This one looks like a jellyfish, and it's one of the better graphs we've seen. If you've got a LinkedIn profile, you can try it out yourself.
The Standard Issue Facebook Graph
This one looks like an erector set gone wrong — which is the look Facebook usually goes with when they're trying to explain the Open Graph.
Sometimes, your network looks like a Plinko board, at least according to this (sadly defunct) Facebook app from Paul Irish.
This is most computery of the bunch, either an air traffic control screen or some kind of DOS-era war game. Try it out if you like.
This Twitter map has been stripped down until it looks like a chandelier.
The Homemade Twitter Graph
If you give it enough space, the graph starts to look astronomical, like planets or satellites in orbit, as in this open-source Python project from Martin Laprise.