Cry Havoc! And Let Slip the Maths of War

20110402_std001 In The Economist:

IN 1948 Lewis Fry Richardson, a British scientist, published what was probably the first rigorous analysis of the statistics of war. Richardson had spent seven years gathering data on the wars waged in the century or so prior to his study. There were almost 300 of them. The list runs from conflicts that claimed a thousand or so lives to the devastation of the two world wars. But when he plotted his results, he found that these diverse events fell into a regular pattern. It was as if the chaos of war seemed to comply with some hitherto unknown law of nature.

At first glance the pattern seems obvious. Richardson found that wars with low death tolls far outnumber high-fatality conflicts. But that obvious observation conceals a precise mathematical description: the link between the severity and frequency of conflicts follows a smooth curve, known as a power law. One consequence is that extreme events such as the world wars do not appear to be anomalies. They are simply what should be expected to occur occasionally, given the frequency with which conflicts take place.

The results have fascinated mathematicians and military strategists ever since. They have also been replicated many times. But they have not had much impact on the conduct of actual wars. As a result, there is a certain “so what” quality to Richardson’s results. It is one thing to show that a pattern exists, another to do something useful with it.

In a paper currently under review at Science, however, Neil Johnson of the University of Miami in Coral Gables, Florida, and his colleagues hint at what that something useful might be. Dr Johnson’s team is one of several groups who, in previous papers, have shown that Richardson’s power law also applies to attacks by terrorists and insurgents. They and others have broadened Richardson’s scope of inquiry to include the timing of attacks, as well as the severity. This prepared the ground for the new paper, which outlines a method for forecasting the evolution of conflicts.

Progress, of a sort

Dr Johnson’s proposal rests on a pattern he and his team found in data on insurgent attacks against American forces in Afghanistan and Iraq. After the initial attacks in any given province, subsequent fatal incidents become more and more frequent. The intriguing point is that it is possible, using a formula Dr Johnson has derived, to predict the details of this pattern from the interval between the first two attacks.

The formula in question (T(n) = T(1)n^-b) is one of a familiar type, known as a progress curve, that describes how productivity improves in a range of human activities from manufacturing to cancer surgery. T(n) is the number of days between the nth attack and its successor. (T(1) is therefore the number of days between the first and second attacks.) The other element of the equation, b, turns out to be directly related to T(1). It is calculated from the relationship between the logarithms of the attack number, n, and the attack interval, T(n). The upshot is that knowing T(1) should be enough to predict the future course of a local insurgency. Conversely, changing b would change both T(1) and T(n), and thus change that future course.