Bio-inspired paradigms are proving to be useful in analyzing propagation and dissemination of information in networks. In this paper we explore the use of multi-type branching processes to analyse viral properties of content in a social network, with and without competition from other sources. We derive and compute various virality measures, e.g., probability of virality, expected number of shares, or the rate of growth of expected number of shares etc. They allow one to predict the emergence of global macro properties (e.g., viral spread of a post in the entire network) from the laws and parameters that determine local interactions. The local interactions, greatly depend upon the structure of the timelines holding the content and the number of friends (i.e., connections) of users of the network. We then formulate a non-cooperative game problem and study the Nash equilibria as a function of the parameters. The branching processes modelling the social network under competition turn out to be decomposable, multi-type and continuous time variants. For such processes types belonging to different sub-classes evolve at different rates and have different probabilities of extinction etc. We compute content provider wise extinction probability, rate of growth etc. We also conjecture the content-provider wise growth rate of expected shares.
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