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Recently I ran into a test using both zipf's and self-similar generated datasets. I followed the description from Jim Gray's paper on generating such datasets (Quickly Generating Billon-Record Synthetic Databases). In that paper it mentions:

It is commonly thought that self-similarity and the Zipf distribution are the same, or at least close. This misconception ... In particular, other statistics can yield very different results for the self-similar and Zipf distributions.

So my question is, from a statistic point of view, what is the actual difference between these two distributions intuitively? I am not a statistician, but (probably wrong) my thought is that they are all some power-law shaped distribution...

Just clarify: when I am talking about "the same", I am meaning that there is a (mathematic or statistic) way to bridge these two distributions... or one can be converted (mapped) into another...

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