PageRank model of opinion formation on Ulam networks

We consider a PageRank model of opinion formation on Ulam networks, generated by the intermittency map and the typical Chirikov map. The Ulam networks generated by these maps have certain similarities with such scale-free networks as the World Wide Web (WWW), showing an algebraic decay of the PageRank probability. We find that the opinion formation process on Ulam networks has certain similarities but also distinct features comparing to the WWW. We attribute these distinctions to internal differences in network structure of the Ulam and WWW networks. We also analyze the process of opinion formation in the frame of generalized Sznajd model which protects opinion of small communities.     

An evolutionary game theory model of binary opinion formation

A basic characteristic of most opinion models is that people tend to agree or compromise in the opinion interaction, which could be hopefully described by cooperative games in the evolutionary game theory framework. This paper presents game theory methods to model the formation of binary opinions: cooperative games are proposed to model the interaction rules of general people who tend to find an agreement; minority games are proposed to model the behaviors of contrarians; opinion preference is considered by varying the payoff values. The Majority Voter model could be restored from the proposed games. The game theory models show evolutionary results similar to traditional opinion models. Specially, the evolution of opinions with consideration of contrarians is in accordance with the Galam model. Furthermore, influences of evolving rule, network topology and initial distribution of opinions are studied through numerical simulations. Discussions about methods to promote or hinder the consensus state at the best equilibrium point are given.     

Ising model on evolution networks and its application on opinion formation

Many phenomena show that in a favorable circumstance an agent still has an updating possibility, and in an unfavor- able circumstance an agent also has a possibility of holding its own state and reselecting its neighbors. To describe this kind of phenomena an Ising model on evolution networks was presented and used for consensus formation and separation of opinion groups in human population. In this model the state-holding probability p and selection-rewiring probability q were introduced. The influence of this mixed dynamics of spin flips and network rewiring on the ordering behavior of the model was investigated, p hinders ordering of opinion networks and q accelerates the dynamical process of networks. Influence of q on the ordering and separating stems from its effect on average path length of networks.     

An opinion formation model with dissipative structure

We propose an opinion formation model which includes both the influence of internal motivation and dissipation. Active agents adopt their neighbors’ opinions according to a simplified set of rules, but they may gradually lose their interests in the discussion and drop out of it. On the other hand, inert agents can become active by initial activation or internal motivation from neighbors, and participate in the discussion. The initial activation is usually due to occurrence of a social event. The internal motivation, opinion update and dissipation procedure take place simultaneously. We apply the voter rule in our model, and carry out analysis and numerical simulations. Results show if nonzero dissipation stays below a threshold value, the system evolves to a balance state where the average concentration of one opinion is equal to that of the other. With dissipation, a number of small-size opinion clusters exist in the end, but the system gains a fast relaxation rate. The final average opinion is closely related to the dissipation intensity and the length of time for initial activation.     

Dynamics of bounded confidence opinion in heterogeneous social networks: Concord against partial antagonism

Bounded confidence models of opinion dynamics in social networks have been actively studied in recent years, in particular, opinion formation and extremism propagation along with other aspects of social dynamics. In this work, after an analysis of limitations of the Deffuant-Weisbuch (DW) bounded confidence, relative agreement model, we propose the mixed model that takes into account two psychological types of individuals. Concord agents (C-agents) are friendly people; they interact in a way that their opinions always get closer. Agents of the other psychological type show partial antagonism in their interaction (PA-agents). Opinion dynamics in heterogeneous social groups, consisting of agents of the two types, was studied on different social networks: Erdös-Rényi random graphs, small-world networks and complete graphs. Limit cases of the mixed model, pure C- and PA-societies, were also studied. We found that group opinion formation is, qualitatively, almost independent of the topology of networks used in this work. Opinion fragmentation, polarization and consensus are observed in the mixed model at different proportions of PA- and C-agents, depending on the value of initial opinion tolerance of agents. As for the opinion formation and arising of “dissidents”, the opinion dynamics of the C-agents society was found to be similar to that of the DW model, except for the rate of opinion convergence. Nevertheless, mixed societies showed dynamics and bifurcation patterns notably different to those of the DW model. The influence of biased initial conditions over opinion formation in heterogeneous social groups was also studied versus the initial value of opinion uncertainty, varying the proportion of the PA- to C-agents. Bifurcation diagrams showed an impressive evolution of collective opinion, in particular, radical changes of left to right consensus or vice versa at an opinion uncertainty value equal to 0.7 in the model with the PA/C mixture of population near 50/50.     

