Since I don't know what kind of social network you would like to build and what kinds of features it will have, I can't tell you exactly how much developing a social network would cost. But I can give you some insight into the items that you need to consider that would affect the overall price.
The first thing is finding the right web developer. Good web developers charge anywhere from between $50 and $200 an hour (and many times a bit more than that). Make sure the web developer has a good understanding of exactly what you're trying to build so they can give you an accurate quote.
The second thing is your platform. Typically (again depending on levels of customization, etc.) going with an open source platform like Drupal or Ruby on Rails is the better choice. There are some drawbacks to using open source platforms, chief of which is the code base really isn't "yours" totally.
But, more and more companies are using open source platforms so this is less of an issue now, but it's still something you need to know about going in.
You also need to budget to have a set of wireframes developed. Don't skimp here. Wireframes can save you tons of time and money in development costs. Think of wireframes like blueprints for a house.
You'll also need QA testing to make sure the social network operates how you want it to operate.
Also, someone needs to handle front-end design or GUI. Sometimes your web developer will have someone to design the graphics (how the site looks) for you, but you want to be clear about this and ask if there are any additional costs associated.
That said, the company that I know does the best work in web development are the folks at Null Variable. They're not cheap, but they do an excellent job and the CEO knows his stuff inside and out.
Lena West is the CEO and Chief Strategist of xynoMedia Technology, a New York-based firm that helps high-growth companies leverage the power of social media, blogs, podcasts and online communities.