The spreading of opposite opinions on online social networks with authoritative nodes

The study of opinion dynamics, such as spreading and controlling of rumors, has become an important issue on social networks. Numerous models have been devised to describe this process, including epidemic models and spin models, which mainly focus on how opinions spread and interact with each other, respectively. In this paper, we propose a model that combines the spreading stage and the interaction stage for opinions to illustrate the process of dispelling a rumor. Moreover, we set up authoritative nodes, which disseminate positive opinion to counterbalance the negative opinion prevailing on online social networking sites. With analysis of the relationship among positive opinion proportion, opinion strength and the density of authoritative nodes in networks with different topologies, we demonstrate that the positive opinion proportion grows with the density of authoritative nodes until the positive opinion prevails in the entire network. In particular, the relationship is linear in homogeneous topologies. Besides, it is also noteworthy that initial locations of the negative opinion source and authoritative nodes do not influence positive opinion proportion in homogeneous networks but have a significant impact on heterogeneous networks. The results are verified by numerical simulations and are helpful to understand the mechanism of two different opinions interacting with each other on online social networking sites.