In this paper, we study a novel problem of influence maximization on social networks: Given a period of promotion time, a set of target users and a network in which each node can be activated by its neighbors multiple times, we aim at determining the k most influential seeds to maximize the total frequency of activations received by these target users. The promising viral marketing paradigm on social network is different from the current research in two main aspects. First, instead of maximizing the message spread over the entire social network, we focus on the target market since the business vendors almost specify the group of target users before designing its marketing strategy. Second, the status of a user is no longer a binary indicator representing either active or inactive. In the new model, the user status turns to be an integer value reflecting the amount of influences delivered to that user. In this paper, we prove the NP-hard nature of this challenging problem. We further present several strategies, including an efficient heuristic algorithm based on the simulated annealing optimization concept and a greedy algorithm as the baseline, to select the initial k seeds in pursuit of resulting quality close to the optimal one. As demonstrated in the empirical study on real data, instead of only providing the flexibility of striking a compromise between the execution efficiency and the resulting quality, our proposed heuristic algorithm can achieve high efficiency and meanwhile can obtain the target acceptance frequency even better than the greedy result in some cases, demonstrating its prominent feasibility to resolve the challenging problem efficiently.
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