In addition to the sources listed below, be sure to look at the information about data formats.
The University of Florida Sparse Matrix Collection (from Tim Davis)
Voteview website (contains Congressional roll call data; maintained by Keith Poole)
Voteworld website (contains voting data from several countries)
Charles Stewart's Congressional Data Page
Committee Assignments in the U.S. House of Representatives (as MATLAB files)
Processed roll call voting matrices (102nd-107th Houses) (as MATLAB files)
NCAA Division I-A Football
brain connectivity matrices (also contains .m files for quite a few network tools)
Santo Fortunato's community detection benchmarks (from this paper)
Dataverse (maintained by Harvard University's Institute for Quantitative Social Science)
Data posted by Lazlo Barabasi's group (University of Notre Dame)
Data posted by the Uri Alon group (Weizmann Institute)
Data posted by Duncan Watts' Collective Dynamics group (Columbia University) (as raw edgelist)
Pajek datasets (in pajek formats)
Caida AS datasets
Data posted by Mark Newman (in GML format)
Data posted by School of Library and Information Sciences at University of Indiana
http://data.un.org/
Free and large Irish social network data (there's a competition with this too)
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