the normalcy of power laws (part 3)

04Sep08

part 1 part 2

continuing ever on…

“why are power laws and fractals mentioned in the same breath”?

vaguely: power laws represent a statistical fractal.

fractals are pretty ubiquitous and popular, so i’m assuming every ones familiar with the concept. but the image that comes to mind when discussing fractals are mandelbrot sets, julia curves, koch curves, serpinski gaskets, etc. these are all visually stunning and exemplify the scale symmetry nicely, but they all have a feature that’s a bit contrived: they’re all highly regular.

“nature” is much messier than this. instead of getting an exact copy back again after rescaling, often you get something that’s similar looking, but not exact.

for example, take a graph with a power law degree distribution. maybe this graph was generated by considering the internet web graph. maybe its a graph generated via preferential attachment. maybe its just a graph generated by forcing a degree sequence to be power law and and connecting vertices together as best as you can.

all of these have a power law in degree distribution, but only statistically so. after a suitiable rescale (and this operation might not be as trivial as you think) the resulting, smaller, graph will still have a “look and feel” of the larger graph. this look and feel is the degree distribution, which is again, power law, as was the original graph.

so while images we might normally associate with the term fractal are rigorously symmetric and regularly structured, they can also include this much “messier” version which is the power law.

remember that probability distributions are very general. as a rule of thumb, features of the underlying distributions wash out and whats left is the main feature. in this case, the power law. so don’t be too fooled by the apparent simplicity of the power law. the underlying machinations of the distribution might be quite complex but still give rise to the simple power law. of course, sometimes that’s not the case and simple processes give rise to a simple power law, so always best to be careful. see here for some further discussion.

best to mention that this is most definitely my own take on the subject, so please don’t take the above as gospel truth.

should also be mentioned that there is a multi-fractal concept out there, which i know little about, but that touches on the above subjects. i’m not exactly sure what the connection to power law distributions (or more specifically levy stable distributions) these have, so if anyone has any suggestions on this, i would like to hear them.

for a video on multi-fractals see here. she obviously only has a tenuous grasp of the subjects she’s lecturing on, but the talk is still worthwhile. she completely fumbles all the questions asked her, so you should see here (which she recommends in the lecture) to fill in the gaps on the differences between power laws/multiplicative processes and log-normal distributions . i’ve found here for a deeper, but not very enlightening, paper on the subject. and of course, see here for why we should care in the first place about binomial cascades, fractional brownian motion and multi-fractal stuff (hint, finance) that are mentioned in that video lecture.

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