Akshay Java, from ebiquity group, used the Large Graph Layout (LGL) tool to visualize a large social network on Twitter. The top graph shown here was built using contacts from about 25,000 users. Notice that there is a link connecting two users if either one has the other as a friend and hence it is an undirected graph (of about 250,000 edges). Compare this to the bottom graph that is constructed using only users who are mutually acquainted. i.e. A knows B and also B knows A.
As Akshay reveals in his post: "I find that visualizing such large graphs is quite a challenge and to glean meaningful information from it is even more difficult". However, he goes further in explaining that some insights can still be gained from this project. Akshay points out that a number of users seem to be trying to win a popularity contest of some sort, while a number of bloggers and (perhaps fake) celebrity profiles have a huge fan following in Twitter. He also mentions how the two graphs look very different on account of the fact that users with public profiles get a lot of followers whom they might not really know and would hence never add them as an acquaintance. But to really understand what the differences are one would need to look at the community structure and properties of the two graphs.
ebiquity group has also explored the Twitter API in other projects   in order to get a better understanding of the microblogging trend.