We introduce and analyse a three-body consensus model (3CM) for non-linear consensus dynamics on hypergraphs. Our model incorporates reinforcing group effects, which can cause shifts in the average state of the system even in if the underlying graph is complete (corresponding to a mean-field interaction), a phenomena that may be interpreted as a type of peer pressure. We further demonstrate that for systems with two clustered groups, already a small asymmetry in our dynamics can lead to the opinion of one group becoming clearly dominant. We show that the nonlinearity in the model is the essential ingredient to make such group dynamics appear, and demonstrate how our system can otherwise be written as a linear, pairwise interaction system on a rescaled network.
Monte Carlo simulations mix the opinion dynamics of Deffuant et al with the cultural transfer model of Axelrod, using ten discrete possible opinions on ten different themes. As Jacobmeier's simulations of the pure Deffuant case, people preferably agree on nearly all or nearly no theme.
Usually, opinion formation models assume that individuals have an opinion about a given topic which can change due to interactions with others. However, individuals can have different opinions in different topics and therefore n-dimensional models are best suited to deal with these cases. While there have been many efforts to develop analytical models for one dimensional opinion models, less attention has been paid to multidimensional ones. In this work, we develop an analytical approach for multidimensional models of continuous opinions where dimensions can be correlated or uncorrelated. We s…
Consider a multiplex network formed by two layers indicating social interactions: the first layer is a friendship network and the second layer is a network of business relations. In this duplex network each pair of individuals can be connected in different ways: they can be connected by a friendship but not connected by a business relation, they can be connected by a business relation without being friends, or they can be simultaneously friends and in a business relation. In the latter case we say that the links in different layers overlap. These three types of connections are called multilink…
We study opinion dynamics over multiplex networks where agents interact with bounded confidence. Namely, two neighbouring individuals exchange opinions and compromise if their opinions do not differ by more than a given threshold. In literature, agents are generally assumed to have a homogeneous confidence bound. Here, we study analytically and numerically opinion evolution over structured networks characterised by multiple layers with respective confidence thresholds and general initial opinion distributions. Through rigorous probability analysis, we show analytically the critical thresholds …
We study multidimensional continuous opinion dynamics, where opinions are nonnegative vectors which components sum up to one. Examples of such opinions are budgets or other allocation vectors which display a distribution of a fixed amount of ressource to n projects. We use the opinion dynamics models of Deffuant-Weisbuch and Hegselmann-Krause, which both extend naturally to more dimensional opinions. They both rely on bounded confidence of the agents and differ in their communication regime. We show detailed simulation results regarding $n=2,...,8$ and the bound of confidence $\eps$. Number, l…
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…
We have presented a numerical model of a collective opinion formation procedure to explain political phenomena such as two-party and multi-party systems in politics, political unrest, military coup d'etats and netizen revolutions. Nonlinear interaction with binary and independent decision making processes can yield various collective behaviors or collective political opinions. Statistical physics and nonlinear dynamics may provide useful tools to study various socio-political dynamics.
In recent years networks have gained unprecedented attention in studying a broad range of topics, among them in complex systems research. In particular, multi-agent systems have seen an increased recognition of the importance of the interaction topology. It is now widely recognized that emergent phenomena can be highly sensitive to the structure of the interaction network connecting the system's components, and there is a growing body of abstract network classes, whose contributions to emergent dynamics are well-understood. However, much less understanding have yet been gained about the effect…
Individuals have opinions but can change them under the influence of others. The recent models of Sznajd (missionaries), of Deffuant et al. (negotiators), and of Krause and Hegselmann (opportunists) are reviewed here, while the voter and Ising models, Galam's majority rule and the Axelrod multicultural model were dealt with by other lecturers at this 8th Granada Seminar.
