We introduce stochastic driving in the Sznajd model of opinion spreading. This stochastic effect is meant to mimic a social temperature, so that agents can take random decisions with a varying probability. We show that a stochastic driving has a tremendous impact on the system dynamics as a whole by inducing an order-disorder nonequilibrium phase transition. Interestingly, under certain conditions, this stochastic dynamics can spontaneously lead to agents in the system who are analogous to Galam's contarians.
In this work we consider the presence of contrarian agents in discrete 3-state kinetic exchange opinion models. The contrarians are individuals that adopt the choice opposite to the prevailing choice of their contacts, whatever this choice is. We consider binary as well as three-agent interactions, with stochastic parameters, in a fully-connected population. Our numerical results suggest that the presence of contrarians destroys the absorbing state of the original model, changing the transition to the para-ferromagnetic type. In this case, the consequence for the society is that the three opin…
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. 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 majority and $y$ when B is majority. Third scheme considers…
We assume a community whose members adopt one of two opinions $A$ or $B$. Each member appears as an inflexible, or as a non-contrarian or contrarian floater. An inflexible sticks to its opinion, whereas a floater may change into a floater of the alternative opinion. The occurrence of this change is governed by the local majority rule: members meet in groups of a fixed size, and a floater then changes its opinion provided it is a minority in the group. Subsequently, a non-contrarian floater keeps the opinion as adopted under the local majority rule, whereas a contrarian floater adopts the alter…
In the coevolving voter model, each voter has one of two diametrically opposite opinions, and a voter encountering a neighbor with the opposite opinion may either adopt it or rewire the connection to another randomly chosen voter sharing the same opinion. As we smoothly change the relative frequency of rewiring compared to that of adoption, there occurs a phase transition between an active phase and a frozen phase. By performing extensive Monte Carlo calculations, we show that the phase transition is characterized by critical exponents β=0.54(1) and ν =1.5(1), which differ from the existing me…
We revisit the no tie breaking 2-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, it now depends for each opinion on its global support. Proportionate contrarians are thus found to indeed preserve the former case main results. However, restricting the contrarian behavior to only the current collective majority, makes the dynamics more complex with novel features. For a density a<a_c=1/9 of one-sided contrarians, a chaotic basin is found in the fifty-fifty region separated from tw…
In the voter and many other opinion formation models, agents are assumed to behave as congregators (also called the conformists); they are attracted to the opinions of others. In this study, I investigate linear extensions of the voter model with contrarian agents. An agent is either congregator or contrarian and assumes a binary opinion. I investigate three models that differ in the behavior of the contrarian toward other agents. In model 1, contrarians mimic the opinions of other contrarians and oppose (i.e., try to select the opinion opposite to) those of congregators. In model 2, contraria…
In this work we study opinion formation on a fully-connected population participating of a public debate with two distinct choices, where the agents may adopt three different attitudes (favorable to either one choice or to the other, or undecided). The interactions between agents occur by pairs and are competitive, with couplings that are either negative with probability $p$ or positive with probability $1-p$. This bimodal probability distribution of couplings produces a behavior similar to the one resulting from the introduction of Galam's contrarians in the population. In addition, we consid…
In recent months the figure of Greta Thunberg and the theme of climate changings quickly became the focus of the debate. This has lead to a polarization effect in opinion forming about the climate subject. Starting from the analysis of this phenomenon, we develop an opinion dynamics model in which several types of contrarians agents are considered. Each agent is supposed to have an opinion on several topics related to each other, thus the opinions being formed on these topics are also mutually dependent. The aim of the paper is to investigate the indirect effects of contrarians agents on the c…
We investigate the opinion evolution of outflow dynamics based on the Sznajd model on a complete graph involving contrarian and independence behaviors. We consider a group of four spins representing the social agents with the following scenarios: (1) scenario two-two with contrarian agents or independence agents and (2) scenario three-one with contrarian or independence agents. All of them undergo a second-order phase transition according to our simulation. The critical point decreases exponentially as $λ$ and $f$ increases, where $λ$ and $f$ are contrarian and flexibility factors, respectivel…
