Asymptotic and Stability Analysis of Kinetic Models for Opinion Formation on Networks: An Allen-Cahn Approach

We present the analysis of stationary equilibria and their stability in the case of an opinion formation process in the presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model implementing microscopic interaction rules that can be either binary or ternary for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen-Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation that can be studied by means of general relative entropy tools and integral operators. We extend the analysis to a continuous opinion model and coevolving networks.