“A Cumulative Distribution Function (CDF) is the probability of any outcome less than or equal to x. Sometimes a CDF is defined with a different inequality; e.g. a Pareto distribution is a CDF F(x) which is the probability of any outcome *greater* than or equal to x.”

When we first do the inversion to get the CDF graph, the spike on the left isn’t really low users, it’s *all* users, since the y axis represents how many users have performed t or more taggings, and of course all users have performed 0 or more taggings.

But after we take the derivative to get the final PDF graph, then the spike on the left does indeed represent the large number of low-tagging users, and the “long tail” is then the few users with many taggings. However, the “area” is not meaningful, as it is with the original ranked graph.

Another interesting thing to note is that since we take the derivative, the tail becomes “less long” in the PDF. For example, if the original ranked histogram fits a t ~ 1/u curve (the way most long tails are illustrated), then the final PDF graph is n ~ 1/t^2, which is “taller and shorter tailed” than the original graph. I guess I should have showed this in the figure…

Inverting the graph, on the other hand - mea culpa for missing this yesterday - gives you a long tail with zero on the left; the long tail there actually represents the ’spike’ of heavy taggers, while the spike on the left is the ‘long tail’ of low users. This is in line with the suggestion I made, in rather more naive terms, back here. I’ve since got hold of some numbers & will post further when I get round to it.

For example, if there are 5 ranked users who performed 3, 2, 2, 2, and 1 taggings, this corresponds to 1 user who performed 3 or more taggings, 4 who performed 2 or more, and 5 who performed 1 or more. So the histogram {3, 2, 2, 2, 1} turns into the histogram {3, 0, 0, 2, 1}. Thus, the power law curve that is fit to the data would be the same.

So, as far as I can tell, nothing depends on unique ‘t’s (or unique ‘u’s, which must be unique to put them on the x axis to begin with — although I think you must have been referring to each u having a unique t value). Thanks much for the comment, and please pass on any further thoughts; the transformation issue above was a surprise, I’m sure there’s other issues here…

saying that the u-th user performed t taggings is equivalent to saying that u users performed t or more taggings … if we invert this graph and turn the number of users performing t or more taggings into the percentage of such users, we arrive at a probability distribution

This only really works if you can assume unique ‘u’s - and by implication unique ‘t’s. I’ve been looking at some real figures on inbound links. If you order them by ranking, they do show something like the familiar downward curve (5389, 3849, 3309, 3211, 3183, 2542…) But here are ‘rankings’ 90-104:
816, 816, 812, 811, 792, 789, 786, 779, 779, 778, 778, 767, 758, 753, 753
And so it goes on. Further out, of course, there are many more ‘duplicates’; over 2,000 of them in the case of the lowest non-zero value (one link, in other words). The simple ‘long tail’ histogram could represent these readily enough - as a descending mountain range with a lot of plateaux - but the position of any ‘column’ within one of those plateaux would be meaningless. In effect the X axis would carry information at some points but not others.

Or am I missing something?