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In social network (an unweighted indirect network), is there some measure of importance of a person A on another person B, denote as I(A,B), so that (approximately):

  1. The smaller the shortest distance between A, B in the the network, the higher the I(A,B) (closeness)
  2. If A_1 is 1 of the 100 friends of B_1, A_2 is 1 of the 2 friends of B_2, then I(A_1,B_1) < I(A_2,B_2) (the more friends I have, the less importance of each one)
  3. Some other reasonable properties that resemble the importance of a friend to another friend in real society.
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    $\begingroup$ This may be too broad a question, as I suspect that large parts of social network research is about this exact question. Community votes, please! $\endgroup$ – Raphael Jan 20 '16 at 16:10
  • $\begingroup$ But, what about the bounty, if this question is closed, then will it go wasted ? @Raphael $\endgroup$ – azam Jan 27 '16 at 11:02
  • $\begingroup$ @azam Yes, but that should not influence voting decisions. The author can always improve their question and have it reopened before the period runs out. (Assuming the bounty will stay in place, which I am not completely sure about.) $\endgroup$ – Raphael Jan 27 '16 at 12:02
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    $\begingroup$ Also, possible duplicate? $\endgroup$ – Raphael Jan 27 '16 at 12:10
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yes, social networks have a simple (graph) connectivity property in many or most cases/ situations/ contexts (such as "friend" connections on a social network like Facebook etc), but also a weighted property of the graph in various other situations. a common aspect of this now measured is widely/ generally referred to as the influence of the "friendship". two users may be connected, but influence measures other aspects.

one aspect of high interest to marketers and driving a lot of research is how much "friends" influence each others purchasing decisions ie if person A buys x, does that influence friend B to buy x also, or a y similar to x. here are a few papers/ refs on this subject. there are different ways to attempt to measure influence and in some cases it may be derived/ estimated that some users have more friends and are more thus influential on those friends, in other ways it can be determined from additional data such as looking at buying patterns of A, B if A, B are connected etc.

buying influence is just one of the major/ dominating areas of weighting under study, but there are various others. another key aspect coming up in 2016 in the US is voting influence that is under some study also, ie tendency of connected users to vote in similar ways.

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    $\begingroup$ another way to estimate influence is through eg facebook likes, and look at reciprocal counts of the likes between A, B, or one can look at reciprocal commenting, etc $\endgroup$ – vzn Jan 27 '16 at 16:08

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