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.