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We study the connections between network structure, opinion dynamics, and an adversary's power to artificially induce disagreements. We approach these questions by extending models of opinion formation in the social sciences to represent scenarios, familiar from recent events, in which external actors seek to destabilize communities through sophisticated information warfare tactics via fake news and bots. In many instances, the intrinsic goals of these efforts are not necessarily to shift the overall sentiment of the network, but rather to induce discord. These perturbations diffuse via opinion dynamics on the underlying network, through mechanisms that have been analyzed and abstracted through work in computer science and the social sciences. We investigate the properties of such attacks, considering optimal strategies both for the adversary seeking to create disagreement and for the entities tasked with defending the network from attack. We show that for different formulations of these types of objectives, different regimes of the spectral structure of the network will limit the adversary's capacity to sow discord; this enables us to qualitatively describe which networks are most vulnerable against these perturbations. We then consider the algorithmic task of a network defender to mitigate these sorts of adversarial attacks by insulating nodes heterogeneously; we show that, by considering the geometry of this problem, this optimization task can be efficiently solved via convex programming. Finally, we generalize these results to allow for two network structures, where the opinion dynamics process and the measurement of disagreement become uncoupled, and determine how the adversary's power changes; for instance, this may arise when opinion dynamics are controlled an online community via social media, while disagreement is measured along "real-world" connections.