Continuous Activity Maximization in Online Social Networks

Activity maximization is a task of seeking a small subset of users in a given social network that makes the expected total activity benefit maximized. This is a generalization of many real applications. In this paper, we extend activity maximization problem to that under the general marketing strategy <inline-formula><tex-math notation="LaTeX">$\boldsymbol{x}$</tex-math></inline-formula>, which is a <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula>-dimensional vector from a lattice space and has probability <inline-formula><tex-math notation="LaTeX">$h_u(\boldsymbol{x})$</tex-math></inline-formula> to activate a node <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> as a seed. Based on that, we propose the continuous activity maximization (CAM) problem, where the domain is continuous and the seed set we select conforms to a certain probability distribution. It is a new topic to study the problem about information diffusion under the lattice constraint, thus, we address the problem systematically here. First, we analyze the hardness of CAM and how to compute the objective function of CAM accurately and effectively. We prove this objective function is monotone, but not DR-submodular and not DR-supermodular. Then, we develop a monotone and DR-submodular lower bound and upper bound of CAM, and apply sampling techniques to design three unbiased estimators for CAM, its lower bound and upper bound. Next, adapted from IMM algorithm and sandwich approximation framework, we obtain a data-dependent approximation ratio. This process can be considered as a general method to solve those maximization problem on lattice but not DR-submodular. Last, we conduct experiments on three real-world datasets to evaluate the correctness and effectiveness of our proposed algorithms.