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The recently recognized nature of power–law degree distributions in real networks revived a vast interest in networks and their representations as scale-free ones. In addition, the implication of such configurations in terms of network resilience is a particularly intriguing topic.

Many complex real systems proved to be robust in the face of errors, where local failures in the system rarely lead to a system wide crisis. This stability has been commonly attributed to the redundant connections between the system underlying network components. In 2000, Albert, Jeong and Barab´asi [AJB00] demonstrated that this robustness to random failures is only displayed in one class of complex systems, namely those that can be characterized as scale free. Because of its significant contributions to the study of network resilience, we now highlight the results of their paper.

Complex networks can generally be divided into two types according to their de- gree distribution. The first is the well studied case of random graphs characterized by a degree distribution p(k) peaked around the its average hki which decays expo-

Omneia R.H. Ismail – PhD Thesis – McMaster University – CES

nentially for large k. The most famous is the model of Erd¨os and R´enyi which leads to a homogeneous network where each node has approximately the same number of links. The second type are scale-free networks, where the connectivity distribution is described by a power law distribution. Unlike the previous case, it is an inhomoge- neous network with the probability of nodes with high links statistically significantly different from zero. In comparing their robustness, Albert, Jeong and Barab´asi con- structed two networks with an equal number of nodes: a random network, that is, a Erd¨os-R´enyi random graph (n, p), and a scale–free graph. They then examined the effect of two types of node removal: a random node removal, where the node to be removed is chosen at random (an error or failure of a node), and a removal of nodes chosen according to their degree, such as removing the most connected nodes (an attack). The effect on the network was then examined using its diameter and giant components.

Recall that the diameter is defined as the average length of the shortest paths connecting any two nodes and reflects the nodes’ ability to communicate with one another. The effect of removal of nodes on the network diameter can be summarized as follows:

• Error tolerance (random removal of nodes): when a small fraction f of nodes is removed there is in general an observed increase in the diameter for both types of networks (random and scale free). For the random network the diameter increases monotonically with f , which can be easily explained in terms of the homogeneity of the network nodes: since all nodes have approximately the same number of edges the removal of each node causes the same amount of increase in the diameter. On the other hand we find a drastically different behaviour for the scale–free network: the diameter remained unchanged above a certain fraction f (say 5% of node removal). This robustness to node failure is again attributed to the degree distribution of the nodes, where under the power law the majority of nodes have only a small number of edges. Thus nodes with small connectivity

will be the one most likely selected under the random removal scenario. That in turn would not alter the network diameter all that much.

• Attack survivability (removal of most connected nodes): an ‘informed’ network attacker will target the most connected nodes of the network. To simulate such an attack the authors removed nodes in a decreasing order in terms of their degree starting with the most connected one. Since in the random graph the nodes have approximately the same number of connections, there is no witnessed difference between the previous scenario of random removal and the attack scenario. For scale–free networks, the situation is quite different, with the diameter increasing rapidly with each removal, and by the time of removing 5% of the most connected nodes the diameter is doubled. The scale–free network is thus vulnerable to attacks, and the removal of its highest connected nodes causes a drastic alteration of the network configuration.

Viewing a network as clusters of nodes, an outcome of removing a node is the process of cutting nodes from the main (i.e., giant) cluster. In other words, removing a node from a network means removing the edges this node has with other nodes, which in turn might cut these nodes from the giant cluster. Measuring the giant cluster as a fraction S of the total network size (in terms of the number of nodes), the authors obtain the following:

• Error tolerance (random removal of nodes): in random graphs, when a fraction f of nodes is removed, a threshold–like behaviour is observed, in which for a small f there is no major change in S, yet as f increases to reach a threshold fc (found to be fc ≃ 0.28 in their case), the giant cluster falls into pieces

and S ≃ 0. However, for the scale–free network, as expected, S remains close to its is starting value, where the random removal of nodes causes only single nodes isolations, not the isolation of clusters of nodes (extremely delayed critical point).

Omneia R.H. Ismail – PhD Thesis – McMaster University – CES

• Attack survivability (removal of most connected nodes): while the random graph displays the same behaviour under attach as that under error, the scale–free network again displays a different behaviour under the two scenarios. Unlike the case of random removal of node, in the attack scenario there is an early critical value fc, smaller than that observed in random graph (observed to be

fc ≃ 0.18), above which the network breaks apart into isolated clusters.

From the above two measures, the authors concluded that scale–free networks display an exceptionally high degree of robustness (tolerance against random failures), nevertheless that tolerance comes at the expense of its attack vulnerability.

Following the work of Albert, Jeong and Barab´asi, an enormous interest has been devoted to study the robustness of scale–free networks. Crucitti et al. [CLMR04] examined the resilience of random and scale–free networks with respect to error and attack using three criteria for identifying the node importance: the degree of the node, the number of shortest paths that pass through the node, and the number of shortest paths recalculated every time a node is removed. Also they used a slightly different measure for effect of node removal on the network called global efficiency, defined as

E(G) = 1 n(n − 1) X i6=j∈G 1 tij

for tij is the shortest path length between nodes i and j in graph G. The measure is

a modification of using the shortest path length (diameter), with the advantages that it is well-defined for non-connected networks and is a global property of the network. Their results agree with those of Albert, Jeong and Barab´asi for the random removal of nodes and for the intentional attack on nodes with the highest degrees. On the other hand when considering the second and third criteria for choosing the important nodes to remove, they found that the results for the scale–free network do not substantially differ from these reported from choosing important nodes based on their degrees. The difference is more pronounced for the random graph case, where

attacks based on a recalculated number of shortest paths cause a greater damage than that based on degree. They attributed these observations to the fact that in scale– free networks the nodes with the highest degree are also the nodes with the highest number of shortest paths, whereas for the random networks there is not a perfect parallel between them.

The above discussion of error and attack tolerance should not be confused with an- other closely related yet different topic of the cascading failures in complex networks. In our above discussion we examined the effect of node removal on the network con- nectivity, diameter or effectiveness, where it should be emphasized that upon the removal of the node, all its edges are also removed from the network. Then the re- sulting effect on the network is examined. The second class of models of cascading failures is addressing the effect of node removal or failure as the load on that failed node has to be rerouted to other nodes, which may eventually lead to an avalanche of overloads on other nodes that are not able to handle the extra load or traffic. This class of models is sometimes referred to as the dynamic approach of node failures, opposed to the static one represented earlier. The main mechanism responsible for a system-wide crisis rests on the process of load redistribution, a problem well com- prehended in complex communication, transportation, electrical power grids systems and traffic networks. Under such consideration, it has been shown that a single node failure can collapse the entire system due to the dynamics of the redistribution of its loads.