Seed investment bounds for viral marketing under generalized diffusion

This paper attempts to provide viral marketeers guidance in terms of an investment level that could help capture some desired γ percentage of the market-share by some target time t with a desired level of confidence. To do this, we first introduce a 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, disease, etc.) to dominate a pre-specified γ percentage of a social network (denoted by R_γ) is smaller than some pre-selected target time t > 0, i.e., we find a lower bound on the probability of the event {R_γ ≤ t}. Simulation results performed over a wide variety of networks, including random as well as real-world, are then provided to verify that our bound indeed holds in practice. The Kullback-Leibler divergence measure is used to evaluate 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 does worse.