File gwtermsNotes.txt, 5.1 KB (added by morrism, 4 years ago)
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1From morrism@U.WASHINGTON.EDU Mon Nov 10 11:40:54 2014
2Date: Mon, 10 Nov 2014 11:38:18 -0800
3From: martina morris <morrism@U.WASHINGTON.EDU>
4To: SOCNET@LISTS.UFL.EDU
5Subject: Re: Interpretation of GWDEGREE
6
7*****  To join INSNA, visit http://www.insna.org  *****
8
9Hi Mark,
10
12
13There are two parameters in the gw-terms, and the overall effect on the
14odds of a tie is a product of the two, given the way the statistic is
15constructed.  That's why these are called "curved terms".
16
17You can think of the gwdegree term as having the form
18
19beta * f(y, alpha)
20
21This has the usual form parameter*statistic, except that there is a second
22parameter, alpha, in the statistic itself.  The statistic essentially sums
23the number of nodes of each degree, except that alpha modifies the value
24of that number, as a function of degree.
25
26Alpha essentially imposes a rate of decay by degree, so the higher degree
27nodes contribute less to the statistic than the lower.  It can be
28interpreted as the declining marginal return for each additional tie (or
29additional shared partner for gw(e/n/d)sp).  So yes, this does relate to
30the preferential attachment concept (more below).
31
32Beta controls the overall propensity for degree (or shared partners).
33
34A good way to start to interpret the parameters is to set alpha=0, and
35look at the change statistics (you can do this by calculating the f(y,
36alpha) statistic with and without a proposed tie).  Setting alpha=0 has
37the effect of making only the first tie for a node count as a change; so
38the possible values of the change statistic are
39
400 (if both nodes already have other ties),
411 (if one node was an isolate), and
422 (if both nodes were isolates).
43
44Beta then multiplies this, so it can be interpreted as how the odds of a
45tie change, as a function of the change in the number of nodes that are no
46longer isolates when it is toggled on.
47
48Of course, interpretation depends on the other terms in the model, and in
49general you would have an edges term in to control overall density.  In
50that case, beta would reflect a propensity against/for isolates (for
51positive/negative estimates respectively), relative to a random graph
52with this density.
53
54When alpha > 0, there is no discontinuity at 1 vs more, but instead a
55continuous decline in the value of additional partners, where the rate of
56decline falls as alpha increases.  For alpha=inf, there is no declining
57marginal return, the odds of a tie don't depend on the degrees of the
58nodes (and for shared partners, you're back to the triangle term).
59
60So, in answer to your question, it's the alpha parameter that is the
61"anti-preferential attachment" component.  As it varies from 0 to inf., it
62never represents preferential attachment -- at inf., ties are just
63independent of degree.  But the smaller the value of alpha, the more
64anti-preferential the degree distribution will be.
65
66I found it helped me to understand these terms by making up an excel
67spreadsheet to calculate the term itself, and the change statistics.  If
68you think something like this might help, I can clean mine up and make it
69available.
70
71best,
72Martina
73
74On Mon, 10 Nov 2014, Lubell, Mark wrote:
75
76> *****  To join INSNA, visit http://www.insna.org  *****
77>
78> Dear SOCNET:
79>
80> 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.
81>
82> 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?
83>
84> Thanks, Mark Lubell
85> UC Davis
86>
87> _____________________________________________________________________
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94****************************************************************
95  Professor of Sociology and Statistics
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