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Clauset, Aaron, Cosma Rohilla Shalizi, and M. E. J. Newman. 2009.
Power-law distributions in empirical data. arXiv:0706.1062v1
On 2/26/2012 11:33 AM, James Holland Jones wrote:
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> Hi Barry,
> At the risk of coming across as self-serving, I will answer this question since Mark Handcock and I have done quite a bit of work on this topic.
> They're generally right. I don't think that any human network that isn't mediated by technology actually has a power-law degree distribution. Maintaining relationships is too costly. I'm also skeptical about the power-law status of most socio-technical networks, but I keep an open mind. Not surprisingly, they missed (or ignored) some work by non-physicists. Mark Handcock and I noted the statistical weakness of all the power-law degree distribution work in a series of papers and provided an alternative approach (ML estimation of actual probability models rather than OLS fitting of the double-log survival plot).
> Jones, J. H., and M. S. Handcock. 2003. An Assessment of Preferential Attachment as a Mechanism for Human Sexual Network Formation. Proceedings of the Royal Society of London Series B-Biological Sciences. 270:1123-1128.
> Jones, J. H., and M. S. Handcock. 2003. Sexual Contacts and Epidemic Thresholds. Nature. 425:605-606.
> Network Formation. Proceedings of the Royal Society of London Series B-Biological Sciences. 270:1123-1128.
> Handcock, M. S., and J. H. Jones. 2004. Likelihood-Based Inference for Stochastic Models of Sexual Network Evolution. Theoretical Population Biology. 65:413-422.
> Handcock, M. S., and J. H. Jones. 2006. Interval Estimates for Epidemic Thresholds in Two-Sex Network Models. Theoretical Population Biology. 70:125-134.
> In the first two, we show that the variance of the power-law exponent is greatly under-estimated by OLS (and also suggest that the same marginal power-law degree distribution can represent very different networks from an epidemiological perspective). Our estimates (and 95% CIs) of the scaling exponents show that if power laws do fit sexual network data, they fall off so fast (because the exponents a much greater than 2) that all the weird stuff about epidemics on power-law graphs (e.g., no epidemic threshold level of transmission) don't actually apply. In the 2004 paper, we show that the best-fitting model is generally not a power law when we test against multiple models. It's a shifted negative binomial. Longish tail but certainly not fat. Re Stumpf& Porter's point about the weakness mechanistic sophistication, this can be seen in the fact that so a wide variety of models -- with differing underlying stochastic processes leading to the marginal degree distribution -- fit!
> about equally well. The OLS fit to the logged survival plot provides no insight into the behavior driving the evolution of the network.
> There is also a very good critique of power laws (also missed) that doesn't specifically address networks. He suggests that most power laws arise because of truncated data samples drawn from exponential-like distributions, especially lognormals.
> Perline, R. 2005. Strong, Weak and False Inverse Power Laws. Statistical Science. 20 (1):68-88.
> I will present very detailed data gathered using wireless sensor networks on the degree distribution of school contact network at Sunbelt this year and show that it is not even remotely power-law. The weighted degree distribution is very well fit by a normal mixture. Indeed, I think that a mixture interpretation actually works for all the supposedly power-law degree distributions of social networks (e.g., sexual networks): a power law is a highly parsimonious way to fit a widely separated mixture. A negative binomial/Poisson mixture, corresponding to a high-risk/main population, has at least 5 parameters (with the possibility of a shift parameter for the nb). A power-law like a Yule or Riemann distribution has one. With the small samples we get from even nationally representative studies like Sex in Sweden or NSHLS, there isn't enough statistical power to show that the heavily parameterized mixture model actually fits better even though the stochastic mechanism may make mo!
> re sense behaviorally.
> Allometric scaling laws are very well supported empirically. I know that there is a fair degree of controversy over their mechanistic basis however (namely, fractal geometry of biological transport networks within organisms) that is not addressed in this brief note to Nature. I don't know the current status of that debate.
> Hope this helps...
> On Feb 26, 2012, at 6:42 AM, Barry Wellman wrote:
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>> Michael Stumpf& Mason Porter.
>> Scienc 10Feb 2012
>> Strongly critcizes power law modeling.
>> Comments by more knowledgeable than me?
>> Barry Wellman
>> S.D. Clark Professor of Sociology, FRSC NetLab Director
>> Department of Sociology 725 Spadina Avenue, Room 388
>> University of Toronto Toronto Canada M5S 2J4 twitter:barrywellman
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> James Holland Jones
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Distinguished Scholar, Cohen Center for Modern Jewish Studies
Visiting Research Professor Sociology
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