Explaining the prevalence, scaling and variance of urban phenomena

Abstract

The prevalence of many urban phenomena changes systematically with population size 1 . We propose a theory that unifies models of economic complexity 2, 3 and cultural evolution 10 to derive urban scaling. The theory accounts for the difference in scaling exponents and average prevalence across phenomena, as well as the difference in the variance within phenomena across cities of similar size. The central ideas are that a number of necessary complementary factors must be simultaneously present for a phenomenon to occur, and that the diversity of factors is logarithmically related to population size. The model reveals that phenomena that require more factors will be less prevalent, scale more superlinearly and show larger variance across cities of similar size. The theory applies to data on education, employment, innovation, disease and crime, and it entails the ability to predict the prevalence of a phenomenon across cities, given information about the prevalence in a single city.Scaling is ubiquitous across many phenomena 5 , including physical 6 and biological 7 systems, plus a wide range of human 8,9 and urban activities 1,10 . Figure 1 shows, for US Metropolitan Statistical Areas, ten different phenomena classified in five broad types: employment, innovation, crime, educational attainment, and infectious disease. We observe scaling in the sense that the counts of people in each phenomenon scale as a power of population size. This relation takes the form E{Y |N } = Y 0 N β , where E{·|N } is the expectation operator conditional on population size N , Y is the random variable representing the output of a phenomenon in a city, Y 0 is a measure of general prevalence of the activity in the country, and β is the scaling exponent, i.e., the relative rate of change of Y with respect to N . From Fig. 1 we can also observe notable differences in the average prevalence, the slopes of the regression lines and the variance across all ten phenomena. Hence, we seek to explain four empirical facts: Prevalence follows a power-law scaling with population size, different phenomena have different general prevalence, different scaling exponents, and variance for cities of similar size. Remarkably, these observations appear to be pervasive across phenomena as we find them to be present in more than forty different urban activities. In this paper we propose a mechanism to explain them simultaneously.Scaling laws are important in science because they constrain the development of new theories: any theory that attempts to explain a phenomenon should be compatible with the empirical 3 scaling relationships that the data exhibit. A number of mechanisms have been proposed to explain the origins of scaling. Most theories are based on a network description of the underlying phenomena and derive the scaling properties from the way the number of links grow with the number of nodes in the network, under some energy or budget constraints [11][12][13][15][16][17] . Other scaling relationships are the result of how lines relate to surfaces, and...

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