2012 : WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?

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Mathematician, Computer Scientist; CyberPunk Pioneer; Novelist; Author, Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me About Ultimate Reality, the Meaning of Life, and How to be Happy

Inverse Power Laws

I'm intrigued by the empirical fact that most aspects of our world and our society are distributed according to so-called inverse power laws. That is, many distribution curves take on the form of a curve that swoops down from a central peak to have a long tail that asymptotically hugs the horizontal axis.

Inverse power laws are elegantly simple, deeply mysterious, but more galling than beautiful. Inverse power laws are self-organizing and self-maintaining. For reasons that aren't entirely understood they emerge spontaneously in a wide range of parallel computations, both social and natural.

One of the first social scientists to notice an inverse power law was George Kingsley Zipf, who formulated an observation now known as Zipf's Law. This is the statistical fact that, in most documents, the frequency with which a given word is used is roughly proportional to the reciprocal of the word's popularity rank. Thus the second most popular word is used half as much as the most popular word, the tenth most popular word is used a tenth as much as the most popular word, and so on.

In society, similar kinds of inverse power laws govern society's rewards. Speaking as an author, I've noticed, for instance, that the hundredth most popular author sells a hundred-fold fewer books than the author at the top. If the top writer sells a million copies, somone like me might sell ten thousand.

Disgruntled scribes sometimes fantasize about a utopian marketplace in which the naturally arising inverse power law distribution would be forcibly replaced by a linear distribution, that is, a sales schedule that lies along a smoothly sloping line instead of taking the form of the present bent curve that starts at an impudently high peak and then swoops down to dawdle along the horizontal axis.

But there's no obvious way that the authors' sales curve could be changed. Certainly there's no hope of having some governing group try and force a different distribution. After all, people make their own choices as to what books to read. Society is a parallel computation, and some aspects of it are beyond control.

 The inverse-power-law aspects of income distribution are particularly disturbing. Thus the second-wealthiest person in a society might own half as much as the richest, with the tenth richest person possessing only a tenth as much, and—out on in the burbs—the thousandth richest person is making only one thousandth as much as the person on the top.

Putting the same phenomenon a little more starkly, while a company's chief executive officer might earn a hundred million dollars a year, a software engineer at the same company might earn only a hundred thousand dollars a year, that is, a thousandth as much. And a worker in one of the company's overseas assembly plants might earn only ten thousand dollars a year—a ten-thousandth as much as the top exec.

Power law distributions can also be found in the opening weekend grosses of movies, in the number of hits that web pages get, and in the audience shares for TV shows. Is there some reason why the top ranks do so overly well, and the bottom ranks seem so unfairly penalized?

The short answer is no—there's no real reason. There need be no conspiracy to skew the rewards. Galling as it seems, inverse power law distributions are a fundamental natural law about the behavior of systems. They're ubiquitous.

Inverse power laws aren't limited to societies—they also dominate the statistics of the natural world. The tenth smallest lake is likely to be a tenth as large as the biggest one, the hundredth largest tree in a forest may be a hundredth as big as the largest tree, the thousandth largest stone on a beach is a thousandth the size of the largest one.

Whether or not we like them, inverse power laws are as inevitable as turbulence, entropy, or the law of gravity. This said, we can somewhat moderate them them in our social context, and it would be too despairing to say we have no control whatsoever over the disparities between our rich and our poor.

But the basic structures of inverse power law curves will never go away. We can rail at an inverse power law if we like—or we can accept it, perhaps hoping to bend the harsh law towards not so steep a swoop.