In diffusion social learning over weakly-connected graphs, it has been shown that influential agents end up shaping the beliefs of non-influential agents. In this paper, we analyse this control mechanism more closely and reveal some critical properties. In particular, we characterize the set of beliefs that can be imposed on non-influential agents (i.e., the set of attainable beliefs) and how the graph topology of these latter agents helps resist manipulation but only to a certain degree. We also derive a design procedure that allows influential agents to drive the beliefs of non-influential agents to desirable attainable states. We illustrate the results with two examples.