We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.
This research in epidemiology is relevant in a much broader context, since many dynamical processes, such as the diffusion of information and opinions, can be modeled as epidemics. Indeed, the spread of competing products has been modeled in this way as well @cite_12 @cite_28 . @cite_12 the authors study a @math '' model of competing viruses with perfect mutual immunity in a mean-field setting for fixed networks, and conclude that the winner takes all'', i.e., one virus will take over, while in @cite_28 they study what level of partial immunity allows for coexistence of the two viruses. A related model of competing first passage percolation has been studied in probability theory on various network topologies, including random regular graphs @cite_51 and scale-free networks @cite_35 ; the conclusion again is that the winner takes all. In contrast, in many current markets we observe that competing products coexist, even when they are mutually exclusive.
@cite_28 @cite_35 @cite_51 @cite_12
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Suppose we have two competing ideas products viruses, that propagate over a social or other network. Suppose that they are strong virulent enough, so that each, if left alone, could lead to an epidemic. What will happen when both operate on the network? Earlier models assume that there is perfect competition: if a user buys product 'A' (or gets infected with virus 'X'), she will never buy product 'B' (or virus 'Y'). This is not always true: for example, a user could install and use both Firefox and Google Chrome as browsers. Similarly, one type of flu may give partial immunity against some other similar disease. In the case of full competition, it is known that 'winner takes all,' that is the weaker virus product will become extinct. In the case of no competition, both viruses survive, ignoring each other. What happens in-between these two extremes? We show that there is a phase transition: if the competition is harsher than a critical level, then 'winner takes all;' otherwise, the weaker virus survives. These are the contributions of this paper (a) the problem definition, which is novel even in epidemiology literature (b) the phase-transition result and (c) experiments on real data, illustrating the suitability of our results. 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. The strong analogy between biological viruses and their computational counterparts has motivated the authors to adapt the techniques of mathematical epidemiology to the study of computer virus propagation. In order to allow for the most general patterns of program sharing, a standard epidemiological model is extended by placing it on a directed graph and a combination of analysis and simulation is used to study its behavior. The conditions under which epidemics are likely to occur are determined, and, in cases where they do, the dynamics of the expected number of infected individuals are examined as a function of time. It is concluded that an imperfect defense against computer viruses can still be highly effective in preventing their widespread proliferation, provided that the infection rate does not exceed a well-defined critical epidemic threshold. > The systematic study of large-scale networks has unveiled the ubiquitous presence of connectivity patterns characterized by large-scale heterogeneities and unbounded statistical fluctuations. These features affect dramatically the behavior of the diffusion processes occurring on networks, determining the ensuing statistical properties of their evolution pattern and dynamics. In this article, we present a stochastic computational framework for the forecast of global epidemics that considers the complete worldwide air travel infrastructure complemented with census population data. We address two basic issues in global epidemic modeling: (i) we study the role of the large scale properties of the airline transportation network in determining the global diffusion pattern of emerging diseases; and (ii) we evaluate the reliability of forecasts and outbreak scenarios with respect to the intrinsic stochasticity of disease transmission and traffic flows. To address these issues we define a set of quantitative measures able to characterize the level of heterogeneity and predictability of the epidemic pattern. These measures may be used for the analysis of containment policies and epidemic risk assessment.