Well, I agree with Carter too; if I never hear anyone say "small world"
again, it won't be too soon.  (OK, fine, I say it myself sometimes, but
you know what I mean.)  But I don't agree that it's not important, for
two reasons:

1) Sure, the probability of any *particular* short path getting activated
may be small, but what Milgram's experiment failed to reveal is that
there can be very many short paths between two people, not just one.
Speaking totally off the top of my head, I'd guess that for a sparse
graph well above the giant component transition, the number of
paths shorter than a constant times log of the graph volume is probably
exponential in the graph diameter, and hence algebraic in the graph volume.
That could give you a *lot* of short paths.  So even if the probability of
activation of any particular path is small, the probability that at
least one of them gets activated could be quite large.

2) There are some kinds of activation that don't require you to do
anything deliberate.  An obvious and highly salient example is the
spread of disease.  This depends on the physical contact network, rather
than the acquaintance network, of course.  What if Ed Peay had the flu
and gave it to his academic colleague, who in turn passed it on to the
Queen, and as a result she couldn't give her Christmas speech on the
telly this year? That would make a difference.  OK, a small difference.

More to the point, the only reason why a flu epidemic can sweep the
country in six weeks, rather than six years, is because the diameter of
the physical contact network is small.  This has really huge public
health implications of course.  Although it's also not a new result.

Mark Newman.

> >I couldn't agree more with Carter Butts. Another question, however,
> which
> >doesn't appear to receive much attention is whether the "Small World"
> >phenomenon matters very much.  In most of his work, Milgram liked to
> create
> >a splash by producing a flashy and counterintuitive phenomenon and then
> >move on.
> >The point is whether the links in the chain which appears to tie two
> people
> >together are successively activated outside a deliberate experimental
> >context.  I happen to be two steps from Queen Elizabeth (through a
> former
> >academic colleague who became a politician and then a high government
> >official in London) and hence three steps from most world leaders.  The
> >practical significance of this is nil.
> >Ed Peay