For me, "ubiquitous" has always been one of those words. Until recently, when my research led me to see the word so many times that eventually I no longer had any excuse to not remember it.
"Ubiquitous" is a fancy word for "found everywhere." And apparently scientists won't use a simpler word when a fancy one will do, so I've been reading a lot about how power laws are ubiquitous. So far, I've found it's a pretty accurate statement.
What's a power law?
I touched on this very briefly last week, but power laws are probability distributions that generate fat tails. Specifically, the probability law p(x) follows the form p(x)=Cx^-a, where C and a are constants. (The value of the constant a is the most important, as it determines how "fat" the fat tails will be.)
As a quick refresher, "fat tails" in a probability distribution mean that extreme events are far more probable (and remain more probable) than in systems with normal, Gaussian probability distributions, where the probability of extreme events quickly decreases to zero.
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| This image from a few weeks ago shows the fat tails in price fluctuation probabilities, but could be a representation of the probability of many different values |
To understand scale-free distributions on a more descriptive level, we can go back to Pareto's law of income distribution from last week. Sometimes called the 80-20 rule, or Pareto's Principle, Pareto found that 80% of the wealth of the world is held by only 20% of the people.
However, if we look at those top 20%, we'll see that 80% of their wealth is held by the top 20% in that group (and so forth). Hence, no matter what scale you are looking at it, the distribution rule remains the same. (Incidentally, income distribution follows a power law.)
So why are power laws considered ubiquitous?
I talked about this a bit in the comments section of my fat tails post from a few weeks ago, but I thought I'd do a quick recap here. Examples of power-law distributions in nature include, but are not limited to:
- Magnitude of earthquakes and avalanches
- Diameter of moon craters
- Intensity of solar flares
- Models of Van der Waals forces
- Volume of water flowing through river branches
- Initial Mass Function of stars
Power laws are also very prevalent in economics, showing up in:
- Intensity of economic recessions
- Income and wealth distribution
- Stock market indices (and price fluctuations)
- Population of cities
- Urban areas of cities
- Company size
- Number of books sold in U.S.
Heck, they even show up in disciplines that seem fairly unrelated, including:
- Emails received
- Frequency of word usage
- Frequency of family names
- Hits on websites
- Intensity of wars
It's important to note that these are just some of the applications of power law probability distributions. They really are-- to use my new favorite word-- ubiquitous, and we should perhaps begin to pay more attention to their interesting properties. (In particular with regards to the markets.)
Also interesting is that power laws are often found in systems that follow the self-organized criticality that I talked about a few weeks ago; so, these power law distributions are actually really relevant to my work with the sandpiles.
In related news: this week, I read a book called The Black Swan: The Impact of the Highly Improbable by Nassim Taleb. I didn't enjoy it very much (I'm hoping to do an in-depth comparison in a few weeks with more market-related books I've been reading, so I'll explain more then) but the basic premise was that extreme events occur more frequently than we would expect them to. Which, given the ubiquity (heh) of power laws and fat tails, shouldn't really come as much of a surprise.
Hope everyone is having a great week. Comments are welcome!
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