Opinion dynamics in finance and business: a literature review and research opportunities

Opinion dynamics is an opinion evolution process of a group of agents,where the final opinion distribution tends to three stable states:consensus,polarization,and fragmentation.At present,the opinion dynamics models have been extensively studied in differrent fields.This paper provides a review of opinion dynamics in finance and business,such as,finance,marketing,e-commerce,politics,and group decision making.Furthermore,identified research challenges have been proposed to promote the future research of this topic.

Symmetry breaking in the opinion dynamics of a multi-group project organization

A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group＇s opinion evolution is driven by two types of forces：（i） the group＇s cohesive force which tends to restore the opinion back towards the initial status because of its company culture;and（ii） nonlinear coupling forces with other groups which attempt to bring opinions closer due to collaboration willingness.Bifurcation analysis for the case of a two-group project shows a cusp catastrophe phenomenon and three distinctive evolutionary regimes,i.e.,a deadlock regime,a convergence regime,and a bifurcation regime in opinion dynamics.The critical value of initial discord between the two groups is derived to discriminate which regime the opinion evolution belongs to.In the case of a three-group project with a symmetric social network,both bifurcation analysis and simulation results demonstrate that if each pair has a high initial discord,instead of symmetrically converging to consensus with the increase of coupling scale as expected by Gabbay＇s result（Physica A 378（2007） p.125 Fig.5）,project organization（PO） may be split into two distinct clusters because of the symmetry breaking phenomenon caused by pitchfork bifurcations,which urges that apart from divergence in participants＇ interests,nonlinear interaction can also make conflict inevitable in the PO.The effects of two asymmetric level parameters are tested in order to explore the ways of inducing dominant opinion in the whole PO.It is found that the strong influence imposed by a leader group with firm faith on the flexible and open minded follower groups can promote the formation of a positive dominant opinion in the PO.

OPINION DYNAMICS ON SOCIAL NETWORKS

Opinion dynamics has recently attracted much attention,and there have been a lot of achievements in this area.This paper first gives an overview of the development of opinion dynamics on social networks.We introduce some classical models of opinion dynamics in detail,including the DeGroot model,the Krause model,O-1 models,sign networks and models related to Gossip algorithms.Inspired by some real life cases,we choose the unit circle as the range of the individuals'opinion values.We prove that the individuals'opinions of the randomized gossip algorithm in which the individuals'opinion values are on the unit circle reaches consensus almost surely.

Symmetry breaking in the opinion dynamics of a multi-group project organization

A bounded confidence model of opinion dynamics in multi-group projects is presented in which each group’s opinion evolution is driven by two types of forces:(i) the group’s cohesive force which tends to restore the opinion back towards the initial status because of its company culture;and(ii) nonlinear coupling forces with other groups which attempt to bring opinions closer due to collaboration willingness.Bifurcation analysis for the case of a two-group project shows a cusp catastrophe phenomenon and three distinctive evolutionary regimes,i.e.,a deadlock regime,a convergence regime,and a bifurcation regime in opinion dynamics.The critical value of initial discord between the two groups is derived to discriminate which regime the opinion evolution belongs to.In the case of a three-group project with a symmetric social network,both bifurcation analysis and simulation results demonstrate that if each pair has a high initial discord,instead of symmetrically converging to consensus with the increase of coupling scale as expected by Gabbay’s result(Physica A 378(2007) p.125 Fig.5),project organization(PO) may be split into two distinct clusters because of the symmetry breaking phenomenon caused by pitchfork bifurcations,which urges that apart from divergence in participants’ interests,nonlinear interaction can also make conflict inevitable in the PO.The effects of two asymmetric level parameters are tested in order to explore the ways of inducing dominant opinion in the whole PO.It is found that the strong influence imposed by a leader group with firm faith on the flexible and open minded follower groups can promote the formation of a positive dominant opinion in the PO.

Opinion Dynamics in Two-Layer Networks with Hypocrisy

We propose a general concealed voter model(GCVM),in which individuals interact in two layers and can exchange their opinions in the internal layer.This interaction is not allowed in a CVM.By exchanging opinions in the internal layer,we mean that individuals share their real or internal opinions with their close friends.The process of opinion formation in GCVM is presented in the paper.We make the series of numerical simulations of GCVM with different network structures(both external and internal)and get some counterintuitive conclusions.For instance,we find out that sometimes with a relatively simple network structure of an external layer the consensus within the individuals’opinions cannot be reached,and if individuals in the network are not good at expressing their opinions publicly(in an external layer),exchanging opinions with their close friends(in an internal layer)is almost useless.

Partial convergence of heterogeneous Hegselmann-Krause opinion dynamics

In opinion dynamics,the convergence of the heterogeneous Hegselmann-Krause(HK) dynamics has always been an open problem for years which looks forward to any essential progress.In this short note,we prove a partial convergence conclusion of the general heterogeneous HK dynamics.That is,there must be some agents who will reach static states in finite time,while the other opinions have to evolve between them with a minimum distance if all the opinions does not reach consensus.And this result leads to the convergence of several special cases of heterogeneous HK dynamics,including when the minimum confidence bound is large enough,the initial opinion difference is small enough,and so on.

