We've done a lot of graphical / path-finding analyses, and we've found that they're basically unfeasable.
Graphing: Even graphing the relationships between just the list of all of the people on my friends list was a 2000x2000 pixel image littered with lines, and it brought my 800mhz Athlon to its knees. (I was using graphviz, which is pretty state-of-the-art in terms of graphing software.)
Shortest path: Unless you can propose a previously-unheard of algorithm, anything you do will be a horrible hit on the servers.
There are almost 400,000 users, and the hardcore users have over 100 friends each; that's a possible 40,000,000 links.
Most importantly (and also a response to your earlier lj_dev post): some people hide their friends list, for privacy purposes. (Like me.)
Finally, if you spider the database, I think dormando
will murder you in your sleep.
We used to export friends list data for interested parties like you, but some people started abusing it.
Mail me privately and I'll give you some data to play with.