Viral spread on large graphs has many real-life applications such as malware propagation in computer networks and rumor (or misinformation) spread in Twitter-like online social networks. Although viral spread on large graphs has been intensively analyzed on classical models such as Susceptible-Infectious-Recovered, there still exits a deficit of effective methods in practice to contain epidemic spread once it passes a critical threshold. Against this backdrop, we explore methods of containing viral spread in large networks with the focus on sparse random networks. The viral containment strategy is to partition a large network into small components and then to ensure the sanity of all messages delivered across different components. With such a defense mechanism in place, an epidemic spread starting from any node is limited to only those nodes belonging to the same component as the initial infection node. We establish both lower and upper bounds on the costs of inspecting inter-component messages. We further propose heuristic-based approaches to partition large input graphs into small components. Finally, we study the performance of our proposed algorithms under different network topologies and different edge weight models.
The percolation theory has established the critical threshold for wide-scale epidemic spreading and has been widely applied to study epidemic spreading in diverse network structures, such as small-world networks @cite_29 , heterogeneous networks @cite_2 , and sensor networks @cite_1 . In the context of scale-free networks, selectively immunizing those highly connected nodes is an effective approach to slowing down epidemic spread in such networks @cite_20 @cite_27 . In both problems of containing social-based malware and misinformation spread in social networks, however, the key challenge that faces node immunization is the difficulty of interacting with individual users due to the distributed nature of social networks. An alternative approach would be to achieve node immunization by sanitizing all messages that come to or come from those nodes to be immunized. In our problem, however, this may not be the most cost-effective approach because highly connected nodes could generate a large number of communication messages.
@cite_29 @cite_1 @cite_27 @cite_2 @cite_20
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Since full coverage treatment is not feasible due to limited resources, we need to utilize an immunization strategy to effectively distribute the available vaccines. On the other hand, the structure of contact network among people has a significant impact on epidemics of infectious diseases (such as SARS and influenza) in a population. Therefore, network-based immunization strategies aim to reduce the spreading rate by removing the vaccinated nodes from contact network. Such strategies try to identify more important nodes in epidemics spreading over a network. In this paper, we address the effect of overlapping nodes among communities on epidemics spreading. The proposed strategy is an optimized random-walk based selection of these nodes. The whole process is local, i.e. it requires contact network information in the level of nodes. Thus, it is applicable to large-scale and unknown networks in which the global methods usually are unrealizable. Our simulation results on different synthetic and real networks show that the proposed method outperforms the existing local methods in most cases. In particular, for networks with strong community structures, high overlapping membership of nodes or small size communities, the proposed method shows better performance. Complex networks such as the sexual partnership web or the Internet often show a high degree of redundancy and heterogeneity in their connectivity properties. This peculiar connectivity provides an ideal environment for the spreading of infective agents. Here we show that the random uniform immunization of individuals does not lead to the eradication of infections in all complex networks. Namely, networks with scale-free properties do not acquire global immunity from major epidemic outbreaks even in the presence of unrealistically high densities of randomly immunized individuals. The absence of any critical immunization threshold is due to the unbounded connectivity fluctuations of scale-free networks. Successful immunization strategies can be developed only by taking into account the inhomogeneous connectivity properties of scale-free networks. In particular, targeted immunization schemes, based on the nodes' connectivity hierarchy, sharply lower the network's vulnerability to epidemic attacks. The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations. We study the optimal control problem of maximizing the spread of an information epidemic on a social network. Information propagation is modeled as a susceptible-infected (SI) process, and the campaign budget is fixed. Direct recruitment and word-of-mouth incentives are the two strategies to accelerate information spreading (controls). We allow for multiple controls depending on the degree of the nodes individuals. The solution optimally allocates the scarce resource over the campaign duration and the degree class groups. We study the impact of the degree distribution of the network on the controls and present results for Erdős-Renyi and scale-free networks. Results show that more resource is allocated to high-degree nodes in the case of scale-free networks, but medium-degree nodes in the case of Erdős-Renyi networks. We study the effects of various model parameters on the optimal strategy and quantify the improvement offered by the optimal strategy over the static and bang-bang control strategies. The effect of the time-varying spreading rate on the controls is explored as the interest level of the population in the subject of the campaign may change over time. We show the existence of a solution to the formulated optimal control problem, which has nonlinear isoperimetric constraints, using novel techniques that is general and can be used in other similar optimal control problems. This work may be of interest to political, social awareness, or crowdfunding campaigners and product marketing managers, and with some modifications may be used for mitigating biological epidemics.