Heterogeneous beliefs, segregation, and extremism in the making of public opinions

Quasi-critical dynamics in large-scale social systems regulated by sudden events

How do heterogeneous individual behaviors arise in response to sudden events and how do they shape large-scale social dynamics? Based on a five-year naturalistic observation of individual purchasing behaviors, we extract the long-term consumption dynamics of diverse commodities from approximately 2.2 million purchase orders. We subdivide the consumption dynamics into trend, seasonal, and random components and analyze them using a renormalization group. We discover that the coronavirus pandemic, a sudden event acting on the social system, regulates the scaling and criticality of consumption dynamics. On a large time scale, the long-term dynamics of the system, regardless of arising from trend, seasonal, or random individual behaviors, is pushed toward a quasi-critical region between independent (i.e., the consumption behaviors of different commodities are irrelevant) and correlated (i.e., the consumption behaviors of different commodities are interrelated) phases as the pandemic erupts. On a small time scale, short-term consumption dynamics exhibits more diverse responses to the pandemic. While the trend and random behaviors of individuals are driven to quasi-criticality and exhibit scale-invariance as the pandemic breaks out, seasonal behaviors are more robust against regulations. Overall, these discoveries provide insights into how quasi-critical macroscopic dynamics emerges in heterogeneous social systems to enhance system reactivity to sudden events while there may exist specific system components maintaining robustness as a reflection of system stability.

Collective group drift in a partial-differential-equation-based opinion dynamics model with biased perception kernels

In the age of technology, individuals accelerate their biased gathering of information, which in turn leads to a population becoming extreme and more polarized. Here we study a partial-differential-equation model for opinion dynamics that exhibits collective behavior subject to nonlocal interactions. We developed an interaction kernel function to represent biased information gathering. Through a linear stability analysis, we show that biased populations can still form opinionated groups. However, a population that is too heavily biased can no longer come to a consensus, that is, the initial homogeneous mixed state becomes stable. Numerical simulations with biased information gathering show the ability for groups to collectively drift towards one end of the opinion space. This means that a small bias in each individual will collectively lead to groups of individuals becoming extreme together. The characteristic time scale for a group's existence is captured from numerical experiments using the temporal correlation function. Supplementing this, we included a measure of how different each population is after regular time intervals using a form of the Manhattan and Euclidean distance metrics. We conclude by exploring how wall boundary conditions induce pattern formation initially on the most extreme sides of the domain.

Bursts of communication increase opinion diversity in the temporal Deffuant model

Human interactions create social networks forming the backbone of societies. Individuals adjust their opinions by exchanging information through social interactions. Two recurrent questions are whether social structures promote opinion polarisation or consensus and whether polarisation can be avoided, particularly on social media. In this paper, we hypothesise that not only network structure but also the timings of social interactions regulate the emergence of opinion clusters. We devise a temporal version of the Deffuant opinion model where pairwise social interactions follow temporal patterns. Individuals may self-organise into a multi-partisan society due to network clustering promoting the reinforcement of local opinions. Burstiness has a similar effect and is alone sufficient to refrain the population from consensus and polarisation by also promoting the reinforcement of local opinions. The diversity of opinions in socially clustered networks thus increases with burstiness, particularly, and counter-intuitively, when individuals have low tolerance and prefer to adjust to similar peers. The emergent opinion landscape is well-balanced regarding groups' size, with relatively short differences between groups, and a small fraction of extremists. We argue that polarisation is more likely to emerge in social media than offline social networks because of the relatively low social clustering observed online, despite the observed online burstiness being sufficient to promote more diversity than would be expected offline. Increasing the variance of burst activation times, e.g. by being less active on social media, could be a venue to reduce polarisation. Furthermore, strengthening online social networks by increasing social redundancy, i.e. triangles, may also promote diversity.

Generation and influence of eccentric ideas on social networks

Studying extreme ideas in routine choices and discussions is of utmost importance to understand the increasing polarization in society. In this study, we focus on understanding the generation and influence of extreme ideas in routine conversations which we label "eccentric" ideas. The eccentricity of any idea is defined as the deviation of that idea from the norm of the social neighborhood. We collected and analyzed data from two sources of different nature: public social media and online experiments in a controlled environment. We compared the popularity of ideas against their eccentricity to understand individuals' fascination towards eccentricity. We found that more eccentric ideas have a higher probability of getting a greater number of "likes". Additionally, we demonstrate that the social neighborhood of an individual conceals eccentricity changes in one's own opinions and facilitates generation of eccentric ideas at a collective level.

