“scale-free.” Say, for example, that
computer files 2 kilobytes in size are
one-fourth as common as files of 1 kilo-byte. Under a power law regime, file sizes
of 2 megabytes would be one-fourth
as common as those of 1 megabyte, and
so on. Like Mandelbrot’s fractal geometry of a coastline or cauliflower whorl,
whether you view from afar or zoom
in with a microscope, the proportions
remain the same.
These data suggest to Stanley that
global financial crashes and the bubbles
that precede them aren’t outliers. The
same mechanisms that cause the smaller
blips occurring in markets daily may
also be generating bigger crashes.
Knowing that such extreme events
will happen doesn’t mean researchers can predict when, Stanley says. But
acknowledging power law behavior may
help investors and regulators pin the
right number on risk. Having a power
law distribution changes how often
you’d expect to see an event sitting far
from the data’s average, a distance measured in “standard deviations.” With a
Gaussian model, an event that’s 100
standard deviations out —so far out
it’s considered impossible—has a
probability of about 1 in 10350. With a
power law distribution, that likelihood
shoots up to 1 in 108, Stanley notes.
Farmer, who made a small fortune
working in the financial sector throughout the 1990s, says knowing how often big
events may hit is crucial for estimating
risk: “You have to understand your tail.”
Recent work by Farmer, Stefan Thurner
of the Medical University of Vienna and
Yale economist John Geanakoplos suggests that some investment strategies
can actually create a power law long tail.
Say you see an underpriced stock. You
buy it, which normally would push the
price up a bit. But if you’re using leverage
(borrowed money) to try to amplify your
returns, and then the bank cuts you off,
you might be forced to sell prematurely or
sell off other assets. This selling can push
prices down and then other outfits may
sell too, because they see the price sliding.
“Heavy-tailed events can be caused by
leverage,” Farmer says. “It can create a
A different distribution
when displayed on a typical graph, data that follow a power law distribution form
a long tail (below, left graph). but graph the same data on a log scale and they fall
onto a straight line (right). by applying several statistical tests, a recent analysis
identified data sets that probably show power law behavior (bottom).
Typical graph Log scale
.004
Percent of sample
.003
.002
.001
0
0
2x105 4x105
Magnitude
Power law behavior
Percent of sample
10-3
10-2
10-4
10-5
10-6
10-7
10-8
104
106 105
Magnitude
107
the number of
sightings of birds
of different species in the north
american breeding bird Survey
in 2003
the intensity
of wars 1816–
1980, measured
as number of
battle deaths
and adjusted for
population size
SourCeS: m. newman/ arxiv.org 2006; a. ClauSet, C.r. SHalizi and m. newman/ arxiv.org 2009
Frequency of
occurrence of
unique words in
Herman melville’s
Moby Dick
the number of
adherents to
religious bodies
and sects, as
published on
adherents.com
crash.” His team’s simulations suggest
that adding leverage to a market tips
the distribution of price changes from
a Gaussian to a power law distribution.
And when banks cut off many borrowers to control risk, the situation can get
worse, the team reported in 2009 in a
Santa Fe Institute working paper.
Power laws in one area of the econ-
omy may lead to others, says economist
Xavier Gabaix of New York University’s
Stern School of Business. The power law
distribution of CEO pay may arise from
the interplay between a phenomenon
known as the economics of superstars
and a power law that exists for firm size,
he and colleague Augustin Landier, who
is now at the Toulouse School of Eco-
nomics in France, reported in 2008 in
the Quarterly Journal of Economics. Even
though there are only slight differences
in talent within the cream of the crop,
firms want the best CEO. Competition
among large firms for really good CEOs
can lead to huge differences in income,
especially when some firms are super-
sized, the researchers contend.
An eye on outliers
Some researchers argue that understanding the whole economic picture
