Network Attacks Based on Vertex Coverings

The robustness of networks is an important problem that has been studied in a variety of situations, see for instance [3] or the survey [5] and the references therein. A milestone in the study of this problem was the discovery by Albert et al. of the dramatic effect of targeted attacks on networks whose vertex degrees follow a power-law distribution, like the Internet and the World Wide Web [1]. Attack strategies are often described as vertex or edge deletion strategies. We focus in this paper on vertex deletion. The vertices (also called nodes) of the network are sorted according to some property such as degree and are removed from the network according to this property. The resilience of a network to such vertex removal is important in areas ranging from vaccination strategies in epidemiology to fighting criminal organizations. In this paper we propose two vertex deletion strategies based on the structure of vertex covers of the network. Given a simple graph, e.g. a network N , a vertex cover C of N is a set of vertices such that each edge of N is incident to at least one vertex of C. We say that C is minimal if it does not contain another vertex cover. C is said to be minimum if it is of minimal cardinality among all vertex covers of N . For a given vertex v of a network N we define its covering degree, cd(v) as the number of minimal vertex covers of N that contain v. We also define the covering index of v, ci(v) as the number of minimum vertex covers of N that contain v plus the quotient of cd(v) divided by the total number of minimal vertex covers of N . We study these degrees and indices in several network models, such as Erdös-Renyi random graphs, Albert-Barabasi networks and the Watts-Strogatz small world model. In our experiments we found that the covering degree of a vertex was correlated to other widely used properties like vertex degree and betweeness centrality, but surprisingly we found that the covering index was not correlated to any of the other measures, including covering degree. We used the connectivity of the network to measure the resilience of the network to attacks based on covering degree and index and found that the covering index broke the studied networks in a bigger number of smaller disconnected components than any of the other strategies studied.