The sums of death

The likelihood and scale of terrorist attacks can be estimated based on a mathematical relationship known as a power law.

Researchers believe insurgent attacks, such as this car bombing in Peshawar, Pakistan, in October, follow a pattern described by a mathematical formula.

Dramatic events often provoke metaphors. In describing last year's financial crisis, commentators talked of a "financial earthquake". When the Taliban launched its lethal offensive in Pakistan in October, the result was described as an "avalanche" of attacks. While such metaphors are certainly vivid, no one would take them too literally. New research suggests, however, that perhaps we should. It turns out that many apparently lawless phenomena, from stock market runs to terrorist attacks, follow the same mathematical laws as avalanches and earthquakes.

The very suggestion that mathematics can be applied to human affairs may seems absurd: people and events surely do not follow the dictates of an equation to four places of decimals. Those making the claims are, of course, saying no such thing. What they are saying, however, is that if used judiciously, mathematics can cast light on events previously thought beyond useful analysis. The first hints of this intriguing possibility emerged 80 years ago in work by an English mathematician named Lewis Fry Richardson. Eclectic in his interests, Richardson is now recognised as one of the pioneers of multidisciplinary research, seeing no barriers between the physical and social sciences.

Trained as a physicist at Cambridge University, he applied his analytical skills to a host of phenomena, from weather forecasting to the geometry of coastlines. A Quaker and pacifist, Richardson spent many years investigating the mathematics of warfare. In work published posthumously in 1960, Richardson examined data on all wars from 1820 onward, looking for mathematical laws underpinning their nature.

Richardson found that - as expected - huge wars with colossal casualties are rarer than skirmishes. But plotting out a graph showing how the numbers of such wars changes with death toll revealed something surprising: a distinct, regular curve connecting the two. Known as a power law, this suggested that the probability of a war producing a given level of casualties, P, decreases according to some simple power, n, of the death toll, D. In other words, P is proportional to 1/Dn.

At the time, this seemed little more than a mathematical curiosity. But exactly the same type of power law has since been found to govern apparently a host of phenomena, where the probability of events exceeding some given magnitude declines according to size. For example, the chances of large stock price changes of a given percentage decline according to the cube of the size of the change. Many natural phenomena, from forest fires to earthquakes to avalanches, follow similar power laws. But what could all these have in common?

The answer can be found by emptying a bag of rice onto a plate. Once the grains have stopped moving, they form a more or less stable heap. Adding a few more grains makes no difference, but as more are added, some of the falling grains start to trigger small avalanches down the sides of the heap. And every so often, just one more grain will cause a whole side of the heap of rice to collapse. Analysis of the precise size and number of these avalanches shows that they, too, follow a power law, with large slides being rarer than small ones. What links them to financial collapses, earthquakes and wars is what researchers call self-organising criticality: they are all systems in a critical state of instability, so that just a small disturbance provokes them to organise themselves into a more stable state.

There is no telling which grain - or financial report or political event - will cause the sudden change, or how big the change will be. All that can be said is that the probabilities of such events are linked to their magnitude by a power law. Such underlying unity may be intriguing, but is it of any use? The power law relationships hold a crucial lesson for anyone trying to predict events like stock market crashes and earthquakes. As the chances of such events decrease rapidly with their magnitude, any forecasting system will have to be extraordinarily reliable if it is to be trusted. That is because the sheer improbability of a big event overwhelms the predictive power of the forecast, leading to a host of false alarms.

Even so, it is still possible to extract some useful information from a phenomenon governed by a power law. And according to a study published in the current issue of the journal Nature, one such phenomenon may be terrorist attacks. An international team of researchers has analysed the size and timing of over 54,000 insurgency events in countries ranging from Afghanistan and Iraq to Peru and Northern Ireland, and shown that, like major wars, they follow a power law.

However, there are subtle differences with conventional warfare, which the researchers, led by Professor Neil Johnson of the University of Miami, claim could be useful in combating the threat. For example, unlike standard wars, terrorist attacks are focused on maximising outrage, in order to attract global media coverage. According to the team, once a group of insurgents feels sufficiently confident of success, they take action - and the scale of the resulting outrage will follow a power law. Prof Johnson and his colleagues have been able to derive the form of power law using a mathematical model of how the members of an insurgent group interact with one another. Differences in the exact form of the power law can then be used to reveal differences in the characteristics of insurgencies. For example, the team argues that the power law for attacks in Afghanistan suggests a weaker level of insurgency than in Iraq, whose own power law points to growing insouciance among the insurgent groups active there.

Other researchers have tried to use the power laws to make predictions - with disturbing results. Aaron Clauset and Maxwell Young at the University of New Mexico have argued that if the current laws of insurgency remain unchanged, we can expect an event at least as terrible as the attack on the Twin Towers in 2001 within the next two years. Not everyone will be convinced by these attempts to crystallise human behaviour using formulas. The mathematics underpinning the insurgency power law is similar to that for financial markets, yet knowing this did not prevent the financial turmoil of recent years.

Even so, the existence of the power laws is underpinned by an impressive amount of historical evidence. As such we would be wrong to dismiss their central lesson: that terrorist outrages like September 11 are not unique events, but the most egregious of an entire spectrum that reaches down to an improvised roadside bomb in Afghanistan. Robert Matthews is Visiting Reader in Science at Aston University, Birmingham, England