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. Firstly, we show the existence and uniqueness of solutions for the graphon model under consideration and provide a rigorous mathematical proof of the graph limits. Then we present sufficient conditions for the emergence of consensus and bipartite consensus within our system. In the case of bipartite consensus, our results indicate that even in the absence of interactions among agents within a subgroup, they can still achieve consensus by collectively opposing the opinions of the other subgroup. Finally, we provide a series of numerical simulations to illustrate our findings.