Centrality Measures for a directed multigraph I have a directed multigraph that is used to represent an online discussion forum, where each actor is able to contribute comments to a discussion and have their comments commented on by other actors. 
Which centrality measures are most appropriate to use for a directed multigraph? Can I use the standard centrality measures used for non-directed binary graphs (e.g closeness, betweeness, eigenvector), or will this cause problems?
 A: There are a few ways to think about the problem. First, do you intend it to be a graph with multiple edges (i.e., an edge for each response) or a weighted graph (i.e., each edge is weighted by the number of interactions)? If it's a graph with weighted edges, there are weighted versions of betweenness, closeness, and eigenvector centrality, while outdegree would refer to the number of responses to a user's comments. Edges can also be normalized in a weighted network to make the interpretation more comparable to dichotomous graphs.
If you're dealing with multiple edges, then I would recommend thinking about the problem as a longitudinal one. One way to do it would be to create a time window (maybe a day), include all the edges that are created during the time window, dichotomize the network to remove the multiple edges (or make it weighted), calculate a centrality measure, and repeat for the next time window.
Though, if this situation were my problem, I would treat the data as a directed two-mode network. There are actors (users) and events (comments). A user creates a comment resulting in an edge directed to the comment. Should another user respond, that user would receive an edge from the original comment and send an edge to the subsequent comment. This method would preserve directionality, the temporal order of communication, as well as the two-mode nature of the relationship. Each actor will then have geodesics to other actors (through the comments) and betweenness as well as closeness centrality can then be measured through those paths. Outdegree would refer to the number of comments a user makes and indegree would refer to the number of comment responses a user makes.
