Connections between the Sznajd model with general confidence rules and graph theory

The Sznajd model is a sociophysics model that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favor bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modeled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We state these results and present comparisons between the mean field and simulations in Barabasi-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims and some graph theory concepts, together with examples. We also show that there is no qualitative difference in the mean-field results if we require that a group of size q > 2, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean field, this would coincide with the q-voter model).

Effect of external fields in Axelrod's model of social dynamics

The study of the effects of spatially uniform fields on the steady-state properties of Axelrod's model has yielded plenty of counterintuitive results. Here, we reexamine the impact of this type of field for a selection of parameters such that the field-free steady state of the model is heterogeneous or multicultural. Analyses of both one- and two-dimensional versions of Axelrod's model indicate that the steady state remains heterogeneous regardless of the value of the field strength. Turning on the field leads to a discontinuous decrease on the number of cultural domains, which we argue is due to the instability of zero-field heterogeneous absorbing configurations. We find, however, that spatially nonuniform fields that implement a consensus rule among the neighborhood of the agents enforce homogenization. Although the overall effects of the fields are essentially the same irrespective of the dimensionality of the model, we argue that the dimensionality has a significant impact on the stability of the field-free homogeneous steady state.