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What follows is a little thing that Pip Pattison and I wrote, and
posted to SOCNET, almost 10
Tom S., Garry R., and Dave K. have written recent replies to the
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
relationship between p* and QAP, and prompted some very useful
Here (belatedly!) we offer a few additional thoughts on what seems to
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
hand. The essential problem in choosing a particular statistical
that there is no consensus on what to believe about a phenomenon of
but rather a multitude of possible stances to consider. And this
is true whether you think about an approach based solely on
random graph distributions (of which QAP is an example) or on
distributions (or, of course, on both).
The stochastic model underlying QAP (usually) is the conditional
distribution which conditions on (fixes) all node-label-independent
of each of one or more graphs (e.g the graph of familial associations
graph corresponding to the formal hierarchy in Carterís example). In
case, all graphs that are isomorphic (as unlabelled graphs) to the
graph have equal (non-zero) probabilities. Graphs of different
(usually) assumed to occur independently. [QAP stands for
"quadratic-assignment programming", and was first used in relational
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
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
among a set of network relations. There are many other conditional
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
association between a particular graph and an hypothesized graph, or
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
blockmodels. We have discussed the choice of conditional uniform
distributions for such situations in Pattison, Wasserman, Robins and
(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
it is always wise, maybe even necessary, to condition on any design
well-supported hypotheses about processes underlying generation of the
such as fixed degrees, but there may be other more contentious factors
In view of this lack of specificity, we argue that the assessment of
relationships is likely to be more informative if conducted using a
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
ties. These assumptions characterize structural assumptions about
local network processes
giving rise to the network observations. In a given empirical
parameters of the model are estimated from observed networks. Here too,
though, there are many possible assumptions of interest, and a similar
arises with respect to choice of specific distribution (although we
that the multivariate version of a homogeneous Markov graph [Frank &
1986, JASA] is a good starting point).
Overall, the choice of statistical models for networks is often
Further, the set of possible choices can be quite large.
This choice confronts us with the need to develop more explicit
about the processes underlying network structures. Our personal
on the strengths and weaknesses of working with conditional uniform
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
well-suited for capturing effects of local processes in
They yield probabilistic models for global network structures
terms of local structural tendencies.
But, third, conditional uniform random distributions such as
one used by QAP still offer the more accessible
approach for conditioning on more global, or "long-range",
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
allows us to address more carefully.
Stanley & Pip
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