The answer to that question really hinges on what the niche of the social network is.
Let's say the niche is a about insects. Perhaps an alternative to advertising, would be to send out free bug spray to all members with a dominant bug spray company sponsoring the giveaway of their product.
Another example of a way to monetize a network without advertising is to ask for donations once every few months via an email broadcast and banners on your site for a limited time to promote the donation.
There are a million ways to monetize any website, it all comes down to identifying your audience and what state of mind they are in while navigating your webpage.
That depends on the social network, subject matter and demographic of audience. But generally if you have an audience then sell them something. A product, a service or information. If you like more specific advice please feel free to give me a call