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Following up:

----- Original Message -----
From: "Martina Morris" <[log in to unmask]>
>
> 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.

Many of the novel "scale free" conclusions rest on properties of their mathematical
approximations to characteristics of the network. As Martina notes the approximations have some unrealistic features. However, almost all models do. The question is: are the claims robust to (even small) deviations from the assumptions.

This is why the process models are important. One mechanism that underlies the (so-called) power law models is something they call "preferential attachment" (i.e., roughly probability of ties proportional to the degree of the nodes, and random mixing otherwise, this would be called a model of
"proportional mixing" by mathematical epidemiologists). This kind of process has testable implications for an observed degree distribution, When we test them against sexual behavior data the support for the "newsworthy" conclusions appears to be slim(for a glimpse see, e.g., www.csss.washington.edu/Papers/wp23.pdf). Keep in mind that even if the degree distribution implications can't be rejected we still need to assume that mixing (by degree) is random. This has not yet been tested, but seems exceedingly unlikely.

In the end, for the "newsworthy" implications to be true, the central issue is not if the networks are "scale free", power law, approximately follow preferential attachment or otherwise, but what can be said about network characteristics that matter for disease transmission. And it's a lot more difficult to say anything about this if expedient simplistic assumptions do not hold.

Mark

Carter Butts said:
>
> 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.

-------------------------------------------------
Mark S. Handcock
Professor of Statistics and Sociology
Department of Statistics, C014-B Padelford Hall
University of Washington, Box 354322 Phone: (206) 221-6930
Seattle, WA 98195-4322. FAX: (206) 685-7419
Web: www.stat.washington.edu/~handcock
internet: [log in to unmask]

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