In the Erdös graph for TeX.SE two people are connected if they have contributed to (asked or answered or commented on or edited) the same question.
What does this graph look like?
Are the data available in a form that makes an answer possible?
Question prompted by the blog entry at http://tex.blogoverflow.com/2013/07/a-new-milestone/?cb=1 : egreg =? Erdös
Edit towards an answer, prompted by the comments.
The Erdos graph is a way to represent TeX.SE as a social network, so what's relevant are the connections between people that occur when they think about the same question, not the connections between questions and people. (That might be an interesting bipartite graph for another day.)
If the average number of answers per question is A then each question will generate (on average) A(A+1)/2 edges joining vertices (users) in the graph (counting self-answers, not counting comments or edits). This allows multiple edges between vertices (which happens frequently on TeX.SE).
There are about 43000 questions and about 32000 users. Since each edge joins two vertices (users) the average valence of a vertex (user) is about
2*(number of edges)/(number of vertices)
= A(A+1)(number of questions)/(number of users)
I'd guess that A is between 1 and 2, which implies an average valence between 3 and 8. Of course the distribution is far from uniform!
Whether or not to count comments or edits as creating a connection could be easily explored by tweaking the software that builds the graph input from the database.
A database query generating a 43,000 line csv file with one line for each question listing its asker and answerers would provide all the data needed to build the graph. Since I don't know SQL I can't follow up
@texenthusiasts hope that a new query at http://data.stackexchange.com/tex/query/new
might provide what's wanted.