This paper studies how a centralized planner can modify the structure of a social or information network to reduce polarization. First, polarization is found to be highly dependent on degree and structural properties of the network -- including the well-known isoperimetric number (i.e., Cheeger constant). We then formulate the planner's problem under full information, and motivate disagreement-seeking and coordinate descent heuristics. A novel setting for the planner in which the population's innate opinions are adversarially chosen is introduced, and shown to be equivalent to maximization of the Laplacian's spectral gap. We prove bounds for the effectiveness of a strategy that adds edges between vertices on opposite sides of the cut induced by the spectral gap's eigenvector. Finally, these strategies are evaluated on six real-world and synthetic networks. In several networks, we find that polarization can be significantly reduced through the addition of a small number of edges.