Social consensus is important for society. Sometimes the success of society depends on a consensus (e.g. the decision to pay taxes or to commit to the constitution). Examples for continuous opinion dynamics are discussions about tax rates or budget plan proposals for governments investments. Another example is a commission of experts which should reach a estimate about a certain issue, e.g. the tax revenues of the next year. In all these situations we got a group of agents which should reach a common agreement either for reaching a good approximation to the truth but on the other hand for the …
It is known that individual opinions on different policy issues often align to a dominant ideological dimension (e.g. "left" vs. "right") and become increasingly polarized. We provide an agent-based model that reproduces these two stylized facts as emergent properties of an opinion dynamics in a multi-dimensional space of continuous opinions. The mechanisms for the change of agents' opinions in this multi-dimensional space are derived from cognitive dissonance theory and structural balance theory. We test assumptions from proximity voting and from directional voting regarding their ability to …
A long line of work in social psychology has studied variations in people's susceptibility to persuasion -- the extent to which they are willing to modify their opinions on a topic. This body of literature suggests an interesting perspective on theoretical models of opinion formation by interacting parties in a network: in addition to considering interventions that directly modify people's intrinsic opinions, it is also natural to consider interventions that modify people's susceptibility to persuasion. In this work, we adopt a popular model for social opinion dynamics, and we formalize the op…
In this paper Nowak--Szamrej-Latané model is reconsidered. This computerised model of opinion formation bases on Latané theory of social impact. We modify this model to allow for multi (more than two) opinions. With computer simulations we show that in the modified model the signatures of order/disorder phase transition are still observed. The transition may be observed in the average fraction of actors sharing the $i$-th opinion, its variation and also average number of clusters of actors with the same opinion and the average size of the largest cluster of actors sharing the same opinion. Als…
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. 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 …
Krause's model of opinion dynamics has recently been the object of several studies, partly because it is one of the simplest multi-agent systems involving position-dependent changing topologies. In this model, agents have an opinion represented by a real number and they update it by averaging those agent opinions distant from their opinion by less than a certain interaction radius. Some results obtained on this model rely on the fact that the opinion orders remain unchanged under iteration, a property that is consistent with the intuition in models with simultaneous updating on a fully connect…
How do movements and coalitions which engage with multiple social issues succeed in cross-issue solidarity, and when do they instead become fragmented? To address this, the mechanisms of cross-issue interaction have to be understood. Prior work on opinion dynamics and political disagreement has focused on single-issue consensus and polarization. Inspired by practices of cross-issue movement building, we have developed a general model of multi-issue opinion dynamics where agreement on one issue can promote greater inclusivity in discussing other issues, thereby avoiding the pitfalls of exclusiv…
Many empirical networks are intrinsically pluralistic, with interactions occurring within groups of arbitrary agents. Then the agent in the network can be influenced by types of neighbors, common examples include similarity, opposition, and negligibility. Although the influence of neighbors can be described as an amicable and antagonistic relationship in complex real-world systems accurately, and the research on the dynamic process of public opinion evolution with different types of influence is valuable, few studies have mentioned that issue. In this paper, we develop a novel model on network…
Opinion dynamics concerns social processes through which populations or groups of individuals agree or disagree on specific issues. As such, modelling opinion dynamics represents an important research area that has been progressively acquiring relevance in many different domains. Existing approaches have mostly represented opinions through discrete binary or continuous variables by exploring a whole panoply of cases: e.g. independence, noise, external effects, multiple issues. In most of these cases the crucial ingredient is an attractive dynamics through which similar or similar enough agents…
We here propose a model to simulate the process of opinion formation, which accounts for the mutual affinity between interacting agents. Opinion and affinity evolve self-consistently, manifesting a highly non trivial interplay. A continuous transition is found between single and multiple opinion states. Fractal dimension and signature of critical behaviour are also reported. A rich phenomenology is presented and discussed with reference to corresponding psychological implications.
The noisy voter model is a stylised representation of opinion dynamics. Individuals copy opinions from other individuals, and are subject to spontaneous state changes. In the case of two opinion states this model is known to have a noise-driven transition between a unimodal phase, in which both opinions are present, and a bimodal phase in which one of the opinions dominates. The presence of zealots can remove the unimodal and bimodal phases in the model with two opinion states. Here, we study the effects of zealots in noisy voter models with M>2 opinion states on complete interaction graphs. W…
We consider an idealized model in which individuals' changing opinions and their social network coevolve, with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordant edge. Specifically, an interaction between $x$ and one of its neighbors $y$ leads to $x$ imitating $y$ with probability $(1-α)$ and otherwise (i.e., with probability $α$) $x$ cutting its tie to $y$ in order to instead connect to a randomly chosen individual. Building on previous work about the two-opinion case, we study the multiple-opinion…
In this paper we investigate the possibility of reducing the complexity of a system composed of a large number of interacting agents, whose dynamics feature a symmetry breaking. We consider first order stochastic differential equations describing the behavior of the system at the particle (i.e., Lagrangian) level and we get its continuous (i.e., Eulerian) counterpart via a kinetic description. However, the resulting continuous model alone fails to describe adequately the evolution of the system, due to the loss of granularity which prevents it from reproducing the symmetry breaking of the part…