Discrete Opinion Dynamics on Online Social Networks

This paper focuses on the dynamics of binary opinions {＋1,-1} on online social networks consisting of heterogeneous actors. In our model, actors update their opinions under the interplay of social influence and self- affirmation, which leads to rich dynamical behaviors on online social networks. We find that the opinion leading to the consensus features an advantage of the initially weighted fraction based on actors＇ strength over the other, instead of the population. For the role of specific actors, the consensus converges towards the opinion that a small fraction of high-strength actors hold, and individual diversity of self-amrmation slows down the ordering process of consensus. These indicate that high-strength actors play an essential role in opinion formation with strong social influence as well as high persistence. Further investigations show that the initial fraction of high-strength actors to dominate the evolution depends on the heterogeneity of the strength distribution, and less high-strength actors are needed in the case of a smaller exponent of power-law distribution of actors＇ strength. Our study provides deep insights into the role of social influence and self-affirmation on opinion formation on online social networks.

Continuous Opinion Dynamics in Complex Networks

Many realistic social networks share some universal characteristic properties,such as the small-world effects and the heterogeneous distribution of connectivity degree,which affect the dynamics in society system,especially the opinion dynamics in society.To see this,we study the opinion dynamics of the Improved Deffuant Model(IDM)in complex networks.When the two opinions differ by less than the confidence parameterǫ(0<ǫ<1),each opinion moves partly in the direction of the other with the convergence parameterµ,which is a function of the opposite’s degree k;otherwise,the two refuse to discuss and no opinion is changed.We analyze the evolution of the steady opinion s∗as a function of the confidence parameterǫ,the relation between the minority steady opinion smin∗and the individual connectivity k,and find some interesting results that show the dependence of the opinion dynamics on the confidence parameter and on the system topology.This study provides a new perspective and tools to understand the effects of complex system topology on opinion dynamics.

Voter model on adaptive networks

Voter model is an important basic model in statistical physics.In recent years,it has been more and more used to describe the process of opinion formation in sociophysics.In real complex systems,the interactive network of individuals is dynamically adjusted,and the evolving network topology and individual behaviors affect each other.Therefore,we propose a linking dynamics to describe the coevolution of network topology and individual behaviors in this paper,and study the voter model on the adaptive network.We theoretically analyze the properties of the voter model,including consensus probability and time.The evolution of opinions on dynamic networks is further analyzed from the perspective of evolutionary game.Finally,a case study of real data is shown to verify the effectiveness of the theory.

Structure Balance and Opinions Dynamic in Signed Social Network

In this paper,the authors consider both the nodes’opinions dynamics and signed network edges’evolution.Simulated Annealing Algorithm is applied for searching the minimal global energy function,and bounded confidence model is used for nodes’opinions updating.The authors find that the local and global level of balance of signed network is consistent with collective opinions 2-polarization.This property is explainable in terms of the structure balance of the sign distributions on the nodes and edges.The level of balance and the final opinions polarization pattern are achieved depends on the initial density of signed network,and the percentage of initial positive edges.Numerical simulations of the proposed model display a rich and intuitive behavior of the opinions polarization processes.In particular,the authors show that opinions persistent fluctuations is consistent with minimal global the energy function.This work verify that signed social networks are indeed limited balanced,could be used to explain ubiquitous binary polarization phenomenon of real world.

Convergence Rate of the Asymmetric Deffuant-Weisbuch Dynamics

This paper considers the convergence rate of an asymmetric Deffuant-Weisbuch model.The model is composed by finite n interacting agents.In this model,agent i’s opinion is updated at each time,by first selecting one randomly from n agents,and then combining the selected agent j’s opinion if the distance between j’s opinion and i’s opinion is not larger than the confidence radiusε0.This yields the endogenously changing inter-agent topologies.Based on the previous result that all agents opinions will converge almost surely for any initial states,the authors prove that the expected potential function of the convergence rate is upper bounded by a negative exponential function of time t when opinions reach consensus finally and is upper bounded by a negative power function of time t when opinions converge to several different limits.

Soft-Control for Collective Opinion of Weighted DeGroot Model

The DeGroot model is a classic model to study consensus of opinion in a group of individuals(agents). Consensus can be achieved under some circumstances. But when the group reach consensus with a convergent opinion value which is not what we expect, how can we intervene the system and change the convergent value? In this paper a mechanism named soft control is first introduced in opinion dynamics to guide the group's opinion when the population are given and evolution rules are not allowed to change. According to the idea of soft control, one or several special agents,called shills, are added and connected to one or several normal agents in the original group. Shills act and are treated as normal agents. The authors prove that the change of convergent opinion value is decided by the initial opinion and influential value of the shill, as well as how the shill connects to normal agents. An interesting and counterintuitive phenomenon is discovered: Adding a shill with an initial opinion value which is smaller(or larger) than the original convergent opinion value dose not necessarily decrease(or increase) the convergent opinion value under some conditions. These conditions are given through mathematical analysis and they are verified by the numerical tests. The authors also find out that the convergence speed of the system varies when a shill is connected to different normal agents. Our simulations show that it is positively related to the degree of the connected normal agent in scale-free networks.

