Connectivity Damage to a Graph by the Removal of an Edge or a Vertex
  The approach of quantifying the damage inflicted on a graph in Albert, Jeong and Barabsi's (AJB) report "Error and Attack Tolerance of Complex Networks" using the size of the largest connected component and the average size of the remaining components does not capture our intuitive idea of the damage to a graph caused by disconnections. We evaluate an alternative metric based on average inverse path lengths (AIPLs) that better fits our intuition that a graph can still be reasonably functional even when it is disconnected. We compare our metric with AJB's using a test set of graphs and report the differences. AJB's report should not be confused with a report by Crucitti et al. with the same name. Based on our analysis of graphs of different sizes and types, and using various numerical and statistical tools; the ratio of the average inverse path lengths of a connected graph of the same size as the sum of the size of the fragments of the disconnected graph can be used as a metric about the damage of a graph by the removal of an edge or a node. This damage is reported in the range (0,1) where 0 means that the removal had no effect on the graph's capability to perform its functions. A 1 means that the graph is totally dysfunctional. We exercise our metric on a collection of sample graphs that have been subjected to various attack profiles that focus on edge, node or degree betweenness values. We believe that this metric can be used to quantify the damage done to the graph by an attacker, and that it can be used in evaluating the positive effect of adding additional edges to an existing graph. 
Charles L. Cartledge and Michael L. Nelson
cs.DL cs.DS