In this paper, we study the effects of introducing contrarians in a model of Opinion Dynamics where the agents have internal continuous opinions, but exchange information only about a binary choice that is a function of their continuous opinion, the CODA model. We observe that the hung election scenario still exists here, but it is weaker and it shouldn't be expected in every election. Finally, we also show that the introduction of contrarians make the tendency towards extremism of the original model weaker, indicating that the existence of agents that prefer to disagree might be an important …
We introduce a contrarian opinion (CO) model in which a fraction p of contrarians within a group holds a strong opinion opposite to the opinion held by the rest of the group. At the initial stage, stable clusters of two opinions, A and B exist. Then we introduce contrarians which hold a strong B opinion into the opinion A group. Through their interactions, the contrarians are able to decrease the size of the largest A opinion cluster, and even destroy it. We see this kind of method in operation, e.g when companies send free new products to potential customers in order to convince them to adopt…
The influence of contrarians on the noisy voter model is studied at the mean-field level. The noisy voter model is a variant of the voter model where agents can adopt two opinions, optimistic or pessimistic, and can change them by means of an imitation (herding) and an intrinsic (noise) mechanisms. An ensemble of noisy voters undergoes a finite-size phase transition, upon increasing the relative importance of the noise to the herding, form a bimodal phase where most of the agents shear the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states. By the i…
The relationship between the topology of a network and specific types of dynamics unfolding in networks constitutes a subject of substantial interest. One type of dynamics that has attracted increasing attention because of its several potential implications is opinion formation. A phenomenon of particular importance, known to take place in opinion formation, is echo chambers' appearance. In the present work, we approach this phenomenon, while emphasizing the influence of contrarian opinions in a multi-opinion scenario. To define the contrarian opinion, we considered the Underdog effect, which …
We study models of a society composed of a mixture of conformist and reasonable contrarian agents that at any instant hold one of two opinions. Conformists tend to agree with the average opinion of their neighbors and reasonable contrarians to disagree, but revert to a conformist behavior in the presence of an overwhelming majority, in line with psychological experiments. The model is studied in the mean field approximation and on small-world and scale-free networks. In the mean field approximation, a large fraction of conformists triggers a polarization of the opinions, a pitchfork bifurcatio…
We study the contrarian voter model for opinion formation in a society under the influence of an external oscillating propaganda and stochastic noise. Each agent of the population can hold one of two possible opinions on a given issue --against or in favor, and interacts with its neighbors following either an imitation dynamics (voter behavior) or an anti-alignment dynamics (contrarian behavior): each agent adopts the opinion of a random neighbor with a time-dependent probability $p(t)$, or takes the opposite opinion with probability $1-p(t)$. The imitation probability $p(t)$ is controlled by …
A sociophysical model for opinion dynamics is shown to embody a series of recent western hung national votes all set at the unexpected and very improbable edge of a fifty-fifty score. It started with the Bush-Gore 2000 American presidential election, followed by the 2002 Stoiber-Schrőder, then the 2005 Schrőder-Merkel German elections, and finally the 2006 Prodi-Berlusconi Italian elections. In each case, the country was facing drastic choices, the running competing parties were advocating very different programs and millions of voters were involved. Moreover, polls were given a substantial ma…
People are often divided into conformists and contrarians, the former tending to align to the majority opinion in their neighborhood and the latter tending to disagree with that majority. In practice, however, the contrarian tendency is rarely followed when there is an overwhelming majority with a given opinion, which denotes a social norm. Such reasonable contrarian behavior is often considered a mark of independent thought, and can be a useful strategy in financial markets. We present the opinion dynamics of a society of reasonable contrarian agents. The model is a cellular automaton of Isin…
We investigate mean-field dynamics of a nonlinear opinion formation model with congregator and contrarian agents. Each agent assumes one of the two possible states. Congregators imitate the state of other agents with a rate that increases with the number of other agents in the opposite state, as in the linear voter model and nonlinear majority voting models. Contrarians flip the state with a rate that increases with the number of other agents in the same state. The nonlinearity controls the strength of the majority voting and is used as a main bifurcation parameter. We show that the model unde…
We introduce a novel type of contrarian agent, the balancer, to Galam model of opinion dynamics, in order to account for the existence of social skepticism over one-sidedness. We find that the inclusion of balancers, along with majoritarian floaters and single-sided inflexibles, brings about the emergence of a critical point on parametric plane of the dynamical system. Around the critical point, three distinct phases of opinion dynamics separated by discontinuous changes are found.
