A Generalized Sznajd Model

In the last decade the Sznajd Model has been successfully employed in modeling some properties and scale features of both proportional and majority elections. We propose a new version of the Sznajd model with a generalized bounded confidence rule - a rule that limits the convincing capability of agents and that is essential to allow coexistence of opinions in the stationary state. With an appropriate choice of parameters it can be reduced to previous models. We solved this new model both in a mean-field approach (for an arbitrary number of opinions) and numerically in a Barabasi-Albert network (for three and four opinions), studying the transient and the possible stationary states. We built the phase portrait for the special cases of three and four opinions, defining the attractors and their basins of attraction. Through this analysis, we were able to understand and explain discrepancies between mean-field and simulation results obtained in previous works for the usual Sznajd Model with bounded confidence and three opinions. Both the dynamical system approach and our generalized bounded confidence rule are quite general and we think it can be useful to the understanding of other similar models.Comment: 19 pages with 8 figures. Submitted to Physical Review

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 favour 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 modelled), 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 present some graph theory concepts, together with examples, and comparisons between the mean-field and simulations in Barab\'asi-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims. 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).Comment: 15 pages, 18 figures. To be submitted to Physical Revie

The role of asymmetries in rock-paper-scissors biodiversity models.

The maintenance of biodiversity is a long standing puzzle in ecology. It is a classical result that if the interactions of the species in an ecosystem are chosen in a random way, then complex ecosystems can't sustain themselves, meaning that the structure of the interactions between the species must be a central component on the preservation of biodiversity and on the stability of ecosystems. The rock-paper-scissors model is one of the paradigmatic models that study how biodiversity is maintained. In this model 3 species dominate each other in a cyclic way (mimicking a trophic cycle), that is, rock dominates scissors, that dominates paper, that dominates rock. In the original version of this model, this dominance obeys a 'Z IND 3' symmetry, in the sense that the strength of dominance is always the same. In this work, we break this symmetry, studying the effects of the addition of an asymmetry parameter. In the usual model, in a two dimensional lattice, the species distribute themselves according to spiral patterns, that can be explained by the complex Landau-Guinzburg equation. With the addition of asymmetry, new spatial patterns appear during the transient and the system either ends in a state with spirals, similar to the ones of the original model, or in a state where unstable spatial patterns dominate or in a state where only one species survives (and biodiversity is lost)

Phase transition in the Sznajd model with independence

We propose a model of opinion dynamics which describes two major types of social influence -- conformity and independence. Conformity in our model is described by the so called outflow dynamics (known as Sznajd model). According to sociologists' suggestions, we introduce also a second type of social influence, known in social psychology as independence. Various social experiments have shown that the level of conformity depends on the society. We introduce this level as a parameter of the model and show that there is a continuous phase transition between conformity and independence