Mathematicians have long tried to bring order to chaos—to develop complex models to understand seemingly random events. One particularly versatile model has been the "Lévy flight," a pattern of movement that consists of short, frequent steps all clustered in one area, and is then punctuated by long walks to a separate area. Picture wild animals foraging for food in a field, or searching an ocean for prey: they’ll slowly use up all the resources in one small area, and then move to a new area and start eating or hunting again there.
The Lévy flight, named after French mathematician Paul Lévy, has been used to describe phenomena as wide-ranging as financial markets, earthquakes, and a child’s game of hide-and-seek. In recent years, it has even been used to analyze patterns as dreary as the Web-surfing habits of online consumers, and as fascinating as the drips and streaks of Jackson Pollock’s paintings.
But now, a group of mathematicians argue that the Lévy pattern may have an even more practical, real-world application: mapping—and potentially preventing—crime.
“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.”
A group of math students and professors, writing in a recent issue of SIAM Journal on Applied Mathematics, expand on the previous hypothesis that a human’s daily and weekly commute can often resemble an animal’s foraging path. (For instance, a person may take short walks around her neighborhood on the weekend, and then take a relatively long bus or subway ride to work during the week, followed by short walks in the vicinity of the office.)
The authors of this new article build on that idea, and hypothesize that crime might work the same way. For instance, a burglar might try to break into a group of homes in one localized area over the course of several days, and then, after a while, might travel to a new neighborhood for another cluster of break-ins. Short steps, long leap, short steps: the Lévy flight model.
In this article, authors Sorathan Chaturapruek, Jonah Breslau, Daniel Yazdi, Theodore Kolokolnikov, and Scott McCalla, compared burglary data to the step sizes and lengths in a typical Levy flight pattern.
“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,” said Kolokolnikov and McCalla. They argued that while law enforcement agencies normally record information about the location and times of discrete crimes in an area, they don’t yet have a widely-accepted method for tracking—let alone predicting—the movement of individual criminals. “In our research, we have seen a relationship between the dynamics of burglary hotspots and the way criminals move.”
So what could this all mean for crime prevention, on the local level, in the real world? The authors say that law enforcement agencies need to think bigger.
“Certain policing efforts concentrate on known offenders’ home territories as a predictor of future crimes,” Kolokolnikov and McCalla said. “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 authors do seem aware of how vague this all may sound. It may be a bit far-fetched to imagine the New York Police Department, for example, employing a fractal geometrist to help it stop the next break-in from happening.
“Applying models like ours to reproduce the data is a strong first step, but there is clearly more work to be done,” said Kolokolnikov and McCalla. “This would have clear implications for policing policy, and could have a significant impact on burglary rates.”
Still, further research in this area could yield another small tool in the big belt of law enforcement. And it’s (arguably) a better use of our mathematicians’ valuable time than mapping a Pollock painting.