This is a posting that I did a while back to a now-defunct list, but it still is relevant. I have started a page of network properties, a catalog or glossary of sorts. Let me know of any additions or clarifications, the info below is the kind of stuff I am collecting.

Here's a link to a paper that looks at "different classes of small-world networks":

http://www.pnas.org/cgi/content/full/97/21/11149

The authors identify three kinds of small-world networks, based on the nature of the distribution of links. If I understand what they are saying, the cost of adding links (e.g., adding new routes to an airport vs new links to a website), and ongoing ability to add links ("aging", e.g., movie actor collaborations decay over time as an actor ages, also one might add an organization's ability to attract new members/connections -- I think this is another way of describing the "fitness" of nodes; new nodes may be very desirable to attach to, even though initially they do not have many links) will result in different kinds of small-world networks.

These two factors interfere w/ the "preferential attachment" phenomena of network growth (all things being equal, new nodes tend to attach to already well-connected nodes), resulting in the different classes of "small worlds".

jd