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 ǫ. A randomly chosen agent takes the average of the opinions of all neighbouring agents which are compatible with it. We propose a conjecture, based on numerical evidence, on the value of the consensus threshold ǫc of this model. We claim that ǫc can take only two possible values, depending on the behaviour of the average degree d of the graph representing the social relationships, when the population N goes to infinity: if d diverges when N → ∞, ǫc equals the consensus threshold ǫi ∼ 0.2 on the complete graph; if instead d stays finite when N → ∞, ǫc = 1/2 as for the model of Deffuant et al.