If my social network and that of somebody else have significant overlap (e.g. shared membership), then the odds are good that the two of us know each other and even that we know each other well. And if we share the same friends and colleagues, then we are likely to run in the same social circles - and so are likely to be in each other's social network as well. In a sense, the connection between myself and such a contact is strong, reinforced as it is by the multiple links between ourselves - both the direct one and the others that go through our shared contacts.
But, there would seem to be a paradox here.
For some applications, to accrue benefit from a social network (e.g. get that job recommendation, date, stock tip, etc), there is value in maximizing its radius (or equivalently, minimizing the average distance between myself and others). But strong connections don't contribute much here. There is little value (in the sense of expanding my network) in these tight social links. If I dropped such a contact from my network (let's rhetorically say he broke one of my dining room chairs at a New Year's Eve Party) - my network health & viability would likely not be significantly impacted - the other links we would continue to share would provide a buffer to absorb any significant shrinkage in the radius of my network (and to ensure that we will share ongoing awkward silences at summer BBQs).
Or equivalently, if I add a new contact to my online network, and they bring with them no fresh blood - individuals not already in my network - then I gain nothing (in the sense of maximizing radius) from the addition. I might as well just keep them in my Thunderbird Contacts.
More important to maximizing the size of my network are weak connections, i.e. those to individuals typified by the fact that we have few if any shared members in our respective networks. Such connections are weak in the sense that it is only through the direct link that we stay bound, there are no suppporting secondary links reinforcing them. But it is these weak connections that ultimately allow a network to expand beyond its incestuous (not literally) core of strong connections. Adding (or removing) such a connection can have significant impact on network size.
For me, this is a clear argument that I need to get out and meet new people. My wife remains unconvinced.