Statistical Measure for Bidirectional Relationships I have a karma website where you can create a topic and someone can upvote the topic once.
People who receive upvotes from another individual tend to upvote topics from the other individual.
What is a good statistical measure for determining directional relationships between users who might be giving each other upvotes.
I have access to the data for each user and want to say lay out a table with a score for each pair of users measuring upvote buddies.
 A: Let the users be indexed by $i=1,\dotsc,N$.  Make a square contingency table $C=(c_{ij})$ where $c_{ij}=\text{number of times $i$ upvoted $j$}$.  Since presumably a user cannot upvote himself this table would have zeros along the diagonal. 
Now from this table various descriptives can be calculated:


*

*$c_{\cdot \cdot}=\sum_i\sum_j c_{ij}$ the total number of votes

*$c_{\cdot j}= \sum_i c_{ij}$ total number of votes on user $j$

*$c_{i \cdot}$ is total number of votes from user $i$

*$\frac{c_{ij}}{c_{i\cdot}}$ fraction of $i$'s votes given to $j$

*...


You are asking specifically about directional relationships it is not entirely clear what that is. If you are interested in pairs of users $(i,j)$ where $i$ is "giving" (much) more than "receiving", you could look at maybe $\frac{c_{ij}}{c_{ji}}$, or the corresponding ratio of proportions.  Other interpretations could probably also be given, using the counts in $C$. 
If you want to go deeper, you could represent the table as a graph and look for community detection in graphs. Or you could look for models for square contingency tables. Or correspondence analysis for square tables could be interesting. If you express interest in some of these I could write some more on that.
