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.
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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.
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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.
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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.
