<disclaimer>I do NOT express anything but my deepest respect for the thinkers I quote in this post. They are infinitely smarter and wiser than I will ever be: I am dust beneath their feet. But this is internet, so even the likes of me needs to edit and comment, on the Great and the Good more than on the guy in the next cubicle. So, Ladies and Gentlemen, without further ado I give you my own Top 3 fun math errors made by internet gurus!</disclaimer>
Truth be told, these formulae do not compute a network’s value. A ten-nodes network would be worth 1024… what? Dollars? Peanuts? Lottery tickets? Certainly not. The answer is that 1024 is simply the number of subgroups theoretically possible in a graph of ten nodes, each linked to the other nine. A better formulation would be the one used by David Reed himself: the value of a group-forming network increases exponentially, in proportion to 2 to the nth power. In addition to this, the formulae used by Rheingold are just plain wrong: ten nodes have 10x(10-1)/2 = 45 possible links (not one hundred), and the number of possible subgroups is 2 to the tenth power minus ten minus one, hence 1013 and not 1024.
Second prize: one of my favourite authors, Clay Shirky. In Here comes everybody – a great book – Clay correctly describes the equilibria in the ultimatum game. Then he relates what happens when you run ultimatum game experiments in the lab:
[…]In practice, though, the recipient would refuse to accept a division that was seen as too unequal (less than a $7-to-$3 split, in practice) even though this meant that neither persone received any cash at all. Contrary to classical economic theory, in other words, we have a willingness to punish those who are treating us unfairly, even at a personal cost, […] [p. 134]
This is not exactly an error, but it contains an omission so huge as to jeopardize Clay’s conclusion, namely that these experimental results have a number of well-documented methodological problems and should be taken with extreme care. The main problem is that results are thought to depend not only on the split, but also (and crucially) on the absolute value of the prize. If you play ultimatum with a billion dollars, and player 1 offers you a hundredth of that, are you sure you re going to turn 10K down for the pleasure of taking 990K away from him? The matter is open for debate… whereas Clay dismisses it as settled.
Finally, a special award for the nicest attitude goes to Chris Anderson, that guru of gurus, who has recently devoted a very clever post to the risk inherent in generalization.
But now we’re entering a world of unbounded sets, and it’s messing up our language habits. What is the number of “writers” in the world in an age of blogs, the number of “photographers” in an age of Flickr and cameraphone or “videographers” in the age of YouTube?
Pure guru wisdom. The problem is in the title of the post, “Thirteen words that lose their meaning when the denominator approaches infinity”. The words in question are locutions like “most” (as in “most blogs”) or “average” (as in “the average Youtube video”). As Chris’s readers have not failed to note, it’s certainly true that saying stuff like “most blogs have very few readers” is meaningless, because it attempts to describe the blogs phenomenon through a mean which is just not representative when the population is described by a power law distribution. But this has nothing to do with denominators approaching infinity. A phrase like “For most of time, humans didn’t and won’t exist” makes total sense even if the denominator (the universe’s age at the time of the Big Crunch) is as close to infinity as it gets. After a volley of comments making this point, Chris adds a comment of his own:
Yes, you can count me among those who sometimes use mathematical language sloppily to make a point. But at least I admit it!
How can you not love the guy?