Given the number of articles on wikipedia, it would take a unaffordable time to compute THE shortest (my assumption - I haven't tried).
The real problem is to find an acceptable and efficent short path between two articles.
Algorithms that deal with this kind problem are related to The travelling salesman problem. It could be a good point to start from.
I'm personnally also fond of the genetic algorithms approach to find an acceptable optimum in a certain amount of time.
I have just looked at that image and that sets the number of articles to 4.000.000 for en.wikipedia.com in 2013. Much less than I thought indeed.
EDIT: I first stated it was a NP-Hard problem and commenters explain it's not.
Why would this be NP-hard?
Yeah this is just graph search (of course realistically you need to do this carefully, so you don't run out of memory), which has a poly time solution in the number of vertices and edges in the graph. What NP-Hard reduction are you referring to, I don't think this is TSP actually, since that is asking the shortest path through every node, not the shortest path between two nodes.
@tigger that's right. Re-reading the article on TSP and NP-hard. Our case is not NP-Hard and not a case of TSP. But do you mean that something in complexity bigger than O(n), like graph walking, is a possible approach for browsing the wikipedia graph ? ( I find myself confused with NP hard and complete )
@StephaneRolland NP-hard >= NP-complete. Problem X that is at least as hard as the most complex problem Y in NP is NP-hard; X does not have to be a member of NP. NP-complete means "NP-hard and in NP", so those are the most complex problems that are in NP. Also, I'm curious why you mention the number of articles on wikipedia in the context of complexity.