An Open Source Social Network App – Movim
The application is built with PHP-MySQL and includes features like a wall, an instant messaging system, a contacts list, photo gallery, news feed etc.
It gives users the maximum control over their own data and allows them to import/export all the data from one Movim-powered network to another.
There is a special focus on privacy where all user setting are set to the max-private by default and they can customize them later on.
Movim is multilingual and doesn't require anything complicated to setup an instance.
P.S. You may also want to check out other open source social networking applications.