André M. Timpanaro

Exit probability of the one-dimensional q-voter model: analytical results and simulations for large networks.

We discuss the exit probability of the one-dimensional q-voter model and present tools to obtain estimates about this probability, both through simulations in large networks (around 10(7) sites) and analytically in the limit where the network is infinitely large. We argue that the result E(ρ) = ρ(q)/ρ(q) + (1-ρ)(q), that was found in three previous works [F. Slanina, K. Sznajd-Weron, and P. Przybyła, Europhys. Lett. 82, 18006 (2008); R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008), for the case q = 2; and P. Przybyła, K. Sznajd-Weron, and M. Tabiszewski, Phys. Rev. E 84, 031117 (2011), for q > 2] using small networks (around 10(3) sites), is a good approximation, but there are noticeable deviations that appear even for small systems and that do not disappear when the system size is increased (with the notable exception of the case q = 2). We also show that, under some simple and intuitive hypotheses, the exit probability must obey the inequality ρ(q)/ρ(q) + (1-ρ) ≤ E(ρ) ≤ ρ/ρ + (1-ρ)(q) in the infinite size limit. We believe this settles in the negative the suggestion made [S. Galam and A. C. R. Martins, Europhys. Lett. 95, 48005 (2001)] that this result would be a finite size effect, with the exit probability actually being a step function. We also show how the result that the exit probability cannot be a step function can be reconciled with the Galam unified frame, which was also a source of controversy.

Generalized Sznajd model for opinion propagation.

Instituto de Física, Universidade de São Paulo Caixa Postal 66318, 05314-970 São Paulo São Paulo Brazil (Dated: May 4, 2009) Abstract In the last decade the Sznajd Model has been successfully emp loyed in modeling some properties and scale features of both proportional and majority elections . We propose a new version of the Sznajd model with a generalizedbounded confidence rule a rule that limits the convincing capability of agents and that is essential to allow coexistence of opinions in the stationar y 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 Barabási -Albert network (for three and four opinions), studying the transient and the possible stationary states. W built the phase portrait for the special cases of three and four opinions, defining the attractors and their ba sins of attraction. Through this analysis, we were able to understand and explain discrepancies between meanfield and simulation results obtained in previous works for the usual Sznajd Model with bounded confidence and t hree opinions. Both the dynamical system approach and our generalized bounded confidence rule are qui te general and we think it can be useful to the understanding of other similar models.

Coexistence of interacting opinions in a generalized Sznajd model.

The Sznajd model is a sociophysics model that mimics the propagation of opinions in a closed society, where the interactions favor groups of agreeing people. It is based in the Ising and Potts ferromagnetic models and, although the original model used only linear chains, it has since been adapted to general networks. This model has a very rich transient, which has been used to model several aspects of elections, but its stationary states are always consensus states. In order to model more complex behaviors, we have, in a recent work, introduced the idea of biases and prejudices to the Sznajd model by generalizing the bounded confidence rule, which is common to many continuous opinion models, to what we called confidence rules. In that work we have found that the mean field version of this model (corresponding to a complete network) allows for stationary states where noninteracting opinions survive, but never for the coexistence of interacting opinions. In the present work, we provide networks that allow for the coexistence of interacting opinions for certain confidence rules. Moreover, we show that the model does not become inactive; that is, the opinions keep changing, even in the stationary regime. This is an important result in the context of understanding how a rule that breeds local conformity is still able to sustain global diversity while avoiding a frozen stationary state. We also provide results that give some insights on how this behavior approaches the mean field behavior as the networks are changed.

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 Barabási-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).

Dynamical systems approach to the study of a sociophysics agent-based model

The Sznajd model is a Potts‐like model that has been studied in the context of sociophysics [1,2] (where spins are interpreted as opinions). In a recent work [3], we generalized the Sznajd model to include assymetric interactions between the spins (interpreted as biases towards opinions) and used dynamical systems techniques to tackle its mean‐field version, given by the flow: ησ =  ∑ σ′ = 1Mησησ′(ησρσ′→σ−σ′ρσ→σ′). Where hs is the proportion of agents with opinion (spin) σ′, M is the number of opinions and σ′→σ′ is the probability weight for an agent with opinion σ being convinced by another agent with opinion σ′. We made Monte Carlo simulations of the model in a complex network (using Barabasi‐Albert networks [4]) and they displayed the same attractors than the mean‐field. Using linear stability analysis, we were able to determine the mean‐field attractor structure analytically and to show that it has connections with well known graph theory problems (maximal independent sets and positive fluxes in direc...

Study of the attractor structure of an agent-based sociological model

The Sznajd model is a sociophysics model that is based in the Potts model, and used for describing opinion propagation in a society. It employs an agent-based approach and interaction rules favouring pairs of agreeing agents. It has been successfully employed in modeling some properties and scale features of both proportional and majority elections (see for instance the works of A. T. Bernardes and R. N. Costa Filho), but its stationary states are always consensus states. In order to explain more complicated behaviours, we have modified the bounded confidence idea (introduced before in other opinion models, like the Deffuant model), with the introduction of prejudices and biases (we called this modification confidence rules), and have adapted it to the discrete Sznajd model. This generalized Sznajd model is able to reproduce almost all of the previous versions of the Sznajd model, by using appropriate choices of parameters. We solved the attractor structure of the resulting model in a mean-field approach and made Monte Carlo simulations in a Barab?si-Albert network. These simulations show great similarities with the mean-field, for the tested cases of 3 and 4 opinions. The dynamical systems approach that we devised allows for a deeper understanding of the potential of the Sznajd model as an opinion propagation model and can be easily extended to other models, like the voter model. Our modification of the bounded confidence rule can also be readily applied to other opinion propagation models.