Albert-László Barabási is one of the foremost network scientists and runs the Barabasi Lab at Northeastern University. More information about him can be found at http://barabasilab.com/rs-netdb.php.
The Gephi wiki has many network datasets as well as a number of files that are already in various proprietary formats. These can be found at https://wiki.gephi.org/index.php/Datasets.
KONECT is the Koblenz Network Collection available at the University of Koblenz-Landau. The University's site houses one of the best collections of network data, and provides easily accessible symbols that identify specific network attributes, which can be found at http://konect.uni-koblenz.de/.