Adversarial Perturbations of Opinion Dynamics in Networks

In this paper, 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 mathematical social sciences to represent scenarios, familiar from recent events, in which external actors have sought 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 towards a particular policy, but rather to induce discord. These perturbations will 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. Here 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. By employing spectral techniques, 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; in fact, somewhat surprisingly, the entire spectrum can be relevant, rather than just the extreme eigenvectors. Via the strong connections between spectral and structural properties of graphs, we are able to qualitatively describe which networks are most vulnerable or resilient 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; for instance, this may arise when opinion dynamics are controlled by an online community via social media, while disagreement is measured along "real-world" connections. We characterize conditions on the relationship between these two graphs that will determine how much power the adversary gains when this occurs.