8 JUNE 2006
Ask
a historian about the laws of power, and you'll probably hear two: 1.
Power corrupts; 2. Absolute power corrupts absolutely.
Ask a physicist, though, and you'll get some equations depicting what scientists refer to as "power laws."
In this case, the power is not military or political, but mathematical - "power" referring to an exponent appearing in the equations. But in the end, those mathematical powers may prove helpful in understanding political and military issues, such as terrorism.
Power laws describe the spread of sizes (or some magnitude) in a system with no typical scale. In such a "scale-free" system, individual instances span an extensive range. Take earthquakes, for example. There are small ones, moderate ones and Big Ones, and the strongest are a billion times more powerful than the weakest.
Such distributions of sizes are quite unlike may common qualities, say, the
heights of adult humans or the speeds of cars on a freeway, notes University
of Michigan physicist Mark Newman. 
The
average speed on a typical Interstate highway might be something like
70 mph, and the fastest car is unlikely to be going much more than twice
as fast as the slowest. Average adult height is between 5 feet and 6 feet,
with relatively few shorter than 4 or taller than 7. No power laws there.
But power laws do describe many sorts of natural and social phenomena. Almost every arena of science, from biology and geology to economics and sociology, makes use in some way or another of power law equations. Power-law math has been applied not only to the study of earthquakes but also to income distribution, species extinction, the sizes of cities, the severity of wars and the popularity of pages on the World Wide Web.
"The ubiquity of power-law behavior in the natural world has led many scientists to wonder whether there is a single, simple, underlying mechanism linking all these different systems together," Newman wrote in a paper published last year in the journal Contemporary Physics.
Actually, Newman points out, the conventional wisdom is that different mechanisms may produce power law behavior. In fact, scientists aren't really sure exactly what processes are responsible for power laws, although many ideas have been suggested.
One possibility is that systems with growth governed by random influences simply become scale-free naturally, yielding a wide spread of sizes. How long, for example, does it take you to go broke playing the slots in Las Vegas? There's no typical time frame. Many people go broke quickly, some manage to hang on a little longer, and a few stay ahead all night, just by chance.
Another mechanism, widely publicized in recent years, is the tendency of systems to organize themselves into a scale-free condition (a concept known as self-organized criticality). The textbook example involves forest fires. Large, dense forests are rare because s single lightning bolt can start a fire that burns down the whole thing. Smaller clumps of trees are more common because the lightning usually misses, and if it hits, it takes out only one clump. So lots of little clumps of trees are protected from fires. But because they are protected, they grow and merge into clumps of various sizes, with some big enough to be more susceptible to fire wiping them out. Thus tree clumps self-organize to the edge of danger (the critical state), and over the course of time the resulting distribution of clump sizes will look like just what a power law would predict.
Other explanations for power law behavior have been proposed, as Newman outlined in his paper (available online).
But even without uncertainty about the underlying mechanism, power laws can be valuable guides to public policy. Knowing whether a system obeys a power law is critical for sensible planning. Designing a doorway to be twice the height of the average adult guarantees that nobody entering will bump their head. But to design a bridge to withstand only twice the power of a typical earthquake is dumber even than FEMA.
So
it's also worth knowing that the severity of terrorist attacks follows
a power law pattern, as reported in a new paper by University of New Mexico
computer scientist Aaron Clauset and collaborators, available
online.
Courtesy National Park Service
"An adequate model of terrorism should not only give us indications of where and when terrorist events are likely to occur, but also be able to generate the observed frequency distribution of terrorist events and there severity," Clauset and collaborators write.
Their analysis shows that large-scale terrorism catastrophes (such as September 11) are the analog to the biggest earthquakes, events to be expected in a scale-free system governed by a power law.
Such mathematical knowledge is critical to evaluating terrorism's threat, for forecasting the range of severity of future attacks, and for making wise judgments about the allocation of anti-terror resources (Iraq? New York City? Omaha?). So let's hope people making such decisions are infused with the power of mathematical insight and not corrupted by the power of politics.
E-mail: tsiegfried@nasw.org
