent drives the process, while the scaling of the party with the sublinear exponent is merely the result of probability conservation. In the context of elections we identified the the elements of the GLPLH model and showed that social tolerance and diversity replaces creative diversity in this context. We pointed to new political consequences of the model. We believe that he model and the calculations could further be extended to metropolitan areas in other countries or to electoral systems with multiple choices.
See Sections A-G in S1 File for details.