Can social media sites, like Twitter, be used to analyze financial markets for algorithmic trading? How much research has been done on this topic?
Personally, I am very skeptical of the claims in "Twitter mood predicts the stock market". There are several other papers with similar claims, but not so much good quality research is available. Arguably, the sweet bits of these approaches are not public. A sounder approach is to dig at the relationship between social media activity and relate it to the stock market. One idea is take social media as a proxy for investors' attention and exploit this channel to say something sensible about the stock market.
The typical viewpoint is that social media mining provides info on the interests of individual investors, in contrast to big investment firms. Within this framework the hypothesis of Barber and Odean (2008, RFS, "Do retail trades move markets?") makes a lot of sense: retail investors buy popular and news-making stocks, pushing prices up. Interestingly, their investing activity is asymmetric: buy news-makers, but only sell what they own - rarely on news.
Some fresh good work on the topic, based on google searches, is the paper "In search of attention" by Da, Engelberg, and Gao (2011, JF) (EDIT: this is the same work pointed out by wburzyns in his comment above).
I assume that by "how much research" you mean "could you provide me with some links"....
So, as @Shane mentioned in its comment, a hedge fund recently started and is focusing on twitter analysis (here is another link).
From what I understand they are basically implementing a trend-following strategy based on the "mood" of twitter users (I, of course, don't know the details, but it's surely more complicated than that).
As for the links, we already discussed the subject of speech recognition in trading in this post.
The original paper on using Twitter to predict the Dow can be found here. The hedge fund the authors are working with on trading is called Derwent Capital. An interview with the gentleman in charge can be found here. A quick Googling for "twitter analysis paper" turns up a number of results, although I am not going to post links as I have not reviewed them.
The search term factors for a model would be very interesting but very few people have access to the data in sufficient granularity for it to work. You could try licensing data from network providers such as Comcast or Verizon.
Most of the literature on traditional text mining will be applicable to Twitter although some work will be needed to normalize for document length and the unique vocabulary on Twitter.
The was an article about how ten years ago some option market-makers were crawling online discussions forums (like the Yahoo! stock board) to see what people talked about.
If discussion about some ticker spiked, they assumed the volatility for that ticker will increase and factored this into their pricing models. Of course, most of the time is was because of news related to that stock, but not always (think about pump'n'dump schemes).
I assume that today they are much more sophisticated about this.
A related idea is to text mine news and use it for predicting movements of stock market. See this NYTimes article for more information.