Critical exponents of master-node network model

The dynamics of competing opinions in social network plays an important role in society, with many applications in diverse social contexts such as consensus, election, morality, and so on. Here, we study a model of interacting agents connected in networks in order to analyze their decision stochastic process. We consider a first-neighbor interaction between agents in a one-dimensional network with the shape of ring topology. Moreover, some agents are also connected to a hub, or master node, who has preferential choice or bias. Such connections are quenched. As the main results, we observed a continuous nonequilibrium phase transition to an absorbing state as a function of control parameters. By using the finite-size scaling method we analyzed the static and dynamic critical exponents to show that this model probably cannot match any universality class already known.

Depolarization of echo chambers by random dynamical nudge

In social networks, users often engage with like-minded peers. This selective exposure to opinions might result in echo chambers, i.e., political fragmentation and social polarization of user interactions. When echo chambers form, opinions have a bimodal distribution with two peaks on opposite sides. In certain issues, where either extreme positions contain a degree of misinformation, neutral consensus is preferable for promoting discourse. In this paper, we use an opinion dynamics model that naturally forms echo chambers in order to find a feedback mechanism that bridges these communities and leads to a neutral consensus. We introduce the random dynamical nudge (RDN), which presents each agent with input from a random selection of other agents’ opinions and does not require surveillance of every person’s opinions. Our computational results in two different models suggest that the RDN leads to a unimodal distribution of opinions centered around the neutral consensus. Furthermore, the RDN is effective both for preventing the formation of echo chambers and also for depolarizing existing echo chambers. Due to the simple and robust nature of the RDN, social media networks might be able to implement a version of this self-feedback mechanism, when appropriate, to prevent the segregation of online communities on complex social issues.

Asymmetric Contrarians in Opinion Dynamics

Asymmetry in contrarian behavior is investigated within the Galam model of opinion dynamics using update groups of size 3 with two competing opinions A and B. Denoting x and y the respective proportions of A and B contrarians, four schemes of implementations are studied. The first scheme activates contrarians after each series of updates with probabilities x and y for agents holding respectively opinion A and B. Second scheme activates contrarians within the update groups only against global majority with probability x when A is the majority and y when B is the majority. The third scheme considers in-group contrarians acting prior to the local majority update against both local majority and minority opinions. The last scheme activates in-group contrarians prior to the local majority update but only against the local majority. The main result is the loss of the fifty–fifty attractor produced by symmetric contrarians. Producing a bit less contrarians on its own side than the other side becomes the key to win a public debate, which in turn can guarantee an election victory. The associated phase diagram of opinion dynamics is found to exhibit a rich variety of counterintuitive results.

Voting contagion: Modeling and analysis of a century of U.S. presidential elections

Social influence plays an important role in human behavior and decisions. Sources of influence can be divided as external, which are independent of social context, or as originating from peers, such as family and friends. An important question is how to disentangle the social contagion by peers from external influences. While a variety of experimental and observational studies provided insight into this problem, identifying the extent of contagion based on large-scale observational data with an unknown network structure remains largely unexplored. By bridging the gap between the large-scale complex systems perspective of collective human dynamics and the detailed approach of social sciences, we present a parsimonious model of social influence, and apply it to a central topic in political science—elections and voting behavior. We provide an analytical expression of the county vote-share distribution, which is in excellent agreement with almost a century of observed U.S. presidential election data. Analyzing the social influence topography over this period reveals an abrupt phase transition from low to high levels of social contagion, and robust differences among regions. These results suggest that social contagion effects are becoming more instrumental in shaping large-scale collective political behavior, with implications on democratic electoral processes and policies.

