Using math to track, predict criminals’ next move

Lévy flights are similar, except that step lengths are chosen from a probability distribution, specifically, a power-law distribution, which allows the steps of a random walk to have large jumps. The use of Lévy flights thus enables more efficient exploration of a territory, hence extending the UCLA model to incorporate nonlocal movement.

It has been argued in previous literature that animal movement, including human movement, generates Lévy flights instead of random walks. This sort of movement — long jumps, interspersed with local random walks — is also seen in typical daily commutes in cities. The long jumps or “flights” correspond to long distances covered by perhaps a bus or subway to another part of the city. This allows criminals to move to distant, more attractive burglary sites as opposed to being confined to neighboring sites as in the previous model.

Data available on distance between criminals’ homes and their targets shows that burglars are willing to travel longer distances for high-value targets, and tend to employ different means of transportation to make these long trips. Of course, this tendency differs among types of criminals. Professionals and older criminals may travel further than younger amateurs. A group of professional burglars planning to rob a bank, for instance, would reasonably be expected to follow a Lévy flight.

“There is actually a relationship between how far these criminals are willing to travel for a target and the ability for a hotspot to form,” explain Kolokolnikov and McCalla. The authors calculate the likelihood of hotspot formation based on the distribution of step sizes (or lengths) in Lévy flights. “By computing the theoretical crime hot-spot distribution as a function of stepsize distribution, we found that the ‘optimal’ locomotion strategy for criminals is to occasionally take big jumps but otherwise follow a distribution which is close to Brownian motion,” say Kolokolnikov and McCalla. “Taking an occasional big jump greatly increases the number of crimes. However, taking excessively many big jumps does no better than the regular Brownian motion. In the language of Lévy flights, there is an optimal exponent, which results in the maximum possible number of crime hot-spots, and that regime is actually close to the Brownian motion.”

The underlying math model uses a system of two partial differential equations (PDEs) that define criminal density and attractiveness respectively. The resulting PDE for criminal density is nonlocal, whereas the attractiveness field remains local as in the UCLA model. The authors perform a linear stability analysis around a steady state of crime to illustrate the effect of non-locality on hotspot formation.

Kolokolnikov and McCalla explain that while the location and shape of burglary hotspots are extensively recorded and studied, criminal movements are not tracked, and are hence, not well understood. “In our research, we have seen a relationship between the dynamics of burglary hotspots and the way criminals move.”

Such models can better instruct law enforcement efforts. “Certain policing efforts concentrate on known offenders’ home territories as a predictor of future crimes,” say Kolokolnikov and McCalla. “If the relationship between a burglar’s movement and choice of targets becomes better elucidated, then the police will be better informed when they schedule their nightly patrols.”

“The next major challenge is understanding how criminals move in different cities around the world,” according to Kolokolnikov and McCalla. “Applying models like ours to reproduce the data is a strong first step, but there is clearly more work to be done. This would have clear implications for policing policy, and could have a significant impact on burglary rates.”

“One of the surprising results in our model is that the criminals benefit very significantly by making a few big jumps while otherwise following a Brownian (or random) motion. It would be interesting to examine whether there are other situations, such as predator-prey models, where the optimal strategy is to follow nearly-Brownian motion with few jumps,” they conclude.

— Read more in Sorathan Chaturapruek et al., “Crime Modeling with Lévy Flights,” SIAM Journal on Applied Mathematics 73 no. 4 (online publish date: 15 August 2013): 1703-20