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... in these networks, regardless of transmission probability between individuals, a point that was first made, in the context of models of computer virus epidemiology, by Pastor-Satorras and Vespignani =-=[333, 336]-=-, although, as pointed out by Lloyd and May [267, 277], precursors of the same result can be seen in earlier work of May and Anderson [276]. May and Anderson studied traditional (fully mixed) differen...

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... to k−α , for a constant α typically between 2 and 3. Many real-world networks have this property [20], including the social network defined by blog-to-blog links [19]. Pastor-Satorras and Vespignani =-=[25]-=- analyze an SIS model of computer virus propagation in power-law networks, showing that—in stark contrast to random or regular networks—the epidemic threshold is zero, so an epidemic will always occur...

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...l, but it is very difficult to get a power-law degree distribution from a network without the power-law decaying degree distribution. 4.2 Metrics of interest We begin the analysis of online social network topologies by looking at their degree distributions. Networks of a power-law degree distribution, P (k) ∼ k−γ , where k is the node degree and γ ≤ 3, attest to the existence of a relatively small number of nodes with a very large number of links. These networks also have distinguishing properties, such as vanishing epidemic threshold, ultra-small worldness, and robustness under random errors [11, 12, 13, 14]. The degree distribution is often plotted as a complementary cumulative probability function (CCDF), ℘(k) ≡ R ∞ k P (k′)dk′ ∼ k−α ∼ k−(γ−1). As a power-law distribution shows up as a straight WWW 2007 / Track: Semantic Web Session: Semantic Web and Web 2.0 837 line in a log-log plot, the exponent of a power-law distribution is a representative characteristic, distinguishing one from others. Next, we examine the clustering coefficient. The clustering coefficient of a node is the ratio of the number of existing links over the number of possible links between its neighbors. Given a network G = (...

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...having a small diameter [2]. Finding patterns, laws and regularities in large real networks has numerous applications, from criminology and law enforcement [8] to analyzing virus propagation patterns =-=[19]-=-and understanding networks of regulatory genes and interacting proteins [3] and so on. Discovering and listing such laws is only the first step. Ideally, we would like a generative model with the foll...

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...aspects of the problem studied here. Much of the work has focused on infinite scale-free graphs, establishing conditions under which epidemics either die out or sustain themselves forever; see, e.g., =-=[15]-=-. Some work has been done on finite graphs, using primarily heuristic arguments to obtain conditions under which epidemics spread quickly or not. For example, [16] uses a mean field approximation for ...

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...ly vulnerable to attacks that target the high-degree hub nodes [4]. It has been similarly argued that these high-degree hubs are a primary reason for the epidemic spread of computer worms and viruses =-=[37, 9]-=-. The presence of highly connected central nodes in a network having a power law degree distribution is the essence of the so-called scale-free network models, which have been a popular theme in the s...

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... represent modern networks. Staniford et al. [23] reported a study of the Code Red worm propagation, but did not attempt to create an analytic model. The more recent studies by Pastor-Satorras et al. =-=[16, 17, 18, 19, 20]-=- and Barabási et al. [2, 4] focused on epidemic models for power-law networks. This work aims to develop a general analytic model of virus propagation. Specifically, we are interested in models that c...