Modeling Radicalization Phenomena in Heterogeneous Populations

The phenomenon of radicalization is investigated within a mixed population composed of core and sensitive subpopulations. The latest includes first to third generation immigrants. Respective ways of life may be partially incompatible. In case of a conflict core agents behave as inflexible about the issue. In contrast, sensitive agents can decide either to live peacefully adjusting their way of life to the core one, or to oppose it with eventually joining violent activities. The interplay dynamics between peaceful and opponent sensitive agents is driven by pairwise interactions. These interactions occur both within the sensitive population and by mixing with core agents. The update process is monitored using a Lotka-Volterra-like Ordinary Differential Equation. Given an initial tiny minority of opponents that coexist with both inflexible and peaceful agents, we investigate implications on the emergence of radicalization. Opponents try to turn peaceful agents to opponents driving radicalization. However, inflexible core agents may step in to bring back opponents to a peaceful choice thus weakening the phenomenon. The required minimum individual core involvement to actually curb radicalization is calculated. It is found to be a function of both the majority or minority status of the sensitive subpopulation with respect to the core subpopulation and the degree of activeness of opponents. The results highlight the instrumental role core agents can have to hinder radicalization within the sensitive subpopulation. Some hints are outlined to favor novel public policies towards social integration.

Kinetic models of opinion formation in the presence of personal conviction

We consider a nonlinear kinetic equation of Boltzmann type, which takes into account the influence of conviction during the formation of opinion in a system of agents, which interact through the binary exchanges, introduced by Toscani [G. Toscani, Commun. Math. Sci. 4, 481 (2006)]. The original exchange mechanism, which is based on the human tendency to compromise and change of opinion through self-thinking, is here modified in the parameters of the compromise and diffusion terms, which now are assumed to depend on the personal degree of conviction. The numerical simulations show that the presence of conviction has the potential to break symmetry, and to produce clusters of opinions. The model is partially inspired by the recent work [L. Pareschi and G. Toscani, Phil. Trans. R. Soc. A 372, 20130396 (2014)], in which the role of knowledge in the formation of wealth distribution has been investigated.

Opinion dynamics: inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation

We propose and investigate different kinetic models for opinion formation, when the opinion formation process depends on an additional independent variable, e.g. a leadership or a spatial variable. More specifically, we consider (i) opinion dynamics under the effect of opinion leadership, where each individual is characterized not only by its opinion but also by another independent variable which quantifies leadership qualities; (ii) opinion dynamics modelling political segregation in ‘The Big Sort’, a phenomenon that US citizens increasingly prefer to live in neighbourhoods with politically like-minded individuals. Based on microscopic opinion consensus dynamics such models lead to inhomogeneous Boltzmann-type equations for the opinion distribution. We derive macroscopic Fokker–Planck-type equations in a quasi-invariant opinion limit and present results of numerical experiments.

The Undecided Have the Key: Interaction-Driven Opinion Dynamics in a Three State Model

The effects of interpersonal interactions on individual’s agreements result in a social aggregation process which is reflected in the formation of collective states, as for instance, groups of individuals with a similar opinion about a given issue. This field, which has been a longstanding concern of sociologists and psychologists, has been extended into an area of experimental social psychology, and even has attracted the attention of physicists and mathematicians. In this article, we present a novel model of opinion formation in which agents may either have a strict preference for a choice, or be undecided. The opinion shift emerges, in a threshold process, as a consequence of a cumulative persuasion for either one of the two opinions in repeated interactions. There are two main ingredients which play key roles in determining the steady states: the initial fraction of undecided agents and the change in agents’ persuasion after each interaction. As a function of these two parameters, the model presents a wide range of solutions, among which there are consensus of each opinion and bi-polarization. We found that a minimum fraction of undecided agents is not crucial for reaching consensus only, but also to determine a dominant opinion in a polarized situation. In order to gain a deeper comprehension of the dynamics, we also present the theoretical framework of the model. The master equations are of special interest for their nontrivial properties and difficulties in being solved analytically.

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Freezing period strongly impacts the emergence of a global consensus in the voter model

It is well known that human beings do not always change opinions or attitudes, since the duration of interaction with others usually has a significant impact on one's decision-making. Based on this observation, we introduce a freezing period into the voter model, in which the frozen individuals have a weakened opinion switching ability. We unfold the presence of an optimal freezing period, which leads to the fastest consensus, using computation simulations as well as theoretical analysis. We demonstrate that the essence of an accelerated consensus is attributed to the biased random walk of the interface between adjacent opinion clusters. The emergence of an optimal freezing period is robust against the size of the system and the number of distinct opinions. This study is instructive for understanding human collective behavior in other relevant fields.

