Narrow Results

We introduce a consensus model inspired by the Sznajd Model. The updating is synchronous and memory plays a decisive role in making possible the reaching of total consensus. We study the transition between the state with no-consensus to the state with total consensus.

In this paper, we investigate the so-called "Sznajd Model" (SM) in one dimension, which is a simple cellular automata approach to consensus formation among two opposite opinions (described by spin up or down). To elucidate the SM dynamics, we first provide results of computer simulations for the spatio-temporal evolution of the opinion distribution L(t), the evolution of magnetization m(t), the distribution of decision times P(τ) and relaxation times P(μ). In the main part of the paper, it is shown that the SM can be completely reformulated in terms of a linear voter model (VM), where the transition rates towards a given opinion are directly proportional to frequency of the respective opinion of the second-nearest neighbors (no matter what the nearest neighbors are). So, the SM dynamics can be reduced to one rule, "Just follow your second-nearest neighbor". The equivalence is demonstrated by extensive computer simulations that show the same behavior between SM and VM in terms of L(t), m(t), P(τ), P(μ), and the final attractor statistics. The reformulation of the SM in terms of a VM involves a new parameter σ, to bias between anti- and ferromagnetic decisions in the case of frustration. We show that σ plays a crucial role in explaining the phase transition observed in SM. We further explore the role of synchronous versus asynchronous update rules on the intermediate dynamics and the final attractors. As compared to the original SM, we find three additional attractors, two of them related to an asymmetric coexistence between the opposite opinions.

A Consensus Model according to Deffuant on a directed Barabási–Albert network was simulated. Agents have opinions on different subjects. A multi-component subject vector was used. The opinions are discrete. The analysis concerns distribution and clusters of agents which are in agreement with the opinions of the subjects. Remarkable results shown that there mostly exists no absolute consensus. It depends on the ratio of number of agents to the number of subjects, whether the communication ends in a consensus or a pluralism. Mostly a second robust cluster remains, in its size depending on the number of subjects. Two agents agree either in (nearly) all or (nearly) no subject. The operative parameter of the consensus-formating-process is the tolerance in change of views of the group-members.

We analyze the evolution of Sznajd Model with synchronous updating in several complex networks. Similar to the model on square lattice, we have found a transition between the state with nonconsensus and the state with complete consensus in several complex networks. Furthermore, by adjusting the network parameters, we find that a large clustering coefficient does not favor development of a consensus. In particular, in the limit of large system size with the initial concentration p =0.5 of opinion +1, a consensus seems to be never reached for the Watts–Strogatz small-world network, when we fix the connectivity k and the rewiring probability p_s; nor for the scale-free network, when we fix the minimum node degree m and the triad formation step probability p_t.

A model of the opinion dynamics underlying the political decision is proposed. The analysis is restricted to a bipolar scheme with a possible third political area. The interaction among voters is local but the final decision strongly depends on global effects such as the rating of the governments. As in the realistic case, the individual decision making process is determined by the most relevant personal interests and problems. The phenomenological analysis of the national vote in Italy and Germany has been carried out and a prediction of the next Italian vote as a function of the government rating is presented.

We introduce a new model that mimics the strong and sudden effects induced by conformity in tightly interacting human societies. Such effects range from mere crowd phenomena to dramatic political turmoil. The model is a modified version of the Ising Hamiltonian. We have studied the properties of this Hamiltonian using both a Metropolis simulation and analytical derivations. Our study shows that increasing the value of the conformity parameter, results in a first order phase transition. As a result a majority of people begin to honestly support the idea that may contradict the moral principles of a normal human beings though each individual would support the moral principle without tight interaction with the society. Thus, above some critical level of conformity our society destabilizes with respect to ideas that might be doubtful. Our model includes, in a simplified way, human diversity with respect to loyalty to the moral principles.

