When it comes to the mathematical modelling of social interaction patterns, a number of different models have emerged and been studied over the last decade, in which individuals randomly interact on the basis of an underlying graph structure and shar
e their opinions. A prominent example of the so-called bounded confidence models is the one introduced by Deffuant et al.: Two neighboring individuals will only interact if their opinions do not differ by more than a given threshold $theta$. We consider this model on the line graph $mathbb{Z}$ and extend the results that have been achieved for the model with real-valued opinions by considering vector-valued opinions and general metrics measuring the distance between two opinion values. Just as in the univariate case, there exists a critical value for $theta$ at which a phase transition in the long-term behavior takes place.
A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant. If the matrix fails to be copositive there exists some non-negative certificate for which the quadratic form is ne
gative. Due to the scaling property, we can find such certificates in every neighborhood of the origin but their properties depend on the matrix of course and are hard to describe. If it is an integer matrix however, we are guaranteed certificates of a complexity that is at most a constant times the binary encoding length of the matrix raised to the power 3/2.
