To get new network analysis researchers started I recommend downloading a few of those really great collections of networks of all kind. Those have initially been collected here: http://forum.gephi.org/viewtopic.php?f=7&t=162
See infochimps, they provide an excellent access to already existing Twitter networks.
You are also free to of course crawl your own twitter networks using my Twitterlyzer (Ruby on Rails framework) software or try Nodexl for beginners.
As part of his phd thesis Tore has collected quite a few weighted and directed networks