The critical behavior of Hegselmann-Krause opinion model with smart agents

Abstract: The Hegselmann-Krause (HK) model allows one to characterize the continuous change of agents' opinions with the bounded confidence threshold $\varepsilon$. To consider the heterogeneity of agents in characteristics, we study the HK model on homogeneous and heterogeneous networks by introducing a kind of smart agent. Different from the averaging rule in opinion update of HK model, smart agents will consider, in updating their opinions, the environmental influence following the fact that an agent's behavior is often coupled with environmental changes. The environment is characterized by a parameter that represents the biased resource allocation between different cliques. We focus on the critical behavior of the underlying system. A phase transition point separating a complete consensus from the coexistence of different opinions is identified, which occurs at a critical value $\varepsilon_c$ for the bounded confidence threshold. We state analytically that $\varepsilon_c$ can take only one of two possible values, depending on the behavior of the average degree $k_a$ of a social graph, when agents are homogeneous in characteristics. Results also suggest that the phase transition point weakly depends on the network structure but is strongly correlated with the fraction of smart agents and the environmental parameter. We finally establish the finite size scaling law that stresses the role that the system size has in the underlying opinion dynamics. Meanwhile, introducing smart agents does not change the functional dependence between the time to reach a complete consensus and the system size. However, it can drive a complete consensus to be reached faster, for homogeneous networks that are far from the mean field limit.