PageRank centrality for temporal networks

In this paper, we propose a new centrality measure for ranking the nodes and time layers of temporal networks simultaneously, referred to as the f-PageRank centrality. The f-PageRank values of nodes and time layers in temporal networks are obtained by solving the eigenvector of a multi-homogeneous map. The existence and uniqueness of the proposed centrality measure are also guaranteed by existing results, under some reasonable conditions. The numerical experiments on a synthetic temporal network and two real-world temporal networks (i.e., Email-Eu-core and CollegeMsg temporal networks) show that the proposed centrality outperforms some existing centrality measures.     

Continuous opinion model in small-world directed networks

In the compromise model of continuous opinions proposed by Deffuant et al., the states of two agents in a network can start to converge if they are neighbors and if their opinions are sufficiently close to each other, below a given threshold of tolerance ?. In directed networks, if agent i is a neighbor of agent j,j need not be a neighbor of i. In Watts-Strogatz networks we performed simulations to find the averaged number of final opinions 〈F〉 and their distribution as a function of ? and of the network structural disorder. In directed networks 〈F〉 exhibits a rich structure, being larger than in undirected networks for higher values of ?, and smaller for lower values of ?.     

Fairness in opinion dynamics

The opinion dynamics studies how a final consensus emerges from a diversified initial configuration. The final result can be an artifact arising in the voting processes and overlook the features of the initial configuration, which leads to the unfair result. We explore the concept of fairness in opinion dynamics and propose a quantitative measurement in a model system, which allows us to reach a final consensus reflecting impartially the major opinion. For a two-choice system, the unfairness alternates with the increase of meeting agents M. With an odd M, the fair results can be expected; with an even M, the unfairness decays monotonically with an increasing M. When the number of choices is larger than two, such an alternating is smeared out. The fairness at an odd M can no longer be reached. The unfairness increases a bit with the increasing number of choices. Similar M-dependence can be observed for different number of choices. We conclude that the number of choices plays a minor role in reaching a fair final consensus. The fairness is mainly controlled by the meeting size.     

External activation promoting consensus formation in the opinion model with interest decay

We put forward an opinion model that considers internal decay and external activation or deactivation. Agents may withdraw from the discussion, meanwhile, these inactive agents are likely to be motivated by active neighbors. In addition, external influence from outside circumstances is added to the population. We focus on the majority rule of opinion exchange. Our investigations reveal under the impact of external circumstances, the system evolves to different stable states. One opinion can finally be made dominant when the internal motivation is large sufficiently. However, without external activation, consensus is hardly reached in the system with interest decay.     

Social power and opinion formation in complex networks

In this paper we investigate the effects of social power on the evolution of opinions in model networks as well as in a number of real social networks. A continuous opinion formation model is considered and the analysis is performed through numerical simulation. Social power is given to a proportion of agents selected either randomly or based on their degrees. As artificial network structures, we consider scale-free networks constructed through preferential attachment and Watts–Strogatz networks. Numerical simulations show that scale-free networks with degree-based social power on the hub nodes have an optimal case where the largest number of the nodes reaches a consensus. However, given power to a random selection of nodes could not improve consensus properties. Introducing social power in Watts–Strogatz networks could not significantly change the consensus profile.     

Stochastic resonance in a model of opinion formation on small-world networks

We analyze the phenomenon of stochastic resonance in an Ising-like system on a small-world network. The system, which is subject to the combined action of noise and an external modulation, can be interpreted as a stylized model of opinion formation by imitation under the effects of a “fashion wave”. Both the amplitude threshold for the detection of the external modulation and the width of the stochastic-resonance peak show considerable variation as the randomness of the underlying small-world network is changed. Received 19 December 2001     

具有空间因素的社会网络上的舆论形成

舆论形成是个体因素与外部影响共同作用的结果. 个体因素包括个体理性的认知与非理性的情绪因素, 外部影响则包括文化、认知以及行为模式的作用和社会网络上个体之间的影响, 而文化、认知等背景因素往往与空间地域相关联. 本文通过构建一个同时具有空间位置特征与小世界特征的无标度网络以及网络上的Ising模型, 全面考察了空间与网络共同作用对舆论形成的影响, 研究了舆论形成中的相变行为, 重点分析了与区域文化或认知背景以及与整体社会情绪等相关的参数在舆论形成过程中的作用. 计算机数值模拟结果表明, 区域间的认知背景差异足够大时, 会导致公共舆论形成明显的区域性特征, 此时, 空间因素和社会网络具有同样的影响强度时, 会有利于全局一致舆论的形成. 同时, 非理性因素引发的社会情绪可以通过提高个体影响力及降低背景差异使得一致性公共舆论快速形成.     

