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Again, we are puzzling in my research group about the relationship between GWDEGREE terms in ERGM and network centralization, preferential attachment, and variance in degree distribution.

Our general heuristic understanding (see Mike Levy's useful shiny app: is that a positive GWDEGREE coefficient  (add links but higher benefits for low degree links) represents a low variance degree distribution, less centralized network, or anti-preferential attachment.  A negative GWDEGREE coefficient (penalizes new links, but penalty is greatest for low degree links and thus preserves high degree nodes) represents a high variance degree distribution (more "fat tailed"), more centralized network, and stronger tendency for preferential attachment.

However, we are looking at two networks that seem to contradict this general heuristic.  In network 1, we have a less centralized network  (degree centralization, comparing to a "star" network) with a negative gwdegree degree parameter in ERGM. In network 2, we have a more centralized network with a positive GWDEGREE parameter in ERGM.  The issue seems to be that in network 1 (geographic island network), you have a bunch of geographic homophily that creates regional "islands" in the network anchored by a central actor in each island.  In network 2 (call it a "regional network"), there are no islands and all the high degree nodes are more in the center of the global network.  I believe that in estimating GWDEGREE, ERGM doesn't care if the variance in the degree distribution appears in local subgroups or is spread more globally through the network. In other words,  if one were to take all the high-degree nodes in the  "geographic island" network and rewire the network so that you have no homophily effects that create subgroups, you would actually get the exact same GWDEGREE estimate because the variance in the degree distribution is exactly the same regardless if the high degree nodes are distributed locally versus globally.

If this is correct, it raises two important questions:

1.       What is the relationship between GWDEGREE and parameters/forces such as homophily that create subgroups?  Especially if you have both homophily and GWDEGREE in the same ERGM model, do you interpret GWDGREE as capturing the remaining variance in degree distribution conditional on homophily?

2.       If it is the case that GWDEGREE and homophily have an important interaction (and I think the same issue might occur for 2-star and 3-star parameters if they are entered jointly in the model to capture open structures with a possible decreasing marginal benefit), what ERGM parameter can we put in a model to represent a tendency to create networks that are more centralized regardless of subgroup formation processes?

I know that is a lot to read but I'm hoping the networks/ERGM community can help here.

Mark Lubell
Department of Environmental Science and Policy
UC Davis
Center for Environmental Science and Policy:
Twitter: @EnvPolicyCenter

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