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Re: Comparison of QAP and ERGMs (really, p*)

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Fri, 30 May 2008 12:53:52 -0400

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 ```***** To join INSNA, visit http://www.insna.org ***** What follows is a little thing that Pip Pattison and I wrote, and posted to SOCNET, almost 10 years ago. Tom S., Garry R., and Dave K. have written recent replies to the question mentioned in the Subject, first posed by Param last week. This note should help clarify what statistical models for networks are all about ... ******************************** QAP & p* August, 1999 Some time ago (May 12), Balazs Vedres raised the question on SOCNET of the relationship between p* and QAP, and prompted some very useful discussion. Here (belatedly!) we offer a few additional thoughts on what seems to us an interesting (and open) question. As Carter Butts observed, the key issue is what makes the most sense in a given context, conditional on what we believe about the nature of the phenomenon at hand. The essential problem in choosing a particular statistical approach is that there is no consensus on what to believe about a phenomenon of interest, but rather a multitude of possible stances to consider. And this latter point is true whether you think about an approach based solely on conditional uniform random graph distributions (of which QAP is an example) or on multivariate p* distributions (or, of course, on both). The stochastic model underlying QAP (usually) is the conditional uniform (multi)graph distribution which conditions on (fixes) all node-label-independent properties of each of one or more graphs (e.g the graph of familial associations and the graph corresponding to the formal hierarchy in Carterís example). In each case, all graphs that are isomorphic (as unlabelled graphs) to the observed graph have equal (non-zero) probabilities. Graphs of different "kinds" are (usually) assumed to occur independently. [QAP stands for "quadratic-assignment programming", and was first used in relational data situations by Larry Hubert and Frank Baker. Actually, the standard social network application of QAP is to simply test the "conformity" of two relations using a permutation test, an old idea from statistics (see Wasserman, 1987, Psychometrika, where permutation tests are compared to conditional likelihood ratio tests of the parameters from p1). The important thing to note is that the model underlying QAP is but one of many multigraph distributions that can, and have, been used to study the associations among a set of network relations. There are many other conditional uniform multigraph distributions, including multigraph versions of the well-known U|MAN distribution which conditions on the dyad census. There are also distributions based on a variety of blockmodelling ideas. One could use any of these distributions to study the association between a particular graph and an hypothesized graph, or between two observed graphs (often the goal of QAP, and discussed in Chapter 16 of Wasserman & Faust 1994). One can also examine conditional associations among graphs, conditioning, for instance, on other graphs or blockmodels. We have discussed the choice of conditional uniform multigraph distributions for such situations in Pattison, Wasserman, Robins and Kanfer (1999), which was published in the Journal of Mathematical Psychology. There, we argue that the choice of distribution is not necessarily straightforward and that it is generally made in the context of a broader theoretical argument about likely or possible structural issues. This argument is likely to specify (although possibly only in part) the set of relational features on which a particular multigraph distribution should be conditioned. (Of course, it is always wise, maybe even necessary, to condition on any design constraints or well-supported hypotheses about processes underlying generation of the data, such as fixed degrees, but there may be other more contentious factors to consider). In view of this lack of specificity, we argue that the assessment of hypothesized relationships is likely to be more informative if conducted using a carefully selected series of random graph distributions (possibly nested in terms of the properties on which they condition) rather than just one. Another multivariate graph distribution is p*. For p* models, probabilities are assigned to every member of a space of multigraphs according to a parametric multigraph model. The model is derived from a set of assumed conditional dependencies among possible network ties. These assumptions characterize structural assumptions about local network processes giving rise to the network observations. In a given empirical situation, the parameters of the model are estimated from observed networks. Here too, though, there are many possible assumptions of interest, and a similar problem arises with respect to choice of specific distribution (although we would argue that the multivariate version of a homogeneous Markov graph [Frank & Strauss, 1986, JASA] is a good starting point). Overall, the choice of statistical models for networks is often complicated. Further, the set of possible choices can be quite large. This choice confronts us with the need to develop more explicit assumptions about the processes underlying network structures. Our personal observations on the strengths and weaknesses of working with conditional uniform versus p* random multigraph distributions are as follows:          First, p* distributions are often somewhat easier to work with          than conditional uniform random graph distributions --          the framework is much more general.          Second, in their current manifestations, p* models are particularly          well-suited for capturing effects of local processes in networks.          They yield probabilistic models for global network structures in          terms of local structural tendencies.          But, third, conditional uniform random distributions such as the          one used by QAP still offer the more accessible          approach for conditioning on more global, or "long-range", network          features. Of course, it remains an open theoretical question as to how best conceptualize the problem of network modelling (and we think the local p* approach is very promising, both theoretically and empirically) Fortunately, it is at least, a question that the current generation of models allows us to address more carefully. Stanley & Pip _____________________________________________________________________ SOCNET is a service of INSNA, the professional association for social network researchers (http://www.insna.org). To unsubscribe, send an email message to [log in to unmask] containing the line UNSUBSCRIBE SOCNET in the body of the message.```