Yesterday I tried to answer a question on LinkedIn about the value we can assign to high-level connections (3rd or greater): a quite good question from several points of views.
I think that the main problem of “Small Worlds” theory is brought about the huge amount of information needed in order to get a fast propagation through a certain social network. A world can be really small, but such a property may be easily tested by its members only if there’s a global and widespread knowledge about both the structure and the nature of the majority of nodes which a network is made up of.
Thus the value of a 3rd level connection has to be considered not only from an intrinsic point of view (a person we can contact directly just like other “nearer” friends) but also as an additional set of information that we can exploit in order to speed up our “network-trip”. Of course, a utility like an Internet-based social aggregator extends this possibility by a general strengthening of many cognitive functions (memory, first of all) that allows a person to “remember” thousands of complex profiles and, above all, to find out any kind of information in a very short time.
I believe that such a reality can reconfigure the very “perceived” topology of network, so the only difference between a 1st level connection and a 3rd level one would be caused by an aware will to keep a sham gap in order to separate “real” contacts from virtual (in this case, potential but outside) ones.