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To add a few words to Tom's comments: The ERGM (exponential random graph model), referred to as "p*" here, is a way to parameterize arbitrary random graph models. In principle it can represent any probabilistic structure via appropriate choice of the network statistics in the model. However this is less than it seems, as determining the statistics to represent a given structure can be difficult and non-parsimonious.
The fundamental conceptual difference between the ERGM and p2 modeling perspectives is the use of fixed effects in the former and random effects in the latter. The choice of perspective is similar to that in other statistical settings (e.g., ANOVA and hierarchical models). Roughly: If the characteristics of these particular individuals in the network are of primary interest then a fixed effects (ERGM) approach is usually preferable. If the social process that generated the individual characteristics is of primary interest then the random effects (p2 and generalizations) is often preferred.
In this sense the choice between ERGM and p2 perspectives is based on the primary scientific questions. These perspectives aside: the p2 and generalizations can be viewed as a further hierarchical specification of an ERGM, and the model forms at the highest level can be very similar. Most statisticians see the two as variants on a theme and mix random and fixed effects as appropriate.
Regards to all,
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
internet: [log in to unmask]
From: Social Networks Discussion Forum
[mailto:[log in to unmask]] On Behalf Of T.A.B.SNIJDERS
Sent: Thursday, December 15, 2005 12:30 AM
To: [log in to unmask]
Subject: Re: Does p* contain p2?
> ***** To join INSNA, visit http://www.insna.org *****
> Dear Kim and social networkers,
> p* and p2 do not contain each other either way. p* has an in
> principle unlimited possibility for network effects (and p2
> doesn't); the practical limitation resides in the
> estimability of these effects (cf.
> Snijders, Pattison, Robins, Handcock, New specifications for
> exponential random graph models; To be published,
> Sociological Methodology; downloadable from
> p* now is often referred to as ergm (exponential random graph model).
> On the other hand, p2 has random effects for representing
> between-actor differences (and p* doesn't). However,
> in/out/mixed-twostars effects in
> p* have to some degree a similar functionality; but the
> interpretability of unexplained variances at the actor level,
> which is a feature of p2, is not available in p*.
> The focus of p2 is on testing effects of covariates, which
> can be actor-bound or dyad-bound, and which can interact with
> reciprocity; while controlling for differential actor
> tendencies to sending and receiving ties, and for
> reciprocity. The focus of p*/ergm is on structurally modeling
> networks (which may include covariate effects).
> For p2, better estimation methods have been an advantage;
> recently there have been advances in estimation methods for
> both p*/ergm and p2, and p*/ergm now is in much better shape
> than with the earlier pseudolikelihood methods. I think that
> with recent and almost finished work on multivariate and
> multilevel p2, p2 retains an advantage concerning
> availability of good estimation methods across a range of
> model extensions. (The new January 2006 release of Stocnet
> will contain a new p2 version with multivariate and
> multilevel options.) p*/ergm has the advantage of a much
> better representation of structural network effects.
> Best regards,
> Kim Se Kwon wrote:
>> ---------------------- Information from the mail header -----------------------
>> Sender: Social Networks Discussion Forum <[log in to unmask]>
>> Poster: Kim Se Kwon <[log in to unmask]>
>> Subject: Does p* contain p2?
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>> I've read some papers about p* and one paper about p2.
>> If I've read correctly, Equation 4 of reference 2 ;extends ;first p*
>> (equation 1) to contain covariates. And p2 is 'the extension of p1
>> that doesn't assume dyadic independence and accounts for nodal & dyadic covariates.'
>> Does p* (equation 4 in reference 2) contain p2? Because p* has p1 as
>> submodels, p2's model equation is very similar to p*'s. If not, what is the unique features of p2?
>> Thanks in advance for any comments.
>> --- Equations ---
>> Equation 1 of Reference 2 ; : ; P(X = x) ;= exp(theta' * ;t(x)) /
>> c(theta) Equation 4 of Reference 2 ; : ; P(X = x) = exp(vec(x)' Z beta
>> + sum of alpha_k * d_k(x) + theta' * t(x)) / c(alpha, beta, theta) ;
>> (d_k : the number of individuals with exactly k links, Z : exogenous dyadic
>> --- References ---
>> 1. Logit models and logistic regressions for social networks : I An
>> Introduction to markov graphs and p*. Stanley ;Wasserman...
>> 2. statnet: An R package for the Statistical Analysis and Simulation
>> of Social Networks, Mark S. Handcock...
>> 3. p2: a random effects model with covariates for directed graphs. Marijtje A. J. van Duijn...
>> Best Regards
>> Se Kwon, Kim
>> KAIST (Korean Advanced Institute of Science & Technology) Mathematics
>> Student http://math.kaist.ac.kr
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