The study of power laws has become a significant part of modern science. Power laws, it seems, are ubiquitous in the way they describe size distribution of everything from earthquakes to forest fires to financial crashes.
But there’s a curious phenomenon associated with power laws that statisticians until now have missed, says Didier Sornette at the Swiss Federal Institute of Technology. And this provides an interesting new way to look at extreme events.
Let’s look at what he’s claiming. Sornette gives as an example the distribution of city sizes in France, which follows a classic power law, meaning that there are many small cities and only a few large ones. On a log-to-log scale, this distribution gives a straight line–except for Paris, which is an outlier and many times larger than it ought to be if it were to follow the power law.
Paris is an outlier because it has been hugely influential in the history of France and so has benefited from various positive feedback mechanisms that have ensured its outsize growth. Apparently, London occupies a similarly outlying position in the distribution of cities in the United Kingdom.
Sornette goes on to identify a number of data sets showing power laws with outliers that he says are the result of positive feedback mechanisms that make them much larger than their peers. He calls these events dragon kings. What’s interesting about them is that they are entirely unaccounted for by a current understanding of power laws, from which Nassim Nicholas Taleb built the idea of black swans.
The special characteristic of dragon kings is that a positive feedback mechanism creates faster-than-exponential growth, making them larger than expected.
So what to make of this? Sornette offers one interesting observation. The seemingly ubiquitous presence of these dragon kings in all kinds of data sets means that extreme events are significantly more likely than power laws suggest.
That’s important. If you’ve ever wondered why we’ve experienced two or three “once in a century” financial crises in the last couple of decades, here’s your answer. It also implies that you’ll experience a few more before your time is up.
But Sornette goes further. He argues that dragon kings may have properties that make them not only identifiable in real time but also predictable. He puts it like this: “These processes provide clues that allow us to diagnose the maturation of a system towards a crisis.”
That’s much more speculative. It’s one thing to identify the feedback mechanisms that cause faster-than-exponential growth (and it’s not clear that Sornette can do even this) but quite another to spot the event that triggers a crash.
Sornette looks to be on to something interesting with his notion of dragon kings: outliers that exist beyond the usual realm of power laws. That could be a hugely influential. But his contention that these outliers are in some way more easily predictable than other events smacks more of wishful thinking than of good science.
Ref: arxiv.org/abs/0907.4290: Dragon-Kings, Black Swans and the Prediction of Crises