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## December 14, 2010

### Statistical Analysis of Terrorism

Yesterday, my work on global statistical patterns in terrorism [1] was featured in a long article in the magazine Miller-McCune called The Physics of Terrorism, written by Michael Haederle [2].

Much of the article focuses on the weird empirical fact that the frequency of severe terrorist attacks is well described by a power-law distribution [3,4], although it also discusses my work on robust patterns of behavior in terrorist groups, for instance, showing that they typically increase the frequency of their attacks as they get older (and bigger and more experienced), and moreover that they do it in a highly predictable way. There are several points I like most about Michael's article. First, he emphasizes that these patterns are not just nice statistical descriptions of things we already know, but rather they show that some things we thought were fundamentally different and unpredictable are actually related and that we can learn something about large but rare events by studying the more common smaller events. And second, he emphasizes the fact that these patterns can actually be used to make quantitative, model-based statistical forecasts about the future, something current methods in counter-terrorism struggle with.

Of course, there's a tremendous amount of hard-nosed scientific work that remains to be done to develop these empirical observations into practical tools, and I think it's important to recognize that they will not be a silver bullet for counter-terrorism, but they do show us that much more can be done here than has been traditionally believed and that there are potentially fundamental constraints on terrorism that could serve as leverage points if exploited appropriately. That is, so to speak, there's a forest out there that we've been missing by focusing only on the trees, and that thinking about forests as a whole can in fact help us understand some things about the behavior of trees. I don't think studying large-scale statistical patterns in terrorism or other kinds of human conflict takes away from the important work of studying individual conflicts, but I do think it adds quite a bit to our understanding overall, especially if we want to think about the long-term. How does that saying go again? Oh right, "those who do not learn from history are doomed to repeat it" (George Santayana, 1863-1952) [5].

The Miller-McCune article is fairly long, but here are a few good excerpts that capture the points pretty well:

Last summer, physicist Aaron Clauset was telling a group of undergraduates who were touring the Santa Fe Institute about the unexpected mathematical symmetries he had found while studying global terrorist attacks over the past four decades. Their professor made a comment that brought Clauset up short. "He was surprised that I could think about such a morbid topic in such a dry, scientific way," Clauset recalls. "And I hadn’t even thought about that. It was just … I think in some ways, in order to do this, you have to separate yourself from the emotional aspects of it."

But it is his terrorism research that seems to be getting Clauset the most attention these days. He is one of a handful of U.S. and European scientists searching for universal patterns hidden in human conflicts — patterns that might one day allow them to predict long-term threats. Rather than study historical grievances, violent ideologies and social networks the way most counterterrorism researchers do, Clauset and his colleagues disregard the unique traits of terrorist groups and focus entirely on outcomes — the violence they commit.

“When you start averaging over the differences, you see there are patterns in the way terrorists’ campaigns progress and the frequency and severity of the attacks,” he says. “This gives you hope that terrorism is understandable from a scientific perspective.” The research is no mere academic exercise. Clauset hopes, for example, that his work will enable predictions of when terrorists might get their hands on a nuclear, biological or chemical weapon — and when they might use it.

It is a bird’s-eye view, a strategic vision — a bit blurry in its details — rather than a tactical one. As legions of counterinsurgency analysts and operatives are trying, *24*-style, to avert the next strike by al-Qaeda or the Taliban, Clauset’s method is unlikely to predict exactly where or when an attack might occur. Instead, he deals in probabilities that unfold over months, years and decades — probability calculations that nevertheless could help government agencies make crucial decisions about how to allocate resources to prevent big attacks or deal with their fallout.

-----

[1] Here are the relevant scientific papers:

On the Frequency of Severe Terrorist Attacks, by A. Clauset, M. Young and K. S. Gledistch. *Journal of Conflict Resolution* **51**(1), 58 - 88 (2007).

Power-law distributions in empirical data, by A. Clauset, C. R. Shalizi and M. E. J. Newman. *SIAM Review* **51**(4), 661-703 (2009).

A generalized aggregation-disintegration model for the frequency of severe terrorist attacks, by A. Clauset and F. W. Wiegel. *Journal of Conflict Resolution* **54**(1), 179-197 (2010).

The Strategic Calculus of Terrorism: Substitution and Competition in the Israel-Palestine Conflict, by A. Clauset, L. Heger, M. Young and K. S. Gleditsch *Cooperation & Conflict* **45**(1), 6-33 (2010).

The developmental dynamics of terrorist organizations, by A. Clauset and K. S. Gleditsch. arxiv:0906.3287 (2009).

A novel explanation of the power-law form of the frequency of severe terrorist events: Reply to Saperstein, by A. Clauset, M. Young and K.S. Gleditsch. Forthcoming in *Peace Economics, Peace Science and Public Policy*.