Differentiation without distancing. explaining bi-polarization of opinions without negative influence

Explanations of opinion bi-polarization hinge on the assumption of negative influence, individuals’ striving to amplify differences to disliked others. However, empirical evidence for negative influence is inconclusive, which motivated us to search for an alternative explanation. Here, we demonstrate that bi-polarization can be explained without negative influence, drawing on theories that emphasize the communication of arguments as central mechanism of influence. Due to homophily, actors interact mainly with others whose arguments will intensify existing tendencies for or against the issue at stake. We develop an agent-based model of this theory and compare its implications to those of existing social-influence models, deriving testable hypotheses about the conditions of bi-polarization. Hypotheses were tested with a group-discussion experiment (N = 96). Results demonstrate that argument exchange can entail bi-polarization even when there is no negative influence.

Building up of individual inflexibility in opinion dynamics

Two models of opinion dynamics are entangled in order to build a more realistic model of inflexibility. The first one is the Galam unifying frame (GUF), which incorporates rational and inflexible agents, and the other one is the Continuous Opinions and Discrete Actions model. While initially in GUF, inflexibility is a fixed given feature of an agent, it is now the result of an accumulation for a given agent who makes the same choice through repeated updates. Inflexibility thus emerges as an internal property of agents becoming a continuous function of the strength of its opinion. Therefore, an agent can be more or less inflexible and can shift from inflexibility along one choice to inflexibility along the opposite choice. These individual dynamics of the building up and falling off of agent inflexibility are driven by the successive local updates of the associated individual opinions. New results are obtained and discussed in terms of predicting outcomes of public debates.

Influence of opinion dynamics on the evolution of games

Under certain circumstances such as lack of information or bounded rationality, human players can take decisions on which strategy to choose in a game on the basis of simple opinions. These opinions can be modified after each round by observing own or others payoff results but can be also modified after interchanging impressions with other players. In this way, the update of the strategies can become a question that goes beyond simple evolutionary rules based on fitness and become a social issue. In this work, we explore this scenario by coupling a game with an opinion dynamics model. The opinion is represented by a continuous variable that corresponds to the certainty of the agents respect to which strategy is best. The opinions transform into actions by making the selection of an strategy a stochastic event with a probability regulated by the opinion. A certain regard for the previous round payoff is included but the main update rules of the opinion are given by a model inspired in social interchanges. We find that the fixed points of the dynamics of the coupled model are different from those of the evolutionary game or the opinion models alone. Furthermore, new features emerge such as the independence of the fraction of cooperators with respect to the topology of the social interaction network or the presence of a small fraction of extremist players.

Impact of committed individuals on vaccination behavior

We study how the presence of committed vaccinators, a small fraction of individuals who consistently hold the vaccinating strategy and are immune to influence, impact the vaccination dynamics in well-mixed and spatially structured populations. For this purpose, we develop an epidemiological game-theoretic model of a flu-like vaccination by integrating an epidemiological process into a simple agent-based model of adaptive learning, where individuals (except for those committed ones) use anecdotal evidence to estimate costs and benefits of vaccination. We show that the committed vaccinators, acting as "steadfast role models" in the populations, can efficiently avoid the clustering of susceptible individuals and stimulate other imitators to take vaccination, hence contributing to the promotion of vaccine uptake. We substantiate our findings by making comparative studies of our model on a full lattice and on a randomly diluted one. Our work is expected to provide valuable information for decision-making and design more effective disease-control strategy.

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.

Opinion dynamics of random-walking agents on a lattice

The opinion dynamics of random-walking agents on finite two-dimensional lattices is studied. In the model, the opinion is continuous, and both the lattice and the opinion can be either periodic or nonperiodic. At each time step, all agents move randomly on the lattice, and update their opinions based on those of neighbors with whom the differences of opinion are not greater than a given threshold. Due to the effect of repeated averaging, opinions first converge locally, and eventually reach steady states. As in other models with bounded confidence, steady states in general are those with one or more opinion groups in which all agents have the same opinion. When both the lattice and the opinion are periodic, however, metastable states can emerge, in which the whole spectrum of location-dependent opinions can coexist. This result shows that, when a set of continuous opinions forms a structure like a circle, unlike the typically used linear opinions, rich dynamic behavior can arise. When there are geographical restrictions in real situations, a complete consensus is rarely reached, and metastable states here might be one of the explanations for these situations, especially when opinions are not linear.