Consensus of a two-agent opinion dynamical system with processing delay

A sufficient condition is obtained for a two-agent opinion dynamical system with pro-cessing delay to experience unconditional consensus. To this end, the dynamics proposed is transformed into a general class of delay differential equation and asymptotic stability of the origin is then analyzed. It is also shown that increasing delay can prolong the time needed for the system to realize consensus and even induce a Hopf bifurcation.

Emergence of Community Structure in the Adaptive Social Networks

In this paper,we propose a simple model of opinion dynamics to construct social networks,based on the algorithm of link rewiring of local attachment(RLA)and global attachment(RGA).Generality,the system does reach a steady state where all individuals'opinion and the complex network structure are fixed.The RGA enhances the ability of consensus of opinion formation.Furthermore,by tuning a model parameter p,which governs the proportion of RLA and RGA,we find the formation of hierarchical structure in the social networks for p>p_(c).Here,p_(c) is related to the complex network size N and the minimal coordination number 2K.The model also reproduces many features of large social networks,including the“weak links”property.

Mathematical Models of Self-Appraisal in Social Networks

In social networks where individuals discuss opinions on a sequence of topics,the selfconfidence an individual exercises in relation to one topic,as measured by the weighting given to their own opinion as against the opinion of all others,can vary in the light of the self-appraisal by the individual of their contribution to the previous topic.This observation gives rise to a type of model termed a De Groot-Friedkin model.This paper reviews a number of results concerning this model.These include the asymptotic behavior of the self-confidence(as the number of topics goes to infinity),the possible emergence of an autocrat or small cohort of leaders,the effect of changes in the weighting given to opinions of others(in the light for example of their perceived expertise in relation to a particular topic under discussion),and the inclusion in the model of individual behavioral characteristics such as humility,arrogance,etc.Such behavioral characteristics create new opportunities for autocrats to emerge.

Innovation Governs Everything Eventually:Extensions of the DeGroot Model

We study the DeGroot model for continuous opinion dynamics under the influence of innovations. In the original model, individuals＇ opinions, after given their initial values, evolve merely according to the given learning topology. The main contribution of this paper is that external innovation effects are introduced： each individual is given the opportunity to change her opinion to a randomly selected opinion according to a given distribution on the opinion space and then the external opinion is either adapted by the individual, or combined into her learning process. It turns out that all the classical results of the DeGroot model are violated in this new model. We prove that convergence can still be guaranteed in the expectation sense, regardless of the learning topology. We also study the steady distributions of opinions among the society and the time spent to reach a steady state by means of Monte-Carlo simulations.

A COUPLED DYNAMIC MODEL OF BRAND ACCEPTANCE AND PROMOTIVE INFORMATION SPREADING

When people try to decide to buy or not to, they are often influenced by both their inherentopinions and the social marketing activities e.g. advertising, social news with strong point of view.Then people will make their final choice, or even convince other people to buy. After all, this is thebrand acceptance formation process. Factually, the dynamics of brand acceptance is essentially aninterwoven dynamics of endogenous opinion dynamics disturbed by an information diffusion process.To have a better understanding of the dynamics of brand acceptance, we propose and analyze a coupledagent-based dynamic model that combines the Majority-Rule-based Voter model in opinion dynamicswith the SI Model for information spreading to analyze the dynamics of brand acceptance in socialmedia. We focus on two important parameters in diffusion dynamics： the decayed transmission rate （fl）and the diffusion frequency （f）. When the system is stable, the order parameter of the system is theduration time （r）. In the absence of opinion interaction, the simulation results indicate that, when abrand tries to occupy a larger market share through social marketing approaches, it is always effectiveto let the opponent to be the propaganda target. While with the Majority-Rule-based Voter Modelincluded, we observe that the opinion interaction could have a dual function, which shows that a brandholding a small market share in the first place needs to adopt diverse marketing approaches accordingto different marketing environment types.

Free Information Flow Benefits Truth Seeking

How can we approach the truth in a society？ It may depend on various factors. In this paper, using a well-established truth seeking model, the authors show that the persistent free information flow will bring us to the truth. Here the free information flow is modeled as the environmental random noise that could alter one＇s cognition. Without the random noise, the model predicts that the truth can only be captured by the truth seekers who own actively perceptive ability of the truth and their believers, while the other individuals may stick to falsehood. But under the influence of the random noise, the authors strictly prove that even there is only one truth seeker in the group, all individuals will finally approach the truth.