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

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