Nonlinear Opinion Dynamics Models With Stubborn Individuals Over Signed Graphs

Abstract

In this paper, we use nonlinear continuous opinion dynamics models to investigate how the opinions in social networks are affected by the existence of stubborn individuals who remain attached to their initial opinions. Specifically, we are interested in the case where trust-mistrust relations also exist in the network. We extend the existing models, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn positives</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn extremists</i> , by using signed graphs and allowing negative edges to model mistrust interactions. The effect of stubborn individuals is studied for both structurally balanced and unbalanced graphs. We show that, due to the stubborn individuals, a bipartite consensus (polarization) might not occur even if the associated graph is structurally balanced. Moreover, for structurally unbalanced graphs, the opinion vector does not always converge to 0 as compared to the well-known Altafini's model because of stubborn individuals. We present theorems that provide the above convergence guarantees.