  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.
