If you are Digg or LinkedIn you can build your own speedy graph database to represent your complex social network relationships. For those of more modest means Neo4j, a graph database, is a good alternative.
A graph is a collection nodes (things) and edges (relationships) that connect pairs of nodes. Slap properties (key-value pairs) on nodes and relationships and you have a surprisingly powerful way to represent most anything you can think of. In a graph database "relationships are first-class citizens. They connect two nodes and both nodes and relationships can hold an arbitrary amount of key-value pairs. So you can look at a graph database as a key-value store, with full support for relationships."
A graph looks something like:
For more lovely examples take a look at the Graph Image Gallery.
Here's a good summary by Emil Eifrem, founder of the Neo4j, making the case for why graph databases rule: