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.
 Department of Physics, University of Seoul, Seoul 130-743, Korea(Dated: May 22, 2011)We study the non-equilibrium phase transition in a model for epidemic spreading on scale-freenetworks. The model consists of two particle species A and B, and the coupling between them istaken to be asymmetric; A induces B while B suppresses A. This model describes the spreadingof an epidemic on networks equipped with a reactive immune system. We present analytic resultson the phase diagram and the critical behavior, which depends on the degree exponent γ of theunderlying scale-free networks. Numerical simulation results that support the analytic results arealso presented. Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network. In most pathogens, multiple strains are maintained within host populations. Quantifying the mechanisms underlying strain coexistence would aid public health planning and improve understanding of disease dynamics. We argue that mathematical models of strain coexistence, when applied to indistinguishable strains, should meet criteria for both ecological neutrality and population genetic neutrality. We show that closed clonal transmission models which can be written in an “ancestor-tracing” form that meets the former criterion will also satisfy the latter. Neutral models can be a parsimonious starting point for studying mechanisms of strain coexistence; implications for past and future studies are discussed.