Given a particular undirected graph with n nodes, is there an index that would characterize the vulnerability of each of the nodes? By vulnerability I mean the susceptibility of the graph to fragmentation.
Let's say that I remove a particular node i which results in the graph being fragmented into five parts; and when I remove node j it results in the graph being fragmented into two parts (and there would be nodes which would not fragment the graph)...is there an index which gives me this information for all the nodes?
The goal of the above is to identify vulnerable nodes in a graph (made from healthy subjects) and look at node properties of these nodes in another graph (made from patients suffering from various medical conditions).
For the sake of completeness, my nodes are representatives of different brain regions and the edges between these nodes represent correlation coefficient between some form of time series from each of these brain regions. However, the problem can be generalized to other contexts (the "key player" problem in graph theory...identifying key people in a social network, for example).