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Recent years have witnessed a widespread interest on methods using both link structure and node information for link prediction on graphs. One of the state-of-the-art methods is Link Propagation which is a new semi-supervised learning algorithm for link prediction on graphs based on the popularly-studied label propagation by exploiting information on similarities of links and nodes. Despite its efficiency and effectiveness compared to other methods, its applications were still limited due to the computational time and space constraints. In this paper, we propose fast and scalable algorithms for the Link Propagation by introducing efficient procedures to solve large linear equations that appear in the method. In particular, we show how to obtain a compact representation of the solution to the linear equations by using a non-trivial combination of techniques in linear algebra to construct algorithms that are also effective for link prediction on dynamic graphs. These enable us to apply the Link Propagation to large networks with more than 400,000 nodes. Experiments demonstrate that our approximation methods are scalable, fast, and their prediction qualities are comparably competitive.