Phase coexistence in the fully heterogeneous Hegselmann-Krause opinion dynamics model

We present an extensive study of the joint effects of heterogeneous social agents and their heterogeneous social links in a bounded confidence opinion dynamics model. The full phase diagram of the model is explored for two different network's topologies and compared to two opposed extreme cases: on one hand, the heterogeneous agents constitute a mixed population and on the other, their interactions are modeled by a lattice. The results show that when agents prone to compromise coexist with close-minded ones, the steady state of the dynamics shows coexistent phases. In particular, unlike the case of homogeneous agents in networks, or heterogeneous agents in a fully mixed population, it is possible that the society ends up in consensus around one extreme opinion. Moreover, during the dynamics, the consensus may be overturned from one extreme to the other of the opinion space. We also show that the standard order parameter, the normalized average size of the largest opinion cluster, may be misleading in this case, as it hides the existence of these phases. The phase where the opinion of the society is overturned does not require the presence of agents with special characteristics, (stubborn, extremists, etc.); it results from the interplay of agents which have agreed on an extreme opinion with the remaining group that holds the opposite one. Among the former, some may be prone to compromise with other agents which are out of the majority group, these agents, according to their location in the network, may act like bridges between the two groups and slowly attract the whole society to the other extreme.