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 [3,17,35] (b) the phase-transition result and (c) experiments on real data, illustrating the suitability of our results.

Given two competing products (or memes, or viruses etc.) spreading over a given network, can we predict what will happen at the end, that is, which product will 'win', in terms of highest market share? One may naïvely expect that the better product (stronger virus) will just have a larger footprint, proportional to the quality ratio of the products (or strength ratio of the viruses). However, we prove the surprising result that, under realistic conditions, for any graph topology, the stronger virus completely wipes-out the weaker one, thus not merely 'winning' but 'taking it all'. In addition to the proofs, we also demonstrate our result with simulations over diverse, real graph topologies, including the social-contact graph of the city of Portland OR (about 31 million edges and 1 million nodes) and internet AS router graphs. Finally, we also provide real data about competing products from Google-Insights, like Facebook-Myspace, and we show again that they agree with our analysis.