Continuous Opinion Dynamics formulated in Matrix Language and its Relation to Swarms

Jan Lorenz

University of Bremen, Vor dem Steintor 90, 28203 Bremen, Germany

We discuss models of opinion dynamics introduced by Krause-Hegselmann and Deffuant-Weisbuch. Both models treat n agents each having an opinion which is a real number. The agents revise their opinions after considering opinions of other agents by means of averaging. The driving force in both model is, that we assume a bounded confidence for each agent. This means that each agent only averages opinions that are not too far away from his own opinion. Due to this simple driving force the agents agglomerate (find a consensus or consensus in a subgroup) but probably segregate into clusters. The model can be formulated as a repeated multiplication of matrices times the opinion vector. In mathematical terms we treat a time-discrete state-dependent dynamical system. It is related to a non-homogeneous Markov chain. Using theory of non-negative matrices it is possible to derive a stabilization theorem for a wide class of models. The idea of bounded confidence has a similarity to the attraction area used for modelling of flock building. A flock can be seen as a consensus group concerning the velocity vector. A swarm can be seen as a consensus group concerning the position (but only roughly). The talk should point out the formulation in matrix language and its usefulness for analysis. This should introduce another tool for reformulating models to get analytical results additionally to deriving differential equations as common in statistical mechanics. The matrix formulation is much closer to the implementation of simulations for agent based models, because each agent is related to an index of the matrix. It is also easy to implement social networks into the model, because a social network is completely described by its adjacency matrix.