Opinion dynamics for the signed social networks with private and expressed opinions

In this paper, an opinion dynamic model with private and expressed opinions (PAEOs) in the presence of stubborn individuals is proposed on the signed social networks. In this model, the individual's private opinion evolution depends on individual's private opinions and expressed opinions of his/her neighbors, while the evolution of expressed opinions is influenced by a pressure to comply with the average expressed opinions of his/her neighbors.. It is found that if there exist no oblivious individuals, PAEOs can always converge exponentially fast to a stable state, regardless of whether the social networks are structurally balanced or not. At the same time, the steady opinions rely on the initial private opinions, while the initial expressed opinions are forgotten during the evolution process. If there exist oblivious individuals, the convergence of the model depends on whether the opinions of all oblivious individuals can converge to a steady state. Whether the steady opinions rely on the initial expressed opinions is related to the structural balance of the subnetwork consisting of all oblivious individuals. The disagreement level at steady opinions is also analyzed, and a method is obtained to estimate the lower bounds of the disagreement level for the final private opinions. Finally, two examples are used to validate the obtained results.