A large volume of research on networks has been devoted to the concept of centrality. This research addresses the question: Which are the most important or central vertices in a network? There are of course many possible definitions of importance, and correspondingly many centrality measures for nodes in a network.
Once the centrality of nodes have been determined, it is possible to make a ranking of the nodes according to their centrality scores. For instance, one might want to obtain the most important Web pages about a certain topic or the most important academic papers covering a given issue. Moreover, one might be interested to know which are the nodes whose removal from the networks would have the most important consequences in the network structure. A network property that is directly influenced by the removal of nodes is connectivity. For instance, which are the Internet routers whose failure would mostly damage the network connectivity?