Importance of a person to another person in social network
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.


Determining Influential Users in Internet
Social Networks / Trusov, Bodapati, Bucklin
Do Friends Influence Purchases in a Social Network? / Iyengar, Han, Gupta
HOW FRIENDS INFLUENCE US ON SOCIAL MEDIA (AND WHAT THAT MEANS FOR MARKETERS) / fast company