Publicly available social network datasets/APIs
A couple of words about social networks APIs. About a year ago I wrote a review of popular social networks’ APIs for researchers. Unfortunately, it is in Russian. Here is a summary:

Twitter (https://dev.twitter.com/docs/api/1.1)


almost all data about tweets/texts and users is available;
lack of sociodemographic data;
great streaming API: useful for real time text processing;
a lot of wrappers for programing languages;
getting network structure (connections) is possible, but time-expensive (1 request per 1 minute).


Facebook (https://developers.facebook.com/docs/reference/api/)


rate limits: about 1 request per second;
well documented, sandbox present;
FQL (SQL-like) and «regular Rest» Graph API;
friendship data and sociodemographic features present;
a lot of data is beyond event horizon: only friends' and friends' of friends data is more or less complete, almost nothing could be investigated about random user;
some strange API bugs, and looks like nobody cares about it (e.g., some features available through FQL, but not through Graph API synonym).


Instagram (http://instagram.com/developer/)


rate limits: 5000 requests per hour;
real-time API (like Streaming API for Twitter, but with photos) - connection to it is a little bit tricky: callbacks are used;
lack of sociodemographic data;
photos, filters data available;
unexpected imperfections (e.g., it’s possible to collect only 150 comments to post/photo).


Foursquare (https://developer.foursquare.com/overview/)


rate limits: 5000 requests per hour;
kingdom of geosocial data :)
quite closed from researches because of privacy issues. To collect checkins data one need to build composite parser working with 4sq, bit.ly, and twitter APIs at once;
again: lack of sociodemographic data.


Google+ (https://developers.google.com/+/api/latest/)


about 5 requests per second (try to verify);
main methods: activities and people;
like on Facebook, a lot of personal data for random user is hidden;
lack of user connections data.


And out-of-competition: I reviewed social networks for Russian readers, and #1 network here is vk.com. It’s translated to many languages, but popular only in Russia and other CIS countries. API docs link: http://vk.com/dev/. And from my point of view, it’s the best choice for homebrew social media research. At least, in Russia. That’s why:


rate limits: 3 requests per second;
public text and media data available;
sociodemographic data available: for random user availability level is about 60-70%;
connections between users are also available: almost all friendships data for random user is available;
some special methods: e.g., there is a method to get online/offline status for exact user in realtime, and one could build schedule for his audience.