I've seen some social networks powered by Elgg, but mostly in academic environments (Universities that want "their" social networks).

Here http://www.nilambar.net/2012/11/social-networking-script-open-source-free-php.html they talk about Mahara as LinkedIn alternative. I don't know anything else about that project.

Finally, I suppose any social network engine (e.g. pump.io) could be fine. The problem is not in the software, is to get the enough people using it (or maybe some "big, important users") to become "mainstream".