Adaptive Influence Maximization: If Influential Node Unwilling to Be the Seed.

Influence maximization problem attempts to find a small subset of nodes that makes the expected influence spread maximized, which has been researched intensively before. They all assumed that each user in the seed set we select is activated successfully and then spread the influence. However, in the real scenario, not all users in the seed set are willing to be an influencer. Based on that, we consider each user associated with a probability with which we can activate her as a seed, and we can attempt to activate her many times. In this paper, we study the adaptive influence maximization with multiple activations (Adaptive-IMMA) problem, where we select a node at each step, observe whether she accepts to be a seed, if yes, wait to observe the influence diffusion process; If no, we can attempt to activate her again with higher cost, or select other node as a seed. We model the multiple activations mathematically, and define it on the domain of integer lattice. We propose a new concept, adaptive DR-submodularity, and show our Adaptive-IMMA is the problem that maximizing an adaptive monotone and DR-submodular function under the knapsack constraint. Adaptive DR-submodular maximization problem is never covered by any existing studies. Thus, we summarize its properties and study its approximability comprehensively, which is a non-trival generalization of existing analysis about adaptive submodularity. Besides, to overcome the difficulty to estimate the influence spread, we combine adaptive greedy policy with sampling techniques without losing approximation ratio but reducing the time complexity. Finally, we conduct experiments on several real datasets to evaluate the effectiveness and efficiency of our proposed algorithms.