The influence of the network characteristics on the virus spread is analyzed in a new-the -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The -intertwined model has been compared with the exact 2 -state Markov model and with previously proposed "homogeneous" or "local" models. The sharp epidemic threshold , which is a consequence of mean field theory, is rigorously shown to be equal to = 1 ( max ( )), where max ( ) is the largest eigenvalue-the spectral radius-of the adjacency matrix . A continued fraction expansion of the steady-state infection probability at node is presented as well as several upper bounds.Index Terms-Epidemic threshold, Markov theory, mean field theory, spectral radius, virus spread.

Abstract-Security breaches and attacks are critical problems in today's networking. A key-point is that the security of each host depends not only on the protection strategies it chooses to adopt but also on those chosen by other hosts in the network. The spread of Internet worms and viruses is only one example. This class of problems has two aspects. First, it deals with epidemic processes, and as such calls for the employment of epidemic theory. Second, the distributed and autonomous nature of decision-making in major classes of networks (e.g., P2P, adhoc, and most notably the Internet) call for the employment of game theoretical approaches. Accordingly, we propose a unified framework that combines the N-intertwined, SIS epidemic model with a noncooperative game model.We determine the existence of a Nash equilibrium of the respective game and characterize its properties. We show that its quality, in terms of overall network security, largely depends on the underlying topology. We then provide a bound on the level of system inefficiency due to the noncooperative behavior, namely, the "price of anarchy" of the game. We observe that the price of anarchy may be prohibitively high, hence we propose a scheme for steering users towards socially efficient behavior.

Abstract-The effect of virus spreading in a telecommunication network, where a certain curing strategy is deployed, can be captured by epidemic models. In the N -intertwined model proposed and studied in [1], [2], the probability of each node to be infected depends on the curing and infection rate of its neighbors. In this paper, we consider the case where all infection rates are equal and different values of curing rates can be deployed within a given budget, in order to minimize the overall infection of the network. We investigate this difficult optimization together with a related problem where the curing budget must be minimized within a given level of network infection. Some properties of these problems are derived and several solution algorithms are proposed. These algorithms are compared on two real world network instances, while Erdös-Rényi graphs and some special graphs such as the cycle, the star, the wheel and the complete bipartite graph are also addressed.

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Abstract-Communication networks are prone to virus and worms spreading and cascading failures. Recently, a number of social networking worms have spread over public Web sites. Another example is error propagation in routing tables, such as in BGP tables. The immunization and error curing applied to these scenarios are not fast enough. There have been studies on the effect of isolating and curing network elements, however, the proposed strategies are limited to node removals. This paper proposes a link isolation strategy based on the quarantining of susceptible clusters in the network. This strategy aims to maximize the epidemic control while minimizing the impact on the clusters performance. We empirically study the influence of clustering on robustness against epidemics in several real-world and artificial networks. Our results show an average curing rate improvement above 50% for the studied real-world networks under analysis.