On the consensus threshold for the opinion dynamics of Krause-Hegselmann

In the consensus model of Krause-Hegselmann, opinions are real numbers between 0 and 1, and two agents are compatible if the difference of their opinions is smaller than the confidence bound parameter epsilon. A randomly chosen agent takes the average of the opinions of all neighboring agents which are compatible with it. We propose a conjecture, based on numerical evidence, on the value of the consensus threshold epsilon(c) of this model. We claim that epsilon(c) can take only two possible values, depending on the behavior of the average degree d of the graph representing the social relationships, when the population N approaches infinity: if d diverges when N -> infinity, epsilon(c) equals the consensus threshold epsilon(i) similar to 0.2 on the complete graph; if instead d stays finite when N -> infinity, epsilon(c) = 1/2 as for the model of Deffuant et al.