Predictable dynamics of opinion forming on social networks with antagonistic interactions

Being able to predict the outcome of an opinion forming process is an important problem in social network theory. For communities of agents which are not necessarily cooperating, distributed processes of opinion forming are naturally represented by signed graphs, with positive edges representing friendly and cooperative interactions among the agents and negative edges the corresponding antagonistic counterpart. Unlike for networks of cooperating agents, the outcome of a dynamical system evolving on such signed graphs is not obvious and it is in general difficult to characterize, even when the dynamics are linear. In this talk I will discuss cases in which predicting the dynamics from the signature of the graph is however possible. These include: (i) structurally balanced graphs, which induce monotone dynamics; (ii) cases in which the Perron-Frobenius theorem can be applied. The latter correspond to linear systems admitting an invariant cone properly contained in one of the orthants of R^n. When the cone is contained in the first orthant, then the social network achieves an unanimous opinion in spite of the presence of antagonism.