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.
In epidemiology, pathogens fight for survival, and a central topic is the spread of diseases @cite_29 @cite_39 . In classic models of epidemic spreading, individuals are characterized by the stage of the disease in them: they can be susceptible, infected, or recovered removed, leading to the SIR, SIRS and SIS models. The main object of study is the epidemic threshold, i.e., under what conditions does the disease die out or take over the population. An important finding is that the network structure underlying the population of individuals greatly affects the epidemic threshold; in particular, on scale-free networks the epidemic threshold vanishes, and diseases can spread even when infection probabilities are tiny @cite_0 @cite_8 @cite_44 @cite_13 .
@cite_8 @cite_29 @cite_39 @cite_0 @cite_44 @cite_13
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The spread of diseases in human and other populations can be described in terms of networks, where individuals are represented by nodes and modes of contact by edges. Similar models can be applied to the spread of viruses on the Internet. In their Perspective, Lloyd and May discuss the similarities and differences between the dynamics of computer viruses and infections of human and other populations. Part 1 Microparasites: biology of host-microparasite associations the basic model - statics static aspects of eradication and control the basic model - dynamics dynamic aspects of eradication and control beyond the basic model - empirical evidence of inhomogeneous mixing age-related transmission rates genetic heterogeneity social heterogeneity and sexually transmitted diseases spatial and other kinds of heterogeneity endemic infections in developing countries indirectly transmitted microparasites. Part 2 Macroparasites: biology of host-macroparasite associations the basic model - statics the basic model - dynamics acquired immunity heterogeneity within the human community indirectly transmitted helminths experimental epidemiology parasites, genetic variability, and drug resistance the ecology and genetics of host-parasite associations. The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks. We discuss properties of infection processes on scale-free networks, relating them to the node-connectivity distribution that characterizes the network. Considering the epidemiologically important case of a disease that confers permanent immunity upon recovery, we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual’s degree of connectivity within the population. As in an earlier study @R. Pastor-Satorras and A. Vesipignani, Phys. Rev. Lett. 86, 3200 2001!; Phys. Rev. E. 63, 006117 2001!# for an infection that did not confer immunity upon recovery, the epidemic process—in contrast with many traditional epidemiological models—does not exhibit threshold behavior, and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale-free network. Finally, we discuss effects that arise from finite population sizes, showing that networks of finite size do exhibit threshold effects: infections cannot spread for arbitrarily low transmission probabilities. The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible infective removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are nonuniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.