Abstract
This article attempts to provide viral marketeers guidance in terms of an investment level that could help capture some desired <inline-formula> <tex-math notation="LaTeX">\gamma </tex-math></inline-formula> percentage of the market share by some target time <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula> with the desired level of confidence. To do this, we first introduce a generalized diffusion model for social networks. A distance-dependent random graph is then considered as a model for the underlying social network, which we use to analyze the proposed diffusion model. Using the fact that vertices degrees have an almost Poisson distribution in distance-dependent random networks, we then provide a lower bound on the probability of the event that the time it takes for an idea (or a product, campaign, disease, and so on) to dominate a prespecified <inline-formula> <tex-math notation="LaTeX">\gamma </tex-math></inline-formula> percentage of a social network (denoted by <inline-formula> <tex-math notation="LaTeX">R_\gamma </tex-math></inline-formula>) is smaller than some preselected target time <inline-formula> <tex-math notation="LaTeX">t > 0 </tex-math></inline-formula>, i.e., we find a lower bound on the probability of the event <inline-formula> <tex-math notation="LaTeX">\{R_\gamma \leq t\} </tex-math></inline-formula>. Simulation results performed over a wide variety of networks, including random and real world, are then provided to verify that our bound indeed holds in practice. The Kullback-Leibler divergence measure is used to evaluate the performance of our lower bound over these groups of networks, and as expected, we note that for networks that deviate more from the Poisson degree distribution, our lower bound weakens. In the case where absolute/full domination of the market-share is desired, under the linear threshold diffusion model, a particular case of our generalized diffusion model, an upper bound on the size of the seed set is derived, and a selection algorithm is developed to show its tightness. This is also extended to the partial market-share situation.