I'm not a mathematician, but the plain english of your case here seems less than convincing to me, but only if you insist that Facebook is a social network and not a community.
So riddle me this, why isn't Facebook a community (with lax general rules around egress and ingress) whose common focus (shared purpose) is status updates. Like many other communities as you define them, I don't know all the members. Like many other communities as you define them, people come and go all the time -- the boundaries are just like facebook boundaries. Everyone in Facebook is a "card-carrying" approved member by virtue of the formal registration process. If you haven't gone through the process you're not in. Therefore, facebook has the same "network" structure, and the same firm boundaries, that you describe for communities. And the Facebook community overlaps with other entities you describe as social networks, but I'd say also fit your defintion of community.
Let me say this same thing one different way. It seems to me you have members of your social network that come from any community to which you are affiliated, and that Facebook is just another community that happens to contain members from many communities including the Facebook community. Surely you would not suggest that people have not added to their social network via connections made within [the community of] facebook.
In any event, thanks for putting in the time and making the case. It was a great read.
Bob.