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We propose an opinion dynamics model that combines processes of vanity andopinion propagation. The interactions take place between randomly chosen pairs.During an interaction, the agents propagate their opinions about themselves andabout other people they know. Moreover, each individual is subject to vanity:if her interlocutor seems to value her highly, then she increases her opinionabout this interlocutor. On the contrary she tends to decrease her opinionabout those who seem to undervalue her. The combination of these dynamics withthe hypothesis that the opinion propagation is more efficient when coming fromhighly valued individuals, leads to different patterns when varying theparameters. For instance, for some parameters the positive opinion linksbetween individuals generate a small world network. In one of the patterns,absolute dominance of one agent alternates with a state of generaliseddistrust, where all agents have a very low opinion of all the others (includingthemselves). We provide some explanations of the mechanisms behind theseemergent behaviors and finally propose a discussion about their interest

The Leviathan model: Absolute dominance, generalised distrust, small  worlds and other patterns emerging from combining vanity with opinion  propagation