Complexity and Social Networks Blog

Finding data out there that fits the power law is interesting -- but it is interesting in the same sense that learning that the probability that 2 or more people in a room of only 24 randomly selected people will have the same birthday is greater than 0.5. Its a neat curiosity. But can you explain exactly how learning that something fits a power law probability distribution is useful in a scientific sense? If correct, it will, by definition, tell you the probability distribution of the next item (stock, blog, etc.), but will it necessarily tell you anything (or anything important) about the social, political, economic, or other mechanism that gives rise to the observed power law histogram? This does not seem so obvious to me since different underlying mechanisms can give rise to the same aggregate distribution. So why should anyone make a fuss every time a variable seems to fit the power law (whether a distribution _looks_ like a power law statistially fits a power law density is another issue that we can skip over for now).

To echo this point, is there anything intrinsically more interesting about saying that something fits a power law distribution as compared to a normal distribution?

It does seem to me that sometimes it is useful when there is some presumption that the distribution looks different than a power law to say that it is power law distributed. By analogy, this hanukah I was playing dreidel with my kids. The dreidel has four sides, and one assumes that each side has an equal chance of coming up. However, out of a 100+ spins, the gimmel only came up once or so, meaning that it was rather likely we had a weighted dreidel-- good to know, because the game went on forever as a result. For a social science example, Matt Hindman's work on "Googlearchy" is a nice application along these lines, where he finds that for various issue areas a fairly small number of websites dominate. This is different from the factual assertions of some that the Internet would open the doors to a larger number of perspectives. But that's different from asserting that a power law says something very specific about the process that produced that distribution. Indeed, the apparent omnipresence of powerlaws (firm sizes, city sizes, word frequency in Ulysses, degree distribution of websites, forest fires) suggests many processes lead to power laws.

I would suggest that for a given model to produce a power law distribution similar to that observed in real world data is thus a weak test of that model. It is one "peg" of a model to data, but to validate a model should require multiple pegs. (A broader discussion of the issue of model validation will wait for another entry.)

Zipf's law (and relatives) is a partial exception to this, because it specifies not just a power law distribution, but the particular shape of the distribution, which is more powerful. I do not know of a definitive explanation for why city sizes in different countries almost universally fit the same distribution (except, perhaps, the stochastic growth model of Simon's). Again, however, there may turn out to be many ways to produce Zipf's law. (If readers know of recent developments along these lines, please post.)

References:

www.princeton.edu/~mhindman/googlearchy--hindman.pdf

Simon, H.A. (1955). On a class of skew istribution functions. Biometrika, 42, 425-440.

Human Behavior and the Principle of Least Effort
GK Zipf - 1949 - Addison-Wesley