From the top of my mind I can quickly think of one plugin - WP Gra4 Social Network. It does incorporate many of the features that such sites provide - activities, friends, likes etc. Personally, as much as I love WordPress, its overall security is still a concern for me and this is twice as important when you are building a social networking site. So do have that in mind...
Alternative scripts I can suggest, developed exactly for this purpose, would be Dolphin and Elgg. Both are quite user-friendly and there are lots of helpful guides and tutorials for them.
Try WP Symposium with your own customizations to give a Facebook Like Feel
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BuddyPress is another plugin you could try. Social media features for a WordPress site.
However, unless you have a very good reason to do this I'd suggest you forget about it. People are not going to flock to another social media site just because it's there. Maybe if you have a topic with a solid following, and no competing forums or social media groups... I'm not sure if there are many like that. Even then a traditional forum script may be your best bet.