Effects of attachment preferences on coevolution of opinions and networks

In the coevolution of network structures and opinion formation, we investigate the effects of a mixed population with distinctive relinking preferences on both the convergence time and the network structures. It has been found that a heterogeneous network structure is easier to be reached with more high-degree-preferential (HDP) nodes. There exists high correlation between the convergence time and the network heterogeneity. The heterogeneous degree distribution caused by preferential attachment accelerates the convergence to a consensus state and the shortened convergence time inhibits the occurrence of the following disquieting situation that occurs in a continuously evolving network: with preferential attachment and long-time evolvement, most of the nodes would become separated and only a few leaders would have immediate neighbors. Analytical calculations based on mean field theory reveal that both the transition point p_tr and the consensus time τ depend upon the standard deviation of the degree distribution σ_d. p_tr increases while τ decreases with the rise of σ_d. Functions of p_tr=〈k〉/(〈k〉+1) and are found. Theoretical analyses are in accordance with simulation data.     

An opinion diffusion model with clustered early adopters

We propose an opinion dynamic model which studies the adoption process of new opinions or ideas by agents. The proposed model allows the observation range of an agent to be expanded in square lattices. The agent’s opinion update process is not only influenced by a neighbor’s choice but also by the whole environment that can be observed. The model shows a different result with the normal CODA model: if adopters’ initial opinions equal 0.6 and individual observation probability α$\alpha$ equals 0.65, then diffusions with clustered early adopters are 3% faster than those with randomly scattered ones. Introducing the bounded confidence concept into our model leads to appearance of freezing effect in opinion dynamics.     

Effective vaccination strategies for realistic social networks

We consider the effectiveness of targeted vaccination at preventing the spread of infectious disease in a realistic social network. We compare vaccination strategies based on no information (random vaccination) to complete information (PageRank) about the network. The most effective strategy we find is to vaccinate those people with the most unvaccinated contacts. However, this strategy requires considerable information and computational effort which may not be practical. The next best strategies vaccinate people with many contacts who in turn have few contacts.     

A model of the effects of authority on consensus formation in adaptive networks: Impact on network topology and robustness

Opinions of individuals in real social networks are arguably strongly influenced by external determinants, such as the opinions of those perceived to have the highest levels of authority. In order to model this, we have extended an existing model of consensus formation in an adaptive network by the introduction of a parameter representing each agent’s level of ‘authority’, based on their opinion relative to the overall opinion distribution. We found that introducing this model, along with a randomly varying opinion convergence factor, significantly impacts the final state of converged opinions and the number of interactions required to reach that state. We also determined the relationship between initial and final network topologies for this model, and whether the final topology is robust to node removals. Our results indicate firstly that the process of consensus formation with a model of authority consistently transforms the network from an arbitrary initial topology to one with distinct measurements in mean shortest path, clustering coefficient, and degree distribution. Secondly, we found that subsequent to the consensus formation process, the mean shortest path and clustering coefficient are less affected by both random and targeted node disconnection. Speculation on the relevance of these results to real world applications is provided.     

Interplay between social debate and propaganda in an opinion formation model

We introduce a simple model of opinion dynamics in which a two-state agent modified Sznajd model evolves due to the simultaneous action of stochastic driving and a periodic signal. The stochastic effect mimics a social temperature, so the agents may adopt decisions in support for or against some opinion or position, according to a modified Sznajd rule with a varying probability. The external force represents a simplified picture by which society feels the influence of the external effects of propaganda. By means of Monte Carlo simulations we have shown the dynamical interplay between the social condition or mood and the external influence, finding a stochastic resonance-like phenomenon when we depict the noise-to-signal ratio as a function of the social temperature. In addition, we have also studied the effects of the system size and the external signal strength on the opinion formation dynamics.     

Phase transitions and hysteresis in a cellular automata-based model of opinion formation

A particular case of a cellular automata-based model of two-state opinion formation in social groups with a strong leader is studied. We consider a 2D Euclidian geometry of social space and mutual interactions 1/r ⁿ . The model shows an interesting dynamics which can be analytically calculated. There are two stable states of the system: a cluster around the leader and unification. Unstable clusters may also appear. A variation in parameters such as the leader's strength or the social temperature can change the size of a cluster or, when they reach some critical values, make the system jump into another state. For a certain range of parameters the system exhibits bistability and hysteresis phenomena. We obtained explicit formulas for the cluster size, critical leader's strength, and critical social temperature. These analytical results are verified by computer simulations.     

How information influences an individual opinion evolution

We propose an opinion formation model which takes an individual’s opinion transition probability into consideration. In the model, each individual updates his/her opinion based on the probability of reception and acceptance. We describe the process of opinion evolution by using individual’s awareness _Wi$W_i$, message intensity _a0$a_0$ and message credibility _b0$b_0$. Results show that if a message’s view stays below 1 and its credibility takes a value around 2, it can win the trust of individuals with a high value of awareness (_Wi≥1$W_i\ge 1$), leading the public opinion to support the message’s own view within 100 steps and then decide the opinion formation. The final average opinion is closely related to the first message’s credibility and individual’s awareness.