Competitive dynamics between criminals and law enforcement explains the super-linear scaling of crime in cities

While cities have been the engine for innovation and growth for many millennia, they have also endured disproportionately more crime than smaller cities. Similarly to other urban sociological quantities, such as income, gross domestic product (GDP) and number of granted patents, it has been observed that crime scales super-linearly with city size. The default assumption is that super-linear scaling of crime, like other urban attributes, derives from agglomerative effects (that is, increasing returns from potentially more productive connections among criminals). However, crime initiation appears to be generated linearly with the population of a city, and the number of law enforcement officials scales sublinearly with city population. We hypothesize that the observed scaling exponent for net crime in a city is the result of competing dynamics between criminals and law enforcement, each with different scaling exponents, and where criminals win in the numbers game. We propose a simple dynamical model able to accommodate these empirical observations, as well as the potential multiple scaling regimes emerging from the competitive dynamics between crime and law enforcement. Our model is also general enough to be able to correctly account for crime in universities, where university crime does not scale super-linearly, but linearly with enrolment size.