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Is this long tail distribution a power law?

October 11, 2007
The discussion of power law vs long tail came up on Chris Anderson’s Long Tail blog a couple of days ago. 

"Power law or not?" vs "Long-Tail or not?" are separate questions. If I understand Chris’ thesis, Long Tail is the idea that there is a significant population in the "not-hit" part of the distribution, usually of low volume in any rank, but continuing out to very high ranks.

The idea that there is a region where the distribution is essentially "scale free" seems like the key concept. If we start there, interesting questions include: Can we characterize this region with a power law? And (my favorite), what are the dynamics of the system where scale matters? This last question is at the core of the economics of the long tail businesses in general. For example, determining how inexpensive we make "find" and "acquire" activities corresponds with the "knee" in the distribution.

Scale-free is a misleading mathematical idea in that nothing in nature is actually scale free for all domains. For example, an absurdity of assuming scale-free in every domain WRT music or movie hits is that anything created has at least 1 fan (i.e. we don’t have an arbitrarily small hit)–this introduces scale and consequently, the region of arbitrarily small hits with less than 1 fan can’t be modeled by a power law. That’s a toy case, but illustrates how much scale matters.

Instead of including all the points in the power law fit, maybe we can look at the points up to the knee in power law model (it looks like x~3) and then try to understand what interesting dynamics shape the knee for x>3 with the assumption that some scaling has been introduced by cost, potential audience size limits, or whatever…

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