Here are few that I know:
For small networks (up to a million or two million nodes such as Wikipedia Link graph from 2009)
Following libraries provide code to handle and manipulate Network datasets:
1: SNAP by Prof. Jure Leskovec [ http://snap.stanford.edu ] written in C++
2: Networkx by Lanl [ http://networkx.lanl.gov/ ] written in Python, esp. good for fast prototyping
There are few Databases for storing networks, e.g. Neo4J http://neo4j.org/ .
Additionally there is a Graph Processing Language called as Gremlin
http://wiki.github.com/tinkerpop/gremlin/ .
For Large networks with millions and billions of nodes, one can use Hadoop / Map-Reduce or Apache Hama [still in nascent stage]. Google has a special system known as Pregel which it uses to perform scalable computations over large networks.