We study a stochastic model for the coevolution of a process of opinion formation in a population of agents and the network which underlies their interaction. Interaction links can break when agents fail to reach an opinion agreement. The structure of the network and the distribution of opinions over the population evolve towards a state where the population is divided into disconnected communities whose agents share the same opinion. The statistical properties of this final state vary considerably as the model parameters are changed. Community sizes and their internal connectivity are the qua…
In this work we study the opinion formation in a voter-like model defined on a square lattice of linear size $L$. The agents may be in three different states, representing any public debate with three choices (yes, no, undecided). We consider heterogeneous agents that have different convictions about their opinions. These convictions limit the capacity of persuasion of the individuals during the interactions. Moreover, there is a noise $p$ that represents the probability of an individual spontaneously change his opinion to the undecided state. Our simulations suggest that the system reaches st…
We consider a class of models of opinion formation where the dissemination of individual opinions occurs through the spreading of local consensus and disagreement. We study the emergence of full collective consensus or maximal disagreement in one- and two-dimensional arrays. In both cases, the probability of reaching full consensus exhibits well-defined scaling properties as a function of the system size. Two-dimensional systems, in particular, possess nontrivial exponents and critical points. The dynamical rules of our models, which emphasize the interaction between small groups of agents, sh…
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 between married and unmarried people. It seems possible to interpret this marriage gap in terms of the Galam model.
We proposed a new model, which capture the main difference between information and opinion spreading. In information spreading additional exposure to certain information has a small effect. Contrary, when an actor is exposed to 2 opinioned actors the probability to adopt the opinion is significant higher than in the case of contact with one such actor (called by J. Kleinberg "the 0-1-2 effect"). In each time step if an actor does not have an opinion, we randomly choose 2 his network neighbors. If one of them has an opinion, the actor adopts opinion with some low probability, if two - with a hi…
We propose a three-state microscopic opinion formation model for the purpose of simulating the dynamics of financial markets. In order to mimic the heterogeneous composition of the mass of investors in a market, the agent-based model considers two different types of traders: noise traders and contrarians. Agents are represented as nodes in a network of interactions and they can assume any of three distinct possible states (e.g. buy, sell or remain inactive). The time evolution of the state of an agent is dictated by probabilistic dynamics that include both local and global influences. A noise …
We study the dynamical and critical behavior of a model for irreversible opinion spreading on Barabási-Albert (BA) scale-free networks by performing extensive Monte Carlo simulations. The opinion spreading within an inhomogeneous society is investigated by means of the magnetic Eden model, a nonequilibrium kinetic model for the growth of binary mixtures in contact with a thermal bath. The deposition dynamics, which is studied as a function of the degree of the occupied sites, shows evidence for the leading role played by hubs in the growth process. Systems of finite size grow either ordered or…
A general model of opinion dynamics is introduced in which each individual's opinion is measured on a bounded continuous spectrum. Each opinion is influenced heterogeneously by every other opinion in the population. It is demonstrated that consensus, polarisation and a spread of moderate opinions are all possible within this model. Using dynamic mean-field theory, we are able to identify the statistical features of the interactions between individuals that give rise to each of the aforementioned emergent phenomena. The nature of the transitions between each of the observed macroscopic states i…
In this work we consider a model of epidemic spreading coupled with an opinion dynamics in a fully-connected population. Regarding the opinion dynamics, the individuals may be in two distinct states, namely in favor or against a vaccination campaign. Individuals against the vaccination follow a standard SIS model, whereas the pro-vaccine individuals can also be in a third compartment, namely Vaccinated. In addition, the opinions change according to the majority-rule dynamics in groups with three individuals. We also consider that the vaccine can give permanent or temporary immunization to the …
Irreversible opinion spreading phenomena are studied on small-world networks generated from 2D regular lattices by means of the magnetic Eden model, a nonequilibrium kinetic model for the growth of binary mixtures in contact with a thermal bath. In this model, the opinion or decision of an individual is affected by those of their acquaintances, but opinion changes (analogous to spin flips in an Ising-like model) are not allowed. Particularly, we focus on aspects inherent to the underlying 2D nature of the substrate, such as domain growth and cluster size distributions. Larger shortcut fraction…
