Opinion limits study for the multi-selection bounded confidence model

In this work, we study the opinion limit states for a generalized bounded confidence agent-based opinion model. Agents can select multiple opinions in the network, and the confidence bound is considered on the distance between the average of the selected opinions and agent opinion itself. The number of selection agents for a certain agent, which is also called the selection number, means the agent opinion interaction degree. It is known that when the confidence bound is large sufficiently, opinions reach consensus almost surely. We mainly study the opinion consensus and the opinion polarization when the confidence bound is small sufficiently. Firstly, we provide and prove the upper and lower bounds for the opinion consensus probability of this bound confidence model. It shows that the opinion consensus probability almost always decreases as the confidence bound decreases. Secondly, the opinion consensus probability is larger than the one for the opinion evolution of the Deffuant-Weisbuch model. Finally, we demonstrate the ultimate probability distribution of one agent opinion and compare it with the gossip form and the general bounded confidence form, and demonstrate how the opinion polarization probabilities evolve as the selection number changes. Specially, different from other studies, we find that the opinion polarization would happen more easily if the opinion interaction degree is strengthened. In a sum, the multiple selection mechanism will increase the opinion consensus probability and the opinion polarization probability, respectively, comparing to the single selection mechanism.