How can we optimize the topology of a networked system to bring a flu under control, propel a video to popularity, or stifle a network malware in its infancy? Previous work on information diffusion has focused on modeling the diffusion dynamics and selecting nodes to maximize/minimize influence. Only a paucity of recent studies have attempted to address the network modification problems, where the goal is to either facilitate desirable spreads or curtail undesirable ones by adding or deleting a small subset of network nodes or edges. In this paper, we focus on the widely studied linear threshold diffusion model, and prove, for the first time, that the network modification problems under this model have supermodular objective functions. This surprising property allows us to design efficient data structures and scalable algorithms with provable approximation guarantees, despite the hardness of the problems in question. Both the time and space complexities of our algorithms are linear in the size of the network, which allows us to experiment with millions of nodes and edges. We show that our algorithms outperform an array of heuristics in terms of their effectiveness in controlling diffusion processes, often beating the next best by a significant margin.