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SOCNET  May 2008

SOCNET May 2008

Subject:

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

From:

Stanley Wasserman <[log in to unmask]>

Reply-To:

Stanley Wasserman <[log in to unmask]>

Date:

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

Content-Type:

text/plain

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text/plain (181 lines)

*****  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

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