[2] It was also slashdotted.

[3] If you're unfamiliar with power-law distributions, here's a brief explanation of how they're weird, taken from my 2010 article in *JCR*:

What distinguishes a power-law distribution from the more familiar Normal distribution is its *heavy tail*. That is, in a power law, there is a non-trivial amount of weight far from the distribution's center. This feature, in turn, implies that events orders of magnitude larger (or smaller) than the mean are relatively common. The latter point is particularly true when compared to a Normal distribution, where there is essentially no weight far from the mean.

Although there are many distributions that exhibit heavy tails, the power law is special and exhibits a straight line with slope *alpha* on doubly-logarithmic axes. (Note that some data being straight on log-log axes is a necessary, but not a sufficient condition of being power-law distributed.)

Power-law distributed quantities are not uncommon, and many characterize the distribution of familiar quantities. For instance, consider the populations of the 600 largest cities in the United States (from the 2000 Census). Among these, the average population is only x-bar =165,719, and metropolises like New York City and Los Angles seem to be "outliers" relative to this size. One clue that city sizes are not well explained by a Normal distribution is that the sample standard deviation *sigma* = 410,730 is significantly larger than the sample mean. Indeed, if we modeled the data in this way, we would expect to see 1.8 times fewer cities at least as large as Albuquerque (population 448,607) than we actually do. Further, because it is more than a dozen standard deviations above the mean, we would never expect to see a city as large as New York City (population 8,008,278), and largest we expect would be Indianapolis (population 781,870).

As a more whimsical second example, consider a world where the heights of Americans were distributed as a power law, with approximately the same average as the true distribution (which is convincingly Normal when certain exogenous factors are controlled). In this case, we would expect nearly 60,000 individuals to be as tall as the tallest adult male on record, at 2.72 meters. Further, we would expect ridiculous facts such as 10,000 individuals being as tall as an adult male giraffe, one individual as tall as the Empire State Building (381 meters), and 180 million diminutive individuals standing a mere 17 cm tall. In fact, this same analogy was recently used to describe the counter-intuitive nature of the extreme inequality in the wealth distribution in the United States, whose upper tail is often said to follow a power law.

Although much more can be said about power laws, we hope that the curious reader takes away a few basic facts from this brief introduction. First, heavy-tailed distributions do not conform to our expectations of a linear, or normally distributed, world. As such, the average value of a power law is not representative of the entire distribution, and events orders of magnitude larger than the mean are, in fact, relatively common. Second, the scaling property of power laws implies that, at least statistically, there is no qualitative difference between small, medium and extremely large events, as they are all succinctly described by a very simple statistical relationship.

[4] In some circles, power-law distributions have a bad reputation, which is not entirely undeserved given the way some scientists have claimed to find them everywhere they look. In this case, though, the data really do seem to follow a power-law distribution, even when you do the statistics properly. That is, the power-law claim is not just a crude approximation, but a bona fide and precise hypothesis that passes a fairly harsh statistical test.

[5] Also quoted as "Those who cannot remember the past are condemned to repeat their mistakes".

posted December 14, 2010 10:39 AM in Scientifically Speaking | permalink

## Comments

Yes, we all need to stop terrorism.

Posted by: Joe the chainsaw guy at December 17, 2010 02:12 AM

Wow this is a great resource.. I’m enjoying it.. good article

Posted by: veterinary technician at December 17, 2010 09:38 AM

Yes, we should be able to prevent terrorism, and it's not just police and military duties, but also our duty. Thanks, by ~ Militer

Posted by: Militer at December 23, 2010 12:41 AM

Aaron - I'm not as familiar with your work as I'd like, so you may have addressed this is one of your publications, but have you noted any instances in which a particular insurgent/terrorist group or conflict zone has deviated greatly from the larger trends- and that deviation has taught you something unique about that group or conflict? I'm wondering if the deviations might be as valuable in understanding conflict as the trends that lead to potential forecasting capabilities... If a group carries out violent action in a much different manner, maybe it is indicative of certain characteristics of that group. Just curious what you've found in this regard..

Also - glad to see that you're representing CU in this area! I'm a former Buff (Aerospace undergrad) and it's always great to see the University represented well. Nice work!

Matt

Posted by: Matt at January 3, 2011 08:09 AM

Matt,

At this point, my work on terrorist organizations has mainly been focused on trying to understand what "typical" actually means for them. But, you're absolutely right that once we know what we expect based on fairly fundamental processes, measuring deviations from that norm could potentially teach us interesting or even useful things about the particularities of individual groups. Future work, for sure.

-Aaron

Posted by: Aaron at January 3, 2011 10:13 AM

Superb blog post, I have book marked this internet site so ideally I’ll see much more on this subject in the foreseeable future!

Posted by: aerospace engineer at January 4, 2011 05:46 AM