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These are complicated terms to understand, and we do need to make a
more intuitive explanation available to folks. Here's how I like to think
There are two parameters in the gw-terms, and the overall effect on the
odds of a tie is a product of the two, given the way the statistic is
constructed. That's why these are called "curved terms".
You can think of the gwdegree term as having the form
beta * f(y, alpha)
This has the usual form parameter*statistic, except that there is a second
parameter, alpha, in the statistic itself. The statistic essentially sums
the number of nodes of each degree, except that alpha modifies the value
of that number, as a function of degree.
Alpha essentially imposes a rate of decay by degree, so the higher degree
nodes contribute less to the statistic than the lower. It can be
interpreted as the declining marginal return for each additional tie (or
additional shared partner for gw(e/n/d)sp). So yes, this does relate to
the preferential attachment concept (more below).
Beta controls the overall propensity for degree (or shared partners).
A good way to start to interpret the parameters is to set alpha=0, and
look at the change statistics (you can do this by calculating the f(y,
alpha) statistic with and without a proposed tie). Setting alpha=0 has
the effect of making only the first tie for a node count as a change; so
the possible values of the change statistic are
0 (if both nodes already have other ties),
1 (if one node was an isolate), and
2 (if both nodes were isolates).
Beta then multiplies this, so it can be interpreted as how the odds of a
tie change, as a function of the change in the number of nodes that are no
longer isolates when it is toggled on.
Of course, interpretation depends on the other terms in the model, and in
general you would have an edges term in to control overall density. In
that case, beta would reflect a propensity against/for isolates (for
positive/negative estimates respectively), relative to a random graph
with this density.
When alpha > 0, there is no discontinuity at 1 vs more, but instead a
continuous decline in the value of additional partners, where the rate of
decline falls as alpha increases. For alpha=inf, there is no declining
marginal return, the odds of a tie don't depend on the degrees of the
nodes (and for shared partners, you're back to the triangle term).
So, in answer to your question, it's the alpha parameter that is the
"anti-preferential attachment" component. As it varies from 0 to inf., it
never represents preferential attachment -- at inf., ties are just
independent of degree. But the smaller the value of alpha, the more
anti-preferential the degree distribution will be.
I found it helped me to understand these terms by making up an excel
spreadsheet to calculate the term itself, and the change statistics. If
you think something like this might help, I can clean mine up and make it
On Mon, 10 Nov 2014, Lubell, Mark wrote:
> ***** To join INSNA, visit http://www.insna.org *****
> Dear SOCNET:
> My research group is having an internal debate about how to interpret the geometrically weighted degree parameter for ERGM models, as implemented in Statnet. If anybody has a good paper or presentation discussing interpretation (beyond the various papers introducing the calculation and estimation), I would love to know about them.
> In particular, is GWDEGREE an anti-preferential attachment term such that a positive coefficient produces a low variance degree distribution, or does a positive coefficient produce a high variance degree distribution with a centralized network? And if you have a low variance degree distribution....what is the best way to think about the social processes generating a decentralized network?
> Thanks, Mark Lubell
> UC Davis
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