A two state model for opinion forming, which has proven heuristic power, is reviewed with a novel emphasis on the existence or absence of a threshold for the dynamics. Monitored by repeated small groups discussions floater agents update their opinion according to a local majority rule. A threshold makes the initial supports to flow towards eitheir one of two opposite attractors with only one single opinion. While odd sizes yield a threshold at fifty percent even sizes, which allow the inclusion of doubt at an opinion tie, produces a threshold shift toward either 0 or 1 giving rise to minority …
In this work we consider the influence of mass media in the dynamics of the two-dimensional Sznajd model. This influence acts as an external field, and it is introduced in the model by means of a probability $p$ of the agents to follow the media opinion. We performed Monte Carlo simulations on square lattices with different sizes, and our numerical results suggest a change on the critical behavior of the model, with the absence of the usual phase transition for $p>\sim 0.18$. Another effect of the probability $p$ is to decrease the average relaxation times $τ$, that are log-normally distribute…
We study a model of opinion formation where the collective decision of group is said to happen if the fraction of agents having the most common opinion exceeds a threshold value, a \textit{critical mass}. We find that there exists a unique, non-trivial critical mass giving the most efficient convergence to consensus. In addition, we observe that for small critical masses, the characteristic time scale for the relaxation to consensus splits into two. The shorter time scale corresponds to a direct relaxation and the longer can be explained by the existence of intermediate, metastable states simi…
The goal of opinion maximization is to maximize the positive view towards a product, an ideology or any entity among the individuals in social networks. So far, opinion maximization is mainly studied as finding a set of influential nodes for fast content dissemination in a social network. In this paper, we propose a novel approach to solve the problem, where opinion maximization is achieved through efficient information spreading. In our model, multiple sources inject information continuously into the network, while the regular nodes with heterogeneous social learning abilities spread the info…
Irreversible opinion spreading phenomena are studied on small-world and scale-free networks by means of the magnetic Eden model, a nonequilibrium kinetic model for the growth of binary mixtures in contact with a thermal bath. In this model, the opinion of an individual is affected by those of their acquaintances, but opinion changes (analogous to spin flips in an Ising-like model) are not allowed. We focus on the influence of advertising, which is represented by external magnetic fields. The interplay and competition between temperature and fields lead to order-disorder transitions, which are …
Rumors have ignited revolutions, undermined the trust in political parties, or threatened the stability of human societies. Such destructive potential has been significantly enhanced by the development of on-line social networks. Several theoretical and computational studies have been devoted to understanding the dynamics and to control rumor spreading. In the present work, a model of rumor-telling in opinion polarized networks was investigated through extensive computer simulations. The key mechanism is the coupling between ones' opinions and their leaning to spread a given information, eithe…
We study the changes of opinions about vaccination together with the evolution of a disease. In our model we consider a multiplex network consisting of two layers. One of the layers corresponds to a social network where people share their opinions and influence others opinions. The social model that rules the dynamic is the M-model, which takes into account two different processes that occurs in a society: persuasion and compromise. This two processes are related through a parameter $r$, $r<1$ describes a moderate and committed society, for $r>1$ the society tends to have extremist opinions, w…
We generalize Galam's model of opinion spreading by introducing three competing choices. At each update, the population is randomly divided in groups of three agents, whose members adopt the opinion of the local majority. In the case of a tie, the local group adopts opinion A, B or C with probabilities alpha, beta and (1-alpha-beta) respectively. We derive the associated phase diagrams and dynamics by both analytical means and simulations. Polarization is always reached within very short time scales. We point out situations in which an initially very small minority opinion can invade the whole…
A universal formula is shown to predict the dynamics of public opinion including eventual sudden and unexpected outbreaks of minority opinions within a generic parameter space of five dimensions. The formula is obtained combining and extending several components of Galam model of opinion dynamics, otherwise treated separately, into one single update equation, which then deploys in a social space of five dimensions. Four dimensions account for a rich diversity of individual traits within a heterogeneous population, including differentiated stubbornness, contrarianism, and embedded prejudices. T…
Opinion formation and disease spreading are among the most studied dynamical processes on complex networks. In real societies, it is expected that these two processes depend on and affect each other. However, little is known about the effects of opinion dynamics over disease dynamics and vice versa, since most studies treat them separately. In this work we study the dynamics of the voter model for opinion formation intertwined with that of the contact process for disease spreading, in a population of agents that interact via two types of connections, social and contact. These two interacting d…