The Sznajd model has been largely applied to simulate many sociophysical phenomena. In this paper, we applied the Sznajd model with more than two opinions on three different network topologies and observed the evolution of surviving opinions after many interactions among the nodes. As result, we obtained a scaling law which depends of the network size and the number of possible opinions. We also observed that this scaling law is not the same for all network topologies, being quite similar between scale-free networks and Sznajd networks but different for random networks.

Involving effects of media, opinion leader and other agents on the opinion of individuals of market society, a trader based model is developed and utilized to simulate price via supply and demand. Pronounced effects are considered with several weights and some personal differences between traders are taken into account. Resulting time series and probabilty distribution function involving a power law for price come out similar to the real ones.

In the present paper, the Sznajd model of opinion formation with multi-valued opinions for the temperature T > 0 has been used for investigation of the election results in a population of N individuals represented by the nodes of a square lattice. Presence of temperature means that the population under consideration is open for the effect of external information. The distribution of opinions during the election campaign was found for different values of temperature and length of election campaign preceding the election. Comparison of the results of election to the Polish Parliament in the town with the population consisting 750000 voters, shows quite good agreement with the results of our calculations.

We modify the model of Deffuant et al. to distinguish true opinion among others in the fashion of Hegselmann and Krause (). The basic features of both models modified to account for truth seekers are qualitatively the same.

This paper treats the opinion dynamics of an unequal, initial opinion distribution. We simulate the Deffuant model on a directed Barabási–Albert network with discrete opinions and several subjects. We notice a focusing of the the resulting opinion distribution during the simulation towards the average value of the initial opinion distribution. A small change of the focusing is seen. A dependency of this change on the number of subjects and opinions is detected and indicates the change as a consequence of discretization of the opinions. Hereby the average value of the initial opinion distribution can be identified as the guide of opinion forming.

A way to simulate the basic interactions between two individuals with different opinions, in the context of strategic game theory, is proposed. Various games are considered, which produce different kinds of opinion formation dynamics. First, by assuming that all individuals (players) are equals, we obtain the bounded confidence model of continuous opinion dynamics proposed by Deffuant et al. In such a model a tolerance threshold is defined, such that individuals with difference in opinion larger than the threshold can not interact. Then, we consider that the individuals have different inclinations to change opinion and different abilities in convincing the others. In this way, we obtain the so-called "Stubborn individuals and Orators" (SO) model, a generalization of the Deffuant et al. model, in which the threshold tolerance is different for every couple of individuals. We explore, by numerical simulations, the dynamics of the SO model, and we propose further generalizations that can be implemented.

Models of continuous opinion dynamics under bounded confidence have been presented independently by Krause and Hegselmann and by Deffuant et al. in 2000. They have raised a fair amount of attention in the communities of social simulation, sociophysics and complexity science. The researchers working on it come from disciplines such as physics, mathematics, computer science, social psychology and philosophy.

In these models agents hold continuous opinions which they can gradually adjust if they hear the opinions of others. The idea of bounded confidence is that agents only interact if they are close in opinion to each other. Usually, the models are analyzed with agent-based simulations in a Monte Carlo style, but they can also be reformulated on the agent's density in the opinion space in a master equation style. The contribution of this survey is fourfold. First, it will present the agent-based and density-based modeling frameworks including the cases of multidimensional opinions and heterogeneous bounds of confidence. Second, it will give the bifurcation diagrams of cluster configuration in the homogeneous model with uniformly distributed initial opinions. Third, it will review the several extensions and the evolving phenomena which have been studied so far, and fourth it will state some open questions.

We review a series of models of sociophysics introduced by Galam and Galam et al. in the last 25 years. The models are divided into five different classes, which deal respectively with democratic voting in bottom-up hierarchical systems, decision making, fragmentation versus coalitions, terrorism and opinion dynamics. For each class the connexion to the original physical model and techniques are outlined underlining both the similarities and the differences. Emphasis is put on the numerous novel and counterintuitive results obtained with respect to the associated social and political framework. Using these models several major real political events were successfully predicted including the victory of the French extreme right party in the 2000 first round of French presidential elections, the voting at fifty–fifty in several democratic countries (Germany, Italy, Mexico), and the victory of the "no" to the 2005 French referendum on the European constitution. The perspectives and the challenges to make sociophysics a predictive solid field of science are discussed.

