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On Mon, 2003-01-27 at 10:29, Martina Morris wrote:
> Intuitively, though, the scale free property implies tail behavior that is
> physically impossible in social contact networks.  Indeed, it implies that
> there is a small, but non-zero, probability that someone can have more
> contacts than there are members of the population.  And this property
> underlies many of the "newsworthy" analytic results that follow -- i.e.,
> that there can be no effective interventions for sexually transmitted
> diseases.

This is a good point.  Further, Pareto-like distributions have
properties in the _lower_ tail which are also clearly false for most
interpersonal networks.  If the world were truly scale-free, then the
population mode would have to consist of the minimum-degree state...more
people would have to have no friends, for instance, than any other
number.[*]  Furthermore, the next most common degree value would have to
be 1, then 2, etc., etc., etc.  This is trivially true for any
monotonically decreasing degree distribution, and is quite incompatible
with most substantively interesting interpersonal relations.  While one
might be able to write-off the limiting behavior in the upper tail as an
approximation, the lower tail behavior accounts for much of the
probability mass of the distribution...thus, it's very difficult to
ignore!  No matter how you slice it, it doesn't fit.

With respect to the question of degree distribution, Elisa Bienenstock
and I have been looking at this too (though we may be too late, which is
OK :-)).  Based on purely preliminary results, the one class of graphs
I've tested for which the power-law could be a vaguely plausible model
is a set of semantic networks.  My guess is that, when the dust settles
on all this, the consensus will be that these models are reasonable for
certain kinds of citation networks and concept networks, but not for
most substantively interesting interpersonal networks.  Time will tell.

-Carter

[*] Unless everyone has at least one friend, in which case 1 would
become the mode.

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