Overall Evaluations on Benefits of Influence When Disturbed by Rivals

Influence maximization (IM) is a representative and classic problem that has been studied extensively before. The most important application derived from the IM problem is viral marketing. Take us as a promoter, we want to get benefits from the influence diffusion in a given social network, where each influenced (activated) user is associated with a benefit. However, there is often competing information initiated by our rivals diffusing in the same social network at the same time. Consider such a scenario, a user is influenced by both my information and my rivals' information. Here, the benefit from this user should be weakened to certain degree. How to quantify the degree of weakening? Based on that, we propose an overall evaluations on benefits of influence (OEBI) problem. We prove the objective function of the OEBI problem is not monotone, not submodular, and not supermodular. Fortunately, we can decompose this objective function into the difference of two submodular functions and adopt a modular-modular procedure to approximate it with a data-dependent approximation guarantee. Because of the difficulty to compute the exact objective value, we design a group of unbiased estimators by exploiting the idea of reverse influence sampling, which can improve time efficiency significantly without losing its approximation ratio. Finally, numerical experiments on real datasets verified the effectiveness of our approaches regardless of performance and efficiency.