Do you know any social networking web sites for specific universities only?
There are a number of universities that run social network sites based on Elgg. You can find some of them on the list here. Sites powered by Elgg
Others, such as the University of Mary Washington and University of Lincoln are using Buddypress.
You didn't specify if price was an option, so you could try Ning
it's a Social Network like-Facebook - there is a FREE package, but also some of them are paid. Depending on what you want / need.