Influence maximization is a well-studied problem of finding a small set of highly influential individuals in a social network such that the spread of influence under a certain diffusion model is maximized. We propose new diffusion models that incorporate the time-decaying phenomenon by which the power of influence decreases with elapsed time. In standard diffusion models such as the independent cascade and linear threshold models, each edge in a network has a fixed power of influence over time. However, in practical settings, such as rumor spreading, it is natural for the power of influence to depend on the time influenced. We generalize the independent cascade and linear threshold models with time-decaying effects. Moreover, we show that by using an analysis framework based on submodular functions, a natural greedy strategy obtains a solution that is provably within (1 − 1/e) of optimal. In addition, we propose theoretically and practically fast algorithms for the proposed models. Experimental results show that the proposed algorithms are scalable to graphs with millions of edges and outperform baseline algorithms based on a state-of-the-art algorithm.