Discover - July/August 2010

Investigation and Popular Education, so he could look for patterns in the con;ict. Johnson hoped the numbers could tell them something about how the individual particles—in this case, insurgents rather than electrons—functioned when put together in large groups.

Soon the new team had a database that included more than 20,000 separate incidents from two and a half decades of FARC attacks. Johnson and Spagat expected that the success of the attacks, measured in the number of people killed, would cluster around a certain ;gure: There would be a few small attacks and a few large ones as outliers on either end, but most attacks would pile up in the middle. Visually, that distribution forms a bell curve, a shape that represents everything from height (some very short people, some very tall, most American men about 5' 10") to rolls of the dice (the occasional 2 or 12, but a lot of 5s, 6s, and 7s). Bell curves are called normal distribution curves because this is how we expect the world to work much of the time. But the Colombia graph looked completely different. When the researchers plotted the number of attacks along the y, or vertical, axis and people killed along the x, or horizontal, axis, the result was a line that plunged down and then levelled off. At the top were lots of tiny attacks; at the bottom were a handful of huge ones.

That pattern, known as a power law curve, is an extremely common one in math. It describes a progression in which the value of a variable (in this case, the number of casualties) is always ramped up or down by the same exponent, or power, as in: two to the power of two ( 2 x 2) equals four, three to the power of two ( 3 x 3) equals nine, four to the power of two

( 4 x 4) equals 16, and so on. If the height of Americans were distributed according to a power law curve rather than a bell curve, there would be 180 million people 7 inches tall, 60,000 people towering at 8' 11", and a solitary giant as tall as the Empire State Building. Although power laws clearly do not apply to human height, they show up often in everyday situations, from income distribution (billions of people living on a few dollars a day, a handful of multibillionaires) to the weather (lots of small storms, just a few hurricane Katrinas).

In Colombia’s case, decades of news reports con;rmed that the number of attacks formed a line that sloped down from left to right. In general, an attack that causes 10 deaths is 316 times as likely as one that kills 100. The larger the event, the rarer it is.

At ;rst the pattern seemed too clear and simple to be true. “ Immediately I thought, ‘We need to look at another war,’” Johnson says. With the U.S. invasion of Iraq in full

One might use the math to argue that the 9/11 attack was bound to happen. And there is reason to believe that an even bigger one is on the way.

swing, he and his collaborators had an obvious second test. In 2005, using data gleaned from sources like the Iraq Body Count project and iCasualties, a Web site that tracks U.S. military deaths, they crunched the numbers on the size and frequency of attacks by Iraqi insurgents. Not only did the data ;t a power curve, but the shape of that curve was nearly identical to the one describing the Colombian con;ict.

Around that time, a Santa Fe Institute computer scientist named Aaron Clauset was applying the same approach to what seemed like a distinctly different problem. Rather than looking at speci;c guerrilla movements, Clauset was examining total deaths caused by global terrorist attacks since 1968. When he plotted nearly 30,000 incidents on a graph, they formed a curve to the power of – 2. 38. (The power number is negative because it re;ects a decrease rather than an increase in the number of events as death tolls rise.) With its characteristic downward slope, the curve was eerily similar to those generated by Johnson and Spagat for Colombia and Iraq.

To rule out coincidence, Johnson, Spagat, and University of Oxford physicist Sean Gourley gathered data on nine other insurgencies. One after another, the curves clicked into place: Peru’s Shining Path guerrilla movement: a curve with a power of – 2. 4. The Indonesian campaign against rebels in East Timor from 1996 to 2001: – 2. 5. The Palestinian second intifada: – 2. 55. Fighting against Afghanistan’s Taliban from 2001 to 2005: – 2. 44. By contrast, traditional con;icts in which two armies squared off against each other (such as the Spanish and American civil wars) yielded graphs that looked a lot more like bell curves than power curves. Although the politics, religion, funding, motives, and strategies of the insurgencies varied, the power trends did not.

In an age of biological weapons and dirty nukes, the implications are chilling. Although truly massive power-law events—like the Great Depression or killer storms—are drastically less common than smaller disruptions, they still occur. In the normal distribution of a bell curve, you never get such extremes, but the pattern underlying the power curve enables a few rare events of extraordinary magnitude. One might use the math to argue that the 9/11 attack that killed more than 2,700 people in New York City was bound to happen. And there is ample reason to believe that an even bigger one is on the way, sooner or later.