Pinax is a stack of loosely-coupled reusable Django applications billed as "a platform for rapidly developing websites".
Elgg is (was?) geared toward higher education and integrated nicely with Moodle as a portfolio system.
Drupal is a CMS written in PHP that has numerous plugin applications that allow it to be configured as a social networking site.
Plone is a CMS on top of Zope that has numerous plugin applications that allow it to be configufed as a social networking site.
Ning and SocialGo are two platforms that allow you to create social network websites TTW (er, "through-the-web") and manage the hosting and database. These are "low-hanging fruit", and can be a great way throw together a proof-of-concept. Ning owns any data you host with them, and I don't know SocialGo's policies, but they are attempting to be less evil than Ning. (That is, if you consider Ning evil.)
Pinax would be my first choice based solely on my own experiences with each. They all have their strengths and weaknesses. YMMV.