Consensus and Bipartite Consensus in Graphon Models for Opinion Dynamics on the Sphere

In this article, we establish an infinite-dimensional model on the sphere for opinion dynamics based on the graph limit procedure and study consensus formation of this graphon model.Initially, we demonstrate the existence and uniqueness of solutions for the graphon model under consideration, and we offer a rigorous mathematical proof to the graph limits.Then we provide the sufficient frameworks for the emergence of consensus and bipartite consensus in our system.In the case of bipartite consensus, our results demonstratethat even when there is no interaction among agents within one subgroup, they can still attain consensus by jointly resisting the opinions of the other subgroup.Finally, we present a series of numerical simulations to illustrate our results.