Sparse resource allocation to shape a network dynamical process is studied. Specifically, we consider allocating limited distributed control resources among a subset of a network's nodes, to minimize the dominant eigenvalue of a linear dynamical spread process associated with the network. Structural characterizations of the closed-loop dynamics at the optimum are obtained. These results are then used to 1) develop constructive algorithms for optimal resource allocation, 2) identify limits on the control performance, and 3) understand the relationship between the network's graph and the optimal resource profile. While the focus here is on a simplified linear model, an exploratory study of the design's applicability to realistic stochastic and nonlinear spread processes is undertaken, via simulation examples. As a whole, this study advances a research thrust on disease spread control in networks, toward the realistic paradigm that control resources can only be allocated at a subset of network locations.