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Is the relationship between population size of city and the number of crimes superlinear? If so, doubling the population size of a city will more than double the number of crimes. More precisely, what is the extent of such increase in the number of crimes when the population size doubles? Do different types of crime display different patterns of increase? Also, do increasing patterns of crime vary by different time periods of analysis? These are some of the questions we try to examine in this paper. For an overall framework of analysis, we are guided by the Urban Scaling Theory, which was recently developed by urban scholars (Bettencourt et al., 2007a; 2007b; 2010; Bettencourt, 2013). The population size of cities typically ranges from hundreds to millions of people. There are many more small cities than big cities, and this scaling reflects competition for resources.
When cities are viewed as living organisms (Deckeret et al., 2000; Deckeret et al., 2007; Samaniego & Moses, 2008) or city ecosystems (Glaeser & Gottlieb 2009; Florida, 2004; Fuzita et al., 2001; Segal, 1976; Henderson, 2007), cities change shape as the population size increases. For example, the economic output and the wealth of a city grow faster than linearly. The faster superlinear growth of economic output may be due to the number of potential human interactions in bigger cities which increases exponentially. For instance, it has been suggested that the number of potential interactions may increase as the square of the population following Metcalf’s law. In short, these economic outputs may be proportional to social interactions in cities.
Thus, Bettencourt et al., (2007a) propose the superlinear scaling relation for social activities underlying the creation of wealth, productivity and ideas. For example, they have presented empirical evidence of measures that show positive impacts, such as new patents, inventors, R&D employment, and GDP to follow the superlinear relationship as a function of population size. At the same time, increasing number of social interactions from bigger cities also contributes to the superlinear relationship of those social and environmental measures, which are usually viewed as negative. They include measures, such as crime, infectious diseases, congestion, poverty, and etc. In other words, both positive and negative impacts from increasing social interactions in cities are expected to display the superlinear relationship.
On the other hand, there are other measures such as the length of road in a city, which grows slower than the increase in population size. These measures that deal with material infrastructure and network, such as the number of gasoline stations or length of electric cable, display the sublinear relation (Watt et al., 2002; Kleinberg, 2001)
Finally, another group of performance measures which increase linearly as a function of the city’s population size has been identified. These measures deal primarily with needs of individuals in a city. They presented empirical evidence on such measures based on the rate of employment, household electricity and water consumption.
The general quantitative relationship between socioeconomic measures (Y), to population size of cities (N) is expressed by a simple power function of:
Y(t) = A N(t)
b(1)Where N (t) is population at time t, A is a constant independent of N (t), and b is the scaling exponent. When the relationship is superlinear, b>1, and when the relationship is sublinear, b<1. The linear relationship is when b=1.