Adaptive models of opinion formation among humans can display a fragmentation transition, where a social network breaks into disconnected components. Here, we investigate this transition in a class of models with arbitrary number of opinions. In contrast to previous work we do not assume that opinions are equidistant or arranged on a one-dimensional conceptual axis. Our investigation reveals detailed analytical results on fragmentations in a three-opinion model, which are confirmed by agent-based simulations. Furthermore, we show that in certain models the number of opinions can be reduced wit…
Modelling efforts in opinion dynamics have to a large extent ignored that opinion exchange between individuals can also have an effect on how willing they are to express their opinion publicly. Here, we introduce a model of public opinion expression. Two groups of agents with different opinion on an issue interact with each other, changing the willingness to express their opinion according to whether they perceive themselves as part of the majority or minority opinion. We formulate the model as a multi-group majority game and investigate the Nash equilibria. We also provide a dynamical systems…
People often face the challenge of choosing among different options with similar attractiveness. To study the distribution of preferences that emerge in such situations, a useful approach is to simulate opinion dynamics on top of complex networks, composed by nodes (individuals) and their connections (edges), where the state of each node can be one amongst several opinions including the undecided state. We use two different dynamics rules: the one proposed by Travieso-Fontoura (TF) and the plurality rule (PR), which are paradigmatic of outflow and inflow dynamics, respectively. We are speciall…
The significance and influence of US Supreme Court majority opinions derive in large part from opinions' roles as precedents for future opinions. A growing body of literature seeks to understand what drives the use of opinions as precedents through the study of Supreme Court case citation patterns. We raise two limitations of existing work on Supreme Court citations. First, dyadic citations are typically aggregated to the case level before they are analyzed. Second, citations are treated as if they arise independently. We present a methodology for studying citations between Supreme Court opini…
We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multi-state voter models with all-to-all interaction and on uncorrelated networks. Individual realisations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a meta-stable state. The density of active links and the entropy in these states varies from realisation to realisation. Making some simple assumptions we are able to analytically calculate the average density of active links and the ave…
Understanding the evolution of collective beliefs is of critical importance to get insights on the political trends as well as on social tastes and opinions. In particular, it is pivotal to develop analytical models that can predict the beliefs dynamics and capture the interdependence of opinions on different subjects. In this paper we tackle this issue also accounting for the individual endogenous process of opinion evolution, as well as repulsive interactions between individuals' opinions that may arise in the presence of an adversarial attitude of the individuals. Using a mean field approac…
We propose a minimal multi-agent model for the collective dynamics of opinion formation in the society, by modifying kinetic exchange dynamics studied in the context of income, money or wealth distributions in a society. This model has an intriguing spontaneous symmetry breaking transition to polarized opinion state starting from non-polarized opinion state. In order to analyze the model, we introduce an iterative map version of the model, which has very similar statistical characteristics. An approximate theoretical analysis of the numerical results are also given, based on the iterative map …
Opinion formation is an important element of social dynamics. It has been widely studied in the last years with tools from physics, mathematics and computer science. Here, a continuous model of opinion dynamics for multiple possible choices is analysed. Its main features are the inclusion of disagreement and possibility of modulating information, both from one and multiple sources. The interest is in identifying the effect of the initial cohesion of the population, the interplay between cohesion and information extremism, and the effect of using multiple sources of information that can influen…
We study a variant of the voter model with multiple opinions; individuals can imitate each other and also change their opinion randomly in mutation events. We focus on the case of a population with all-to-all interaction. A noise-driven transition between regimes with multi-modal and unimodal stationary distributions is observed. In the former, the population is mostly in consensus states; in the latter opinions are mixed. We derive an effective death-birth process, describing the dynamics from the perspective of one of the opinions, and use it to analytically compute marginals of the stationa…
We study an endogenous opinion (or, belief) dynamics model where we endogenize the social network that models the link (`trust') weights between agents. Our network adjustment mechanism is simple: an agent increases her weight for another agent if that agent has been close to truth (whence, our adjustment criterion is `past performance'). Moreover, we consider multiply biased agents that do not learn in a fully rational manner but are subject to persuasion bias - they learn in a DeGroot manner, via a simple `rule of thumb' - and that have biased initial beliefs. In addition, we also study this…
Individuals who interact with each other in social networks often exchange ideas and influence each other's opinions. A popular approach to study the spread of opinions on networks is by examining bounded-confidence models (BCMs), in which the nodes of a network have continuous-valued states that encode their opinions and are receptive to other nodes' opinions when they lie within some confidence bound of their own opinion. In this paper, we extend the Deffuant--Weisbuch (DW) model, which is a well-known BCM, by examining the spread of opinions that coevolve with network structure. We propose …
We examine the tuning of cooperative behavior in repeated multi-agent games using an analytically tractable, continuous-time, nonlinear model of opinion dynamics. Each modeled agent updates its real-valued opinion about each available strategy in response to payoffs and other agent opinions, as observed over a network. We show how the model provides a principled and systematic means to investigate behavior of agents that select strategies using rationality and reciprocity, key features of human decision-making in social dilemmas. For two-strategy games, we use bifurcation analysis to prove con…
