Michael,
Excellent points - and welcome the challenge :-). A friendly cerebral tug-of-war is always good in my opinion as it fosters better learning and knowledge exchange.
So coming back to the topic of direct/indirect connections. We both agree that relationships are key to social networks. Our different point of views I think arise from the different lens we are looking with. You are looking at it from a "connections" perspective, whereas I am looking at from a "connections & interactions" perspective.
My response to some of the points (focusing solely on the online world as offline is another story).
a. You said, "Since the social network connects everyone on this planet, you kind of have to make a cut off some where". Well, I don't think you necessarily have to make a conscious cut-off. While you might be 'connected' to everyone in the planet theoretically, I think your social network is made up of those nodes that are influenced by you. These could be those directly connected with you and those who interact with you but are not directly connected. For example, I constantly get invites from folks in linkedin not part of my direct network to join a certain group. I do think of them as a part of my social network since they are "aware" of me and my interactions.
b. Which brings up two other points: (a)Social networks are not static, and (b) You only have partial control over your network. To the point you made about direct connections and truly 'your' relationships, I agree that this makes up the core and is relatively more stable (though constantly expanding). However, the periphery of your social network is always in fluctuation as you have no control over who and when someone (not directly connected with you) comes under your influence and initiate an interaction. Now I will agree that in many cases, these non-direct connections turn into direct connections if the interactions continue.
c. While connections are important, [I think] your social network is determined by the type of connections you have and the extent of reciprocity. You and me might be connected with 5 individuals but it makes a big difference in my eyes who those individuals are (a John Doe vs Bill Gates for example) and to what extent they identify with and reciprocate your interactions.
Anyway, these are my viewpoints based on observations rather than a rigorous empirical research. And irrespective of whether our opinions converge or not, I think you have an excellent blog with some cool information and so you should continue to pursue your path based on what you believe in.
Regards,
Ned