Given my networks of friends - can I detect my most "Central" friends? Assuming I can download from facebook a list of my friends, and which of them are friends with each other - can I estimate from this information which of them is "central"? (I am not exactly sure how to define what "central" is in this context)
What information/assumptions are needed?
Thanks.
 A: Simplest solution would be to count the number of friends each of your friends has in common with you, and invert that and use this as a measure of centrality.
Anything further than this is bound to require additional assumptions: you can always sum the distances (along the shortest friend path excluding you) between all your friends (and the one with the smallest "total distance" would then be "most central"). But then you have to decide: what is the "weight" of the longer distances: Say you are considering the distance from A and B to persons C, D and E, and these are respectively 3,1,2 and 4,1,1: do you consider the total distance the same?
Also, if you want to avoid totally disconnected people (which make the sum awkward, because you have to specify a hard number for "disconnected" people's distances), you will probably have to allow for connections outside your own friends circle (e.g.: you are friends with 100 people who know me, but you're not befriended with me, and the 100 people don't necessarily are friends either). But even then you may have disconnected nodes in your "friend graph".
Finally, you may also have to weigh the connections themselves: perhaps the date the friendship was established, perhaps the number of messages that have been posted on either's wall (which could even make the "distance" nonsymmetric), the person who initiated the friendship (sent the "request"), or details specified about the relationship (family relations etc) could matter for your "distance of interest".
All in all: you will have to specify what your goals are, and adapt you distance measure to it. There's bound to be quite a bit of literature about distances in graphs, but all will require figuring out which distance you are interested in.
A: There are many, many, many ways to define your "most central" friends. These are called Centrality measures. Probably the three most common are these, with somewhat plain English explanations.


*

*Degree: How many friends does Friend A have?

*Closeness: How many steps does does one have to go through to go from Friend A to any other friend on your network?

*Betweenness: How many paths on your network between friends in your network pass through Friend A?


Sometimes these are very similar, and highlight the same "important" people. Sometimes, they give interesting results where even someone without a huge number of connections is a "friend of a friend" of nearly everyone, or connects two disparate groups. And these are just a few - there are, as I said, a ton of different ways to look at centrality, with lots of twists.
Lots of software will let you look at these measures. My personal favorite, if you know Python, is NetworkX. NodeXL works for Excel, sna is one of the R packages that handles it, etc.
In terms of the information you need...obviously, you need the network itself. One thing you are assuming is that the network you collect (in this case, Facebook), adequately represents the actual friendship network you're asking about. So, for example, people don't have hangers-on in their friend's list they haven't bothered to delete, or there's no one who is important to your network who has opted out of social networking. Or in the case of Facebook, that there's no spoof accounts.
A: Take a look at NodeXL (an simple, but powerful, Excel extension for network analysis) and the book Analyzing Social Media Networks with NodeXL: Insights from a Connected World. Even if you will use some other software, the book definitely discusses the various measures of centrality and their uses very well. I don't have it in front of me, but I seem to remember it addresses the prioritization question in the context of marketing. 
