Attack tolerance  In the context of complex networks, attack tolerance is the network's robustness meaning the ability to maintain the overall connectivity and diameter of the network as nodes are removed. Several graph metrics have been proposed to predicate network robustness. Algebraic connectivity is a graph metric that shows the best graph robustness among them. If an attack was to be mounted on a network, it would not be through random nodes but ones that were the most significant to the network. Different methods of ranking are utilized to determine the nodes priority in the network. This form of attack prioritizes the most connected nodes as the most important ones. This takes into account the network (represented by graph formula_1) changing over time, by analyzing the network as a series of snapshots (indexed by formula_2); we denote the snapshot at time formula_3 by formula_4. The average of the degree of a node, labeled formula_5, within a given snapshot formula_6, throughout a time interval (a sequence of formula_7 snapshots) formula_8, is given by:  formula_9  This form of attack prioritizes nodes that occur most frequently over a period of time. The equation below calculates the frequency that a node (i) occurs in a time interval formula_8. When the node is present during the snapshot then equation is equal to 1, but if the node is not present then it is equal to 0.  formula_11  Where  formula_12  This form of attack prioritizes nodes by the summation of temporal distances from one node to all other nodes over a period of time. The equation below calculates the temporal distance of a node (i) by averaging the sum of all the temporal distances for the interval [t1,tn].  formula_13  Not all networks are the same, so it is no surprise that an attack on different networks would have different results. The common method for measuring change in the network is through the average of the size of all the isolated clusters, , and the fraction of the nodes contained in the largest cluster, S. When no nodes have been attacked, both S and  equal 1. In the ER model, the network generated is homogeneous, meaning each node has the same number of links. This is considered to be an exponential network. When comparing the connectivity of the ER model when it undergoes random failures vs directed attacks, we are shown that the exponential network reacts the same way to a random failure as it does to a directed attack.