Stupid question by someone who finds this very interesting but lacks some background information: What kind of processing are we talking about? What do you do with the graph? Searching? Finding paths? Mutating contents? Changing the graph structure? Aggregating data about the graph?
The purpose seems to be presupposed, so I guess it's some common use case that I should know.
The major motivation for Giraph (Facebook) and GraphJet (Twitter) was the ability to do graph based recommendations such as Friend of Friend scoring (Facebook) or Who To Follow (Twitter). Basically looking at the followers or friends of each of your followers or friends and using some kind of scoring algorithm to pick which users you will most likely be interested in.
Another common example is doing some kind of clustering (community detection) of vertices like for example labeling the users on the social graph by likely political affiliation or likely hobbies. These labels could be used as features for advertising recommendations later.
The purpose seems to be presupposed, so I guess it's some common use case that I should know.