the normalcy of power laws (part 2)

04Sep08

last entry we learned about how power laws come about so naturally. lets continue, shall we?

“what does ‘scale-free’ mean and why are power laws free of scale”?

zoom out from a power-law distribution, rescale and you will get the same distribution back.

so, lets back up a bit:

symmetries are pretty basic. its a way of describing repeated patterns in nature.

take a wallpaper pattern, for example. take it off the wall, shift it by a certain amount and replaster. you’ll find it looks exactly the same. this is translational symmetry.

take a daisy and twirl it around. looks the same. this is rotational symmetry.

look at a tree, then look at one of its branches stuck in the ground. not exact of course, but to a good approximation they look very much alike. the branch being a mini version of the tree. this is scale symmetry.

scale free means that there is a scale symmetry. basically, if you zoom out (or in) of a picture and it looks the same, then its scale-free. the term scale-free comes from the fact that since you get the same picture back again after you zoom in or out, you can’t tell what the characteristic length scale of the picture was, since it didn’t have on in the first place.

some examples will help: consider a square on a piece of paper with a certain length to its side. zoom out by a certain factor. measure the sides and you find that the sides have shrunk. no mystery, you zoom out and the square shrinks. this example is pretty loose but you can see that the “system” in this case has the square’s side as its characteristic length scale and it can be determined by measurement, even after our zooming operation.

now do the same for the coastline of britain, a koch curve, a tree and its branch: you find in each of these cases that its difficult to know whether you zoomed in or out since they all look the same after you’ve done your zooming operation. each of these systems has a scale symmetry, they look the same after rescaling, and are thus scale-free.

now, zoom out of a power law distribution and you’ll find the same distribution again. so power laws are have a scale symmetry and thus are scale free.

in symbols:

$p(x) \propto x^{-\alpha}$
$\to p( x/a ) \propto (x/a)^{-\alpha} = a^{\alpha} x^{-\alpha}$
$\to p(x/a) = a^{\alpha} p(x)$

meaning, besides the factor in front, rescale by a, get a power law distribution back out again.

to see why the above condition is exceptional, consider if $p(x) \propto e^{-x^2}$ or $p(x) \propto e^{-x}$ (two very common distributions). in both of these cases, $p(x/a) \ne a^{\alpha} p(x)$ . both of those two distributions would be said to have characteristic length scale and thus wouldn’t be scale-free.

the normalcy of power laws (part 2)

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