If the function decribes the probability of being greater than x, it is called a power law distribution (or cumulative distribution function - CDF) and is denoted P(>x) = x^-α .  Alternatively, if the power law describes the probability of being exactly equal to x it is called a probability density function (PDF) and is usually denoted p(x) = x^-α.   The PDF and CDF are obviously related:

+∞ ∫ x

p(x) dx

=  P(>x)

Because power laws usually describe systems where the larger events are more rare than smaller events (i.e. magnitude 8 earthquakes happen much less often than magnitude 2) α is positive. This ensures that the the power law is a monotonically decreasing function.

Why are Power Law distributions called 'Heavy-tailed'?

Many processes in nature have density functions which follow a bell-curve, or normal, or gaussian distribtuion.  Heights of adults pulled randomly from a human population are generally gaussian, as are sums of tosses from a weighted die.  You might think that this universal bell curve must indicate universal causes -- human heights must have something in common with weighted dice.   The real reason for this universality, however, is simple statistics. The Central Limit Theorem states that the sum of random variables with finite mean and finite variance will always converge to a gaussian distribution.  The mean is the average value of the random variable and the variance is a measure of how much individuals differ from that average. Therefore, the gaussian is a result of universal statistical processes and not similar causes.

Interestingly, if we relax the constraint that the variance and/or mean be finite (in other words, we allow arbitrarily large steps and/or don't require a characteristic size or mean) then the Central Limit theorem does NOT predict gaussians. Instead, it predicts a variety of sum-stable distributions (such as the cauchy distribution) which all look like power laws as x becomes large.   Gaussian distributions drop off quickly (large events are extremely rare), but power law distributions drop off more slowly.  This means that large events (the events in the tail of the distribution) are more likely to happen in a power law distribution than in a gaussian.  This is why we call power laws heavy-tailed.

This is a linear plot of the tails of a gaussian (blue), cauchy (magenta) and power law(red) distribution. For large x, the cauchy and power law distributions have larger probabilities.

This is a logarithmic plot of  the tails of a gaussian (blue), cauchy(magenta) and power law (red) distribution. The power law and cauchy are linear (on a log-log plot) for large x, while the gaussian drops off much more quickly.

What are the implications for complex systems?

As we have just discussed, large events happen more often than you would expect in systems that exhibit power law distributions.  This means that catastrophically large earthquakes, blackouts, stock market crashes or internet traffic bursts are more likely than predicted by gaussian models, IF those phenomena are correctly described by power laws.  (And most empirical data suggests that this is in fact the case.) Therefore it is important to understand how to model these systems taking into account ideas of infinite variance and possibly infinite mean.  We would like to look for mechanisms that could be used to generate these interesting data sets, (such as SOC, HOT, or allometries) as well as understand how to predict and improve the behavior of these systems.

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