Self organized hotspots and social tomography

A social network often has numerous interesting attributes. When an attribute is quantified, a social tomography would arise from the underlying social network. One of the most interesting attributes is crime hotspots, whose existence has been strongly supported by observations that serious crimes ranging from residential burglary to homicide are strongly patterned in time and space, and by mathematical modeling. So far, however, the structures of hotspots, including their size distributions, have not been adequately studied. Here, we focus on a special type of hotspots, the sex offender clusters, in the United States, and show that their size distribution, where size is defined as the ratio between sex offender population and total population in a 5-digit zip code area, follows a power-law distribution. In contrast, such local total population, both general and sex offenders, do not quite follow power-laws. A heavy-tailed power-law distribution is fundamentally different from a thin-tailed distribution such as a Poisson distribution, and can be used as an objective criterion for defining sex offender clusters. More fundamentally, a power-law is a defining property of self-similarity or fractal behavior. Therefore, our finding indicates that sex offender clusters, size-wise, self-organize into a fractal, due to interplay of economic conditions of offenders, policies and public perceptions.