Why not call them personal network sites?
I’ve often wondered why scholars studying Facebook, MySpace, Bebo, etc., call these platforms ‘social network sites’ instead of personal network sites. After all, these are platforms built around individuals and their personal (or egocentric) networks rather than around ‘whole’ (or sociocentric) networks such as clans, universities, localities or firms. In social network analysis this distinction between personal and whole networks is crucial (see Knox et al 2006), yet it gets conflated in the Internet literature. For instance, Facebook allows for the creation of ‘networks’ (as in whole networks) and ‘groups’ but these are not as central to the site as its immense tangle of 300 million personal networks. It is the ‘logic’ of personal networks that drives Facebook, not the logic of whole networks (Diagram by Peter Timusk).
Reference
Knox, H., Savage, M. & Harvey, P. (2006). Social networks and the study of relations: networks as method, metaphor and form. Economy & Society, 35(1)

Maybe ‘social’ sounds better than, say, ‘selfish networking sites’.
Would you say that facebook is driven by selfishness? That’s not my impression. There’s a fair bit of narcissism and self-promotion but not so sure about selfishness. (I’m speaking anecdotally here).
Let’s say ‘selfish’ means ‘for personal gain’. Does ‘narcissistic’ then mean ‘obsessed with oneself but in a non-purposive, non-gain-oriented way’?
To get back to your original point. ‘Personal networking’ would still have to be ‘social’, otherwise it wouldn’t be networking. I think not all networks on facebook would have ‘self’ in the middle, in the way your diagram positions them. E.g. if you’re an advertiser or event organiser, you might have ‘product’ (or ‘transaction’) in the centre, and ‘selves’ as satellites.
For our launch, Wendy made a person, not a group, called ‘Destination London’. It’s an event masquerading as a self.
1. Yes, personal networks are still social. The term ‘personal networks’ is short for ‘personal social networks’ whilst the term ‘whole network’ stands for ‘whole social network’. The trouble with ‘social network’ is that it does not distinguish between these two radically different sets of social relations. Personal social networks are really odd social formations in that they revolve around individuals and without that individual (say if ego dies) they would cease to exist. Virtually all other social formations centre around a collectivity not an individual.
2. I didn’t argue that all facebook collectivities have a self or ego in the middle, and pointed out that there are also whole social networks (known in facebook parlance by the terms ‘networks’ and ‘groups’). What I suggested was that the *predominant* form is the personal network, i.e. a social network with an individual in the middle. Of course, clever tampering with this basic template occurs, as in the Destination London example.