Dynamics of majority rule with differential latencies

We investigate the dynamics of the majority-rule opinion formation model when voters experience differential latencies. With this extension, voters that just adopted an opinion go into a latent state during which they are excluded from the opinion formation process. The duration of the latent state depends on the opinion adopted by the voter. This leads to a bias toward consensus on the opinion that is associated with the shorter latency. We determine the exit probability and time to consensus for systems of N voters. Additionally, we derive an asymptotic characterization of the time to consensus by means of a continuum model.

Individualization as driving force of clustering phenomena in humans

One of the most intriguing dynamics in biological systems is the emergence of clustering, in the sense that individuals self-organize into separate agglomerations in physical or behavioral space. Several theories have been developed to explain clustering in, for instance, multi-cellular organisms, ant colonies, bee hives, flocks of birds, schools of fish, and animal herds. A persistent puzzle, however, is the clustering of opinions in human populations, particularly when opinions vary continuously, such as the degree to which citizens are in favor of or against a vaccination program. Existing continuous opinion formation models predict “monoculture” in the long run, unless subsets of the population are perfectly separated from each other. Yet, social diversity is a robust empirical phenomenon, although perfect separation is hardly possible in an increasingly connected world. Considering randomness has not overcome the theoretical shortcomings so far. Small perturbations of individual opinions trigger social influence cascades that inevitably lead to monoculture, while larger noise disrupts opinion clusters and results in rampant individualism without any social structure. Our solution to the puzzle builds on recent empirical research, combining the integrative tendencies of social influence with the disintegrative effects of individualization. A key element of the new computational model is an adaptive kind of noise. We conduct computer simulation experiments demonstrating that with this kind of noise a third phase besides individualism and monoculture becomes possible, characterized by the formation of metastable clusters with diversity between and consensus within clusters. When clusters are small, individualization tendencies are too weak to prohibit a fusion of clusters. When clusters grow too large, however, individualization increases in strength, which promotes their splitting. In summary, the new model can explain cultural clustering in human societies. Strikingly, model predictions are not only robust to “noise”—randomness is actually the central mechanism that sustains pluralism and clustering.

Modern societies are characterized by a large degree of pluralism in social, political and cultural opinions. In addition, there is evidence that humans tend to form distinct subgroups (clusters), characterized by opinion consensus within the clusters and differences between them. So far, however, formal theories of social influence have difficulty explaining this coexistence of global diversity and opinion clustering. In this study, we identify a missing ingredient that helps to fill this gap: the striving for uniqueness. Besides being influenced by their social environment, individuals also show a desire to hold a unique opinion. Thus, when too many other members of the population hold a similar opinion, individuals tend to adopt an opinion that distinguishes them from others. This notion is rooted in classical sociological theory and is supported by recent empirical research. We develop a computational model of opinion dynamics in human populations and demonstrate that the new model can explain opinion clustering. We conduct simulation experiments to study the conditions of clustering. Based on our results, we discuss preconditions for the persistence of pluralistic societies in a globalizing world.

Effects of social diversity on the emergence of global consensus in opinion dynamics

We propose a variant of the voter model by introducing the social diversity in the evolution process. Each individual is assigned a weight that is proportional to the power of its degree, where the power exponent alpha is an adjustable parameter that controls the level of diversity among individuals in the network. At each time step, a pair of connected individuals, say i and j , are randomly selected to update their opinions. The probability p(i) of choosing is opinion as their common opinion is proportional to i s weight. We consider the scale-free topology and concentrate on the efficiency of reaching the final consensus, which is significant in characterizing the self-organized systems. Interestingly, it is found that there exists an optimal value of alpha, leading to the shortest consensus time. This phenomenon indicates that, although a strong influence of high-degree individuals is helpful for quick consensus achievement, over strong influence inhibits the convergence process. Other quantities, such as the probability of an individual's initial opinion becomes the final opinion as a function of degree, the evolution of the number of opinion clusters, as well as the relationship between average consensus time and the network size, are also studied. Our results are helpful for better understanding the role of degree heterogeneity of the individuals in the opinion dynamics.

Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders

We propose a mathematical model for opinion formation in a society that is built of two groups, one group of ‘ordinary’ people and one group of ‘strong opinion leaders’. Our approach is based on an opinion formation model introduced in Toscani (Toscani 2006 Commun. Math. Sci. 4 , 481–496) and borrows ideas from the kinetic theory of mixtures of rarefied gases. Starting from microscopic interactions among individuals, we arrive at a macroscopic description of the opinion formation process that is characterized by a system of Fokker–Planck-type equations. We discuss the steady states of this system, extend it to incorporate emergence and decline of opinion leaders and present numerical results.

Mobility and social network effects on extremist opinions

Understanding the emergence of extreme opinions and in what kind of environment they might become less extreme is a central theme in our modern globalized society. A model combining continuous opinions and observed discrete actions (CODA) capable of addressing the important issue of measuring how extreme opinions might be has been recently proposed. In this paper I show extreme opinions to arise in a ubiquitous manner in the CODA model for a multitude of social network structures. Depending on network details reducing extremism seems to be possible. However, a large number of agents with extreme opinions is always observed. A significant decrease in the number of extremists can be observed by allowing agents to change their positions in the network.

Effects of inhomogeneous influence of individuals on an order-disorder transition in opinion dynamics

We study the effects of inhomogeneous influence of individuals on collective phenomena. We focus analytically on a typical model of the majority rule, applied to the completely connected agents. Two types of individuals A and B with different influence activity are introduced. The individuals A and B are distributed randomly with concentrations nu and 1-nu at the beginning and fixed further on. Our main result is that the location of the order-disorder transition is affected due to the introduction of the inhomogeneous influence. This result highlights the importance of inhomogeneous influence between different types of individuals during the process of opinion updating.

Spreading gossip in social networks

We study a simple model of information propagation in social networks, where two quantities are introduced: the spread factor, which measures the average maximal reachability of the neighbors of a given node that interchange information among each other, and the spreading time needed for the information to reach such a fraction of nodes. When the information refers to a particular node at which both quantities are measured, the model can be taken as a model for gossip propagation. In this context, we apply the model to real empirical networks of social acquaintances and compare the underlying spreading dynamics with different types of scale-free and small-world networks. We find that the number of friendship connections strongly influences the probability of being gossiped. Finally, we discuss how the spread factor is able to be applied to other situations.

Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect

We reconsider the no-tie-breaking two-state Galam contrarian model of opinion dynamics for update groups of size 3. While the initial model assumes a constant density of contrarians a for both opinions, the density now depends for each opinion on its global support. Proportionate contrarians are thus found to indeed preserve the main results of the former case. However, restricting the contrarian behavior to only the current collective majority makes the dynamics more complex with additional features. For a density a < a(c) = 1/9 of one-sided contrarians, a chaotic basin is found in the 50-50 region separated from two majority-minority point attractors, one on each side. For 1/9 < a less similar to 0.301 only the chaotic basin survives. In the range a > 0.301 the chaotic basin disappears and the majority starts to alternate between the two opinions with a staggered flow toward two point attractors. We then study the effect of both decoupling the local update time sequence from the contrarian behavior activation and a smoothing of the majority rule. A status quo-driven bias for contrarian activation is also considered. Introduction of unsettled agents driven in the debate on a contrarian basis is shown only to shrink the chaotic basin. The model may shed light on recent apparent contradictory elections with on the one hand very close results as in the United States in 2000 and in Germany in 2002 and 2005, and on the other hand, a huge majority as in France in 2002.

Local versus global interactions in nonequilibrium transitions: A model of social dynamics

A nonequilibrium system of locally interacting elements in a lattice with an absorbing order-disorder phase transition is studied under the effect of additional interacting fields. These fields are shown to produce interesting effects in the collective behavior of this system. Both for autonomous and external fields, disorder grows in the system when the probability of the elements to interact with the field is increased. There exists a threshold value of this probability beyond which the system is always disordered. The domain of parameters of the ordered regime is larger for nonuniform local fields than for spatially uniform fields. However, the zero field limit is discontinous. In the limit of vanishingly small probability of interaction with the field, autonomous or external fields are able to order a system that would fall in a disordered phase under local interactions of the elements alone. We consider different types of fields which are interpreted as forms of mass media acting on a social system in the context of Axelrod's model for cultural dissemination.