Ode to Power Laws

Thursday, March 31, 2011

Anyone who knows me knows that I love power laws. What other distribution even comes close? The normal? Boring. Poisson? Too fishy. Hypergeometric? Ok, that one’s pretty cool, too. But still, power laws reign supreme in my mind.

Evidence of the mystical power of power laws is found by simply looking at references to “power laws” themselves. Parsing the 600,000 papers on the ArXiv,[1] one finds that the number of “power law” mentions in an article follows a power law! (Kudos to Aaron Clauset, Cosma Shalizi, and Mark Newman for the record-breaking 310 mentions of “power law” in their paper, “Power-Law Distributions in Empirical Data.”[2])

Now, some of the power law haters might look skeptically at the graph above, crankily insisting that proper statistical tests be applied before brazenly asserting the data follow a power law. To them I say, pshaw! I think with my gut. Others might say, so what?! — Do you call your BFF every time you see a normal distribution? There are so many ways in which power laws might arise, they argue, what have you learned by simply identifying one? My response: What would you learn if you found a lost Beatles track? Nothing — but it would be beautiful. Science is Truthiness, and Truthiness is Beauty.

As prose alone cannot adequately convey my fondness for power laws, I’ve written a haiku:

scale invariance
a universality
laws full of power

To ensure that the search for power laws continues, I propose that April 1st be national power law day. Go forth, and find yourself a power law!

Bonus Puzzle

30 ordinary dice (see below) are placed in a pitch dark room with the sum of the upward faces totaling 70. How can you divide the dice into two piles such that the sums are the same in each pile?

From Mathworld: “The most common type of die is a six-sided cube with the numbers 1-6 placed on the faces. For the six-sided die, opposite faces are arranged to always sum to seven. This gives two possible mirror image arrangements in which the numbers 1, 2, and 3 may be arranged in a clockwise or counterclockwise order about a corner. Commercial dice may, in fact, have either orientation.”

In other puzzle news, Dan Reeves is the new puzzle editor for SIGecom Exchanges. So if you want something more hardcore, check out his new puzzle in the current issue: Baffling Raffling.

Congrats to Lev Reyzin, who solved what I thought was a pretty challenging brainteaser in a mere 30 minutes! (See his solution below.) The puzzle is our non-googleable version of one of my favorites: You’re in a pitch dark room with 20 coins, exactly 10 of which are heads up. How can you divide the coins into two piles having equal numbers of heads? This is really a great problem, so if you haven’t yet solved the dice version, give this one a shot (it’s a bit easier). Good luck!

And now for some more power law antics. Duncan Watts asked, “Isn’t there some way to combine power laws and puppies? That would be even awesomer.” Well, I couldn’t agree more! So here’s some puppy power law data porn. And, yes, those registration statistics are real—puppy registrations do indeed follow a power law!

Footnotes

[1] Papers on the ArXiv are available for bulk download from Amazon S3.

[2] Counts are for case insensitive match of /power[- ]?law/ anywhere in any LaTeX document on the ArXiv.

Illustration by Kelly Savage

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• http://lev.reyzin@gmail.com Lev Reyzin

Awesome puzzle! (I don’t want to post a solution so quickly.)

I want to add the uniform distribution to your list. It’s nice, simple, and unrealistic — perfect for theory :)

• Kelly

Feel the dice–like braille, add the numbers, and sort :)

• http://5lbs.org/ Mathew Woodyard

This post is full of win. You had me at the haiku. <3 power laws.

:))

• http://www.cs.unm.edu/~aaron/blog/ Aaron

I once also tried to make power-law poetry. The result was gruel, but here it is:

Three Power Laws for the Physicists, mathematics in thrall,
Four for the biologists, species and all,
Eighteen behavioral, our will carved in stone,
One for the Dark Lord on his dark throne.

In the Land of Science where Power Laws lie,
One Paper to rule them all, One Paper to find them,
One Paper to bring them all and in their moments bind them,
In the Land of Science, where Power Laws lie.

Something about my paper having the current most mentions of “power law” in it, of all the power-law articles on the arxiv, fills me with satisfaction.

Aaron, I love how you have power law poetry ready to go on a moment’s notice!

• http://www.levreyzin.com Lev Reyzin

Sharad has informed me my self-imposed blackout period can end, so I’ll post my solution to the puzzle:

Take any 10 dice and say they sum to n. Flip them; now they sum to 70 – n (b/c sum of opposite sides is 7 and there are 10 dice). These make the first pile. The remaining 20 dice make the second pile, which also sums to 70-n (b/c the total was originally 70).

Lev, very nice!