Breaking Consensus in Kinetic Opinion Formation Models on Graphons

In this work, we propose and investigate a strategy to prevent consensus in kinetic models for opinion formation. We consider a large interacting agent system and assume that agent interactions are driven by compromise as well as self-thinking dynamics and also modulated by an underlying static social network. This network structure is included using so-called graphons, which modulate the interaction frequency in the corresponding kinetic formulation. We then derive the corresponding limiting Fokker-Planck equation and analyze its large time behavior. This microscopic setting serves as a starting point for the proposed control strategy, which steers agents away from mean opinion and is characterized by a suitable penalization depending on the properties of the graphon. We show that this minimalist approach is very effective by analyzing the quasi-stationary solution mean-field model in a plurality of graphon structures. Several numerical experiments are also provided to show the effectiveness of the approach in preventing the formation of consensus steering the system toward a declustered state.