When you sample from the posterior and take the vertex marginals, is it
proper to say that we can interpret the marginals for a given vertex as
being the degree of membership in the communities (fuzzy community
membership)?

If so, how does this differ from the overlapping blockstate? I saw in the
mailing list that overlapping is only supported at the base level:

But, even if it were supported at every level, what does this achieve that
the fuzzy model averaging doesn't? Could you do model averaging with the
overlapping state too? E.g., in sample 1 vertex A is in communities c1, c2.
In sample 2 vertex A is in communities c1, c4. Etc. Would this be in some
way a more accurate measure of multiple community membership than the fuzzy?

When you sample from the posterior and take the vertex marginals, is it
proper to say that we can interpret the marginals for a given vertex as
being the degree of membership in the communities (fuzzy community
membership)?

It can be interpreted as the posterior probability of a node belonging
to a particular group.

If so, how does this differ from the overlapping blockstate?

It's different, because the generative model is not the same. In an
overlapping SBM the nodes can belong to multiple groups at the same
time. In the non-overlapping version, this is not possible. The marginal
distribution just conveys the uncertainty of the inference, not joint
membership.

"The marginal distribution just conveys the uncertainty of the inference,
not joint membership."

Is it not the case that those nodes who would be members of multiple
communities (in the overlapping model) would also appear in those
communities during different samples of the posterior?