David Madore's WebLog: Small World experiment

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Small World experiment

The thesis was made popular by John Guare's play Six Degrees of Separation: the idea (very loosely stated) that two people in the world are connected by a “chain of relations” of length at most six. In other words, one has an acquaintance who has an acquaintance who has an acquaintance who has an acquaintance who has an acquaintance in common with the other. Where exactly the number six was found is uncertain, and of course it depends exactly what is meant by acquaintance, and also on whether we require for all pairs of people to be connected this way, or merely most pairs. But the order of magnitude is probably correct.

The naïve explanation is this: one has roughly speaking of the order of one hundred (direct) acquaintances, so at the second level (acquaintances of acquaintances) there should be ten thousand or so, and a million at the third level, a hundred million at the fourth, ten billion at the fifth and a trillion at the sixth; only this is wrong because each acquaintance's circle of acquaintances is not entirely disjoint, quite the contrary, there are many in common, and of course in the end there aren't a trillion people on Earth, but, still, the basic idea is there, that the number of acquaintances at level n should grow exponentially with n until it saturates when mostly everyone has been reached. Even if this is true, some further questions can be asked, for example: whether the linking chain can easily be found in practice (how would I proceed to find a connection between me and someone living in Central Asia whose name I have never heard of?), and whether if follows more or less geographical routes. Also, whether there exist “hubs”, or people who are acquainted to a very large number of people at small degrees, and who serve to shorten the way between two random individuals.

Dunan J. Watts, Peter Sheridan Dodds and Roby Muhamad from Columbia University have attempted to conduct a large-scale experiment on this: their findings have been published in Science's August 2003 issue (An Experimental Study of Search in Global Social Networks). I mention this mainly because I was part of the experiment (and I served to connect Pierre Senellart and my mother in order to get to a certain Monique Laroze-Travers).

Certain Web sites such as friendster.com or tribe.net have attempted to reproduce on a smaller scale the “six degrees of separation” phenomenon. Also nearby in the nootope are mathematical considerations on random graphs; for example: take N points, and for each of the N·(N−1)/2 possible (unordered) pairs of points, connect them with probability p, so as to form a “random graph” with N vertices, and then ask what is the probability (as a function of N and p) that this graph has diameter less than d (meaning that any two points can be connected by a chain of at most d edges); of course, many apparently similar questions could be asked.

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