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.