Modeling and controlling competing opinions in social networks: a continuous density approach and topological intervention

Online social networks have become the core medium for information dissemination and opinion formation in modern society. However, its openness and convenience also come with challenges such as the spread of rumors and polarization of opinions. This study adopts a continuous opinion-density representation described by differential equations, complementing traditional discrete-state models by offering an alternative lens on competing opinion dynamics. The model defines five types of node states (Ignorant I, Latent L, Spreader $$S_1/S_2$$, Silent R) and introduces activation threshold $$\theta _S$$ and ignorance threshold $$\theta _I$$ to control state transitions. The experimental results indicate that the balance between the propagation rate and the forgetting factor is a key factor in determining the macro phase transition of the system. Meanwhile, our model gives rise to emergent complex phenomena such as social polarization, radicalization, and tipping points, which are not trivially predictable from the model’s individual components. We also found that the "Opinion Echo Chamber" elimination strategy based on community detection can effectively reduce the average opinion density on the Internet. This study provides a new perspective for understanding the mechanisms of opposing opinions and offers a topology-based intervention solution for controlling online public opinion.