MDSNIG said:
Basically Boonex Dolphin would do the job you need. If you need further details just drop a line here.
I hope they have their act together these days. I did dolphin a couple of years ago and it was horrible, major security issues. The support was pretty bad, I did get answers but I had to upgrade my account to pay for them. In the end I ditched the whole project because of all the problems. Even after saying all that, I do think dolphin is worth a second look. I keep checking in on it from time to time.
Drupal or Joomla are a good choice if you are willing to put in he dev time to get your site up to par with other social networking sites. Both platforms are stable, but it will take some heavy modification to make a social networking site out of them. You will need to install a ton of plugins that may or may not work well with each other out of the box.
If I was dead set on building a social networking site, I might be tempted to go with dolphin once again as it would be less of a headache early on, but in the long run I think you would be better off with a CMS like sjkstone mentioned.
They only other suggestion I have for your, which is not really related to your question, but I would choose a niche. I would not try and compete with Facebook or MySpace. Honestly, if you could compete with them you would not be here asking how to make a social networking site. So choose a nice niche and do your best to provide a unique and satisfying experience for your users.
On a side note, it cost me on average $1 -$2 in adwords for every new sign up and I spent a lot of time optimizing my adwords campaign. So don't forget to factor in your advertising budget.
Best of luck with your project! I look foward to seeing the finished site! Maybe you will inspire me to give it a go again.