This paper presents four examples of different ways that coordinated influence operations exert pressure on the prominence of ideas on social networks. We argue that these examples illustrate the four archetypical paths to influence: promotion by strengthening, promotion by weakening, demotion by strengthening, and demotion by weakening. We formalize this idea in a stylized economic model of the optimal behavior of the influence operator and derive some predictions about when we should expect each path to be followed. Finally we sketch out how one might go about quantitatively estimating the key parameters of (a variant of) this model and how it applies much more broadly than in the international political influence examples that motivate it.