In 2002, Serge Galam designed a model of a minority opinion spreading. The effect is expected to lead a conservative minority to prevail if the issue is discussed long enough. Here we analyze the marriage gap, i.e., the difference in voting for Bush and Kerry in 2004 and for Bush and Gore in 2000 between married and unmarried people. It seems possible to interpret the data in terms of the Galam model.

A model where agents show discrete behavior regarding their actions, but have continuous opinions that are updated by interacting with other agents is presented. This new updating rule is applied to both the voter and Sznajd models for interaction between neighbors, and its consequences are discussed. The appearance of extremists is naturally observed and it seems to be a characteristic of this model.

In this paper we study urban segregation of two different communities A and B, rich and poor, distributed randomly on finite samples, to check cheap and expensive residences. For this purpose we avoid the complications of the Schelling model which are not necessary and instead we use the Ising model on 500 × 500 square lattices, which gives similar results, with random magnetic field at lower and higher temperatures (k_BT/J = 2.0, 99.0) in finite times equal to 40, 400, 4000 and 40 000. This random-field Ising magnet is a suitable model, where each site of the square lattice carries a magnetic field ±h which is randomly up (expensive) or down (cheap). The resulting addition to the energy prefers up-spins on the expensive and down-spins on the cheap sites. Our simulations were carried out using a 50-line FORTRAN program. We present at a lower temperature (2.0) a time series of pictures, separating growing from non-growing domains. A small random field (h = ±0.1) allows for large domains, while a large random field (h = ±0.9) allows only small clusters. At higher temperature (99.0) we could not obtain growing domains.

We study the asymmetric influential effects in the damage spreading processes, by investigating the opinion dynamics of the Krause–Hegselmann consensus model, wherein the agents sit on the nodes of small-world networks. One agent is randomly selected as the influential agent, which can influence the opinions of other agents with probability p but not vice versa. The damage consists in a sharp change of the opinion of one agent (a randomly selected agent or the influential one) in the initial random opinion configuration. We find that for small values of the influential probability p, there is a damage spreading transition at low values of the confidence bound parameter ε. Interestingly, this transition vanishes for large values of p when the influential agent is damaged in the initial state. We find as well that, there is a critical value of the confidence bound parameter above which the initial perturbation manages to propagate to the whole system. The relationship between the amount of the damage and the influential activity of the perturbed agent is also discussed.

We propose a model that extends the binary "united we stand, divided we fall" opinion dynamics of Sznajd-Weron to handle continuous and multi-state discrete opinions on a linear chain. Disagreement dynamics are often ignored in continuous extensions of the binary rules, so we make the most symmetric continuum extension of the binary model that can treat the consequences of agreement (debate) and disagreement (confrontation) within a population of agents. We use the continuum extension as an opportunity to develop rules for persistence of opinion (memory). Rules governing the propagation of centrist views are also examined. Monte Carlo simulations are carried out. We find that both memory effects and the type of centrist significantly modify the variance of average opinions in the large timescale limits of the models. Finally, we describe the limit of applicability for Sznajd-Weron's model of binary opinions as the continuum limit is approached. By comparing Monte Carlo results and long time-step limits, we find that the opinion dynamics of binary models are significantly different to those where agents are permitted more than 3 opinions.

We embed the behavior of tax evasion into the standard two-dimensional Ising model. In the presence of an external magnetic field, the Ising model is able to generate the empirically observed effect of tax morale, i.e., the phenomenon that in some countries tax evasion is either rather high or low. The external magnetic field captures the agents' trust in governmental institutions. We also find that tax authorities may curb tax evasion via appropriate enforcement mechanisms. Our results are robust for the Barabási–Albert and Voronoi–Delaunay networks.