A classical model for social-influence-driven opinion change is the threshold model. Here we study cascades of opinion change driven by threshold model dynamics in the case where multiple {\it initiators} trigger the cascade, and where all nodes possess the same adoption threshold $ϕ$. Specifically, using empirical and stylized models of social networks, we study cascade size as a function of the initiator fraction $p$. We find that even for arbitrarily high value of $ϕ$, there exists a critical initiator fraction $p_c(ϕ)$ beyond which the cascade becomes global. Network structure, in particul…
In this work, we investigate interactions that simultaneously order a system locally, while keeping it globally disordered. The study is done in the context of the emergence of diversity in opinion propagation models with interactions rooted in conformity, but some suggestions on how this could be extended to other topics (like ecology and neuroscience) are also made. We do this by introducing a generic modification that can be added to different opinion propagation models (and other agent based models) and that seems to introduce a global tendency towards diversity without leading the system …
The adaptive voter model has been widely studied as a conceptual model for opinion formation processes on time-evolving social networks. Past studies on the effect of zealots, i.e. nodes aiming to spread their fixed opinion throughout the system, only considered the voter model on a static network. Here, we extend the study of zealotry to the case of an adaptive network topology co-evolving with the state of the nodes and investigate opinion spreading induced by zealots depending on their initial density and connectedness. Numerical simulations reveal that below the fragmentation threshold a l…
The paper addresses a method for spreading messages in social networks through an initial acceleration by Spreading Groups. These groups start the spread which eventually reaches a larger portion of the network. The use of spreading groups creates a final flow which resembles the spread through the nodes with the highest level of influence (opinion leaders). While harnessing opinion leaders to spread messages is generally costly, the formation of spreading groups is merely a technical issue, and can be done by computerized bots. The paper presents an information flow model and inspects the mod…
We investigate the formation of opinion against authority in an authoritarian society composed of agents with different levels of authority. We explore a "dissenting" opinion, held by lower-ranking, obedient, or less authoritative people, spreading in an environment of an "affirmative" opinion held by authoritative leaders. A real-world example would be a corrupt society where people revolt against such leaders, but it can be applied to more general situations. In our model, agents can change their opinion depending on their authority relative to their neighbors and their own confidence level.…
Many studies have explored spreading and diffusion through complex networks. The following study examines a specific case of spreading of opinions in modern society through two spreading schemes, defined as being either through word-of-mouth (WOM), or through online search engines (WEB). We apply both modelling and real experimental results and compare the opinions people adopt through an exposure to their friend`s opinions, as opposed to the opinions they adopt when using a search engine based on the PageRank algorithm. A simulated study shows that when members in a population adopt decisions…
This work consists of an epidemic model with vaccination coupled with an opinion dynamics. Our objective was to study how disease risk perception can influence opinions about vaccination and therefore the spreading of the disease. Differently from previous works we have considered continuous opinions. The epidemic spreading is governed by a SIS-like model with an extra vaccinated state. In our model individuals vaccinate with a probability proportional to their opinions. The opinions change due to peer influence in pairwise interactions. The epidemic feedback to the opinion dynamics acts as an…
Election results are determined by numerous social factors that affect the formation of opinion of the voters, including the network of interactions between them and the dynamics of opinion influence. In this work we study the result of proportional elections using an opinion dynamics model similar to simple opinion spreading over a complex network. Erdos-Renyi, Barabasi-Albert, regular lattices and randomly augmented lattices are considered as models of the underlying social networks. The model reproduces the power law behavior of number of candidates with a given number of votes found in rea…
We study a model of a population making a binary decision based on information spreading within the population, which is fully connected or covering a square grid. We assume that a fraction of the population wants to make the choice of the majority, while the rest want to make the minority choice. This resembles opinion spreading with "contrarian" agents, but has the game theoretic aspect that agents try to optimize their own situation in ways that are incompatible with the common good. When this fraction is less than 1/2, the population can efficiently self-organize to a state where agents ge…
We introduce a three-state model to study the effects of a neutral party on opinion spreading, in which the tendency of agents to agree with their neighbors can be tuned to favor either the neutral party or two oppositely polarized parties, and can be disrupted by social agitation mimicked as temperature. We study the equilibrium phase diagram and the non-equilibrium stochastic dynamics of the model with various analytical approaches and with Monte Carlo simulations on different substrates: the fully-connected (FC) graph, the one-dimensional (1D) chain, and Erdös-Rényi (ER) random graphs. We s…
