Power laws

A friend’s wife is an astrophysicist, and so one day I found myself chatting with one of her astrophysicist friends, and the subject of what counts as a planet came up… I said something like “So, there’s basically a whole lot of junk floating around out there, and the planet size cutoff is kind of arbitrary, right?” He said “Well, it follows a power law….”

So what he meant was that there’s a whole lot of junk out there, and that the size distribution of that junk is such that, say, objects that are twice as large might be twice as uncommon. (Or ten times as uncommon – but whatever the proportion is, it persists over a range of sizes). Distributions like that look like straight lines when you plot them on exponential scales, like this:

So, although there are other reasonable definitions of where to make the planet size cutoff (like: large enough to have become spherical due to its own gravity), there’s no natural size gap that separates planets from smaller things.

My manager was trained as a condensed matter physicist before moving over to web search, and he said he was stunned when he first saw one of the graphs showing how some host attribute or another (like number of pages on a host) varied with number of hosts. It was a perfect power-law line, spanning almost ten orders of magnitude (like from 0 to 1000000000). He says that there is no way to get a power law like that in condensed matter physics – it’s basically the study of crystals, and crystals aren’t available in such a wide range of sizes. He wasn’t trained in cosmology, but guessed that cosmology probably does have power laws over that range.

So for ten-orders-of-magnitude-power-law-phenomena our candidates are: cosmology and the web. Are there any others?

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7 Responses to Power laws

  1. Do combinatorics count? Lots of real world examples here, such as:

    – Game branching factors (Chess for example)

    – Grammatical sentences with a length of n words.

    Found this example on the web: brightness in radiation equipment

    – Body size in terrestial life (http://www.pnas.org/cgi/content/full/102/1/140)

    – I assume many fractal things, by definition

  2. Tim Converse says:

    Hmm, yeah, I guess I was implicitly ruling out math, and thinking only of “natural” phenomena (like the web 🙂 ). I mean, probably you can come up with number-theory examples that span an arbitrary number of orders-of-magnitude….

    (I’m not sure that I follow the grammatical sentences example. What are the two quantities? If it’s numbers of words vs. number of grammatical sentences, then the latter is exponentially related to the former, not a power law…)

    Body length looks like a good one, spanning ten orders of magnitude: whales about 10^3 meters? bacteria about 10^-6?

  3. You’re right about the sentences. I was making a bad analogy from n-gram statistics, which do seem to follow a power law (‘Zipf’s law’), and probably span 10 orders of magnitude I have empirical data with character 2-grams spanning frequency counts from 1 to 5*10^6 using a corpus of 75*10^6 characters; it wouldn’t be hard to extend this to larger corpora.

    Here’s another: wealth (“Pareto Curve”) Gate’s net worth is 50 billion. Probably true for corporate valuation as well?

  4. Tim Haynes says:

    `whales about 10^3 meters?’

    I hope not. 10-20m is more than enough of a ball-park range.

    However, http://www.microbe.org/news/giant_fungus.asp is rather interesting if you want to consider large life-forms.

  5. Tim Haynes says:

    Additionally, there’s Mark Buchanan’s book _Ubiquity_: http://www.amazon.co.uk/exec/obidos/ASIN/0753812975/ that tries to present the whole power-law business as a “new science”. A bit grandiose, but note-worthy on the whole, anyway.

  6. Tim Converse says:

    Tim Haynes – Heh, OK. But they did record a blue whale 33.27 meters long, just about the geometric mean of 10^2 and 10^3….

  7. Tim, we’re just all your followers. New York Magazine goes off on the power laws of blogging today.

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