PageRank model of opinion formation on social networks

We propose the PageRank model of opinion formation and investigate its rich properties on real directed networks of the Universities of Cambridge and Oxford, LiveJournal, and Twitter. In this model, the opinion formation of linked electors is weighted with their PageRank probability. Such a probability is used by the Google search engine for ranking of web pages. We find that the society elite, corresponding to the top PageRank nodes, can impose its opinion on a significant fraction of the society. However, for a homogeneous distribution of two opinions, there exists a bistability range of opinions which depends on a conformist parameter characterizing the opinion formation. We find that the LiveJournal and Twitter networks have a stronger tendency to a totalitarian opinion formation than the university networks. We also analyze the Sznajd model generalized for scale-free networks with the weighted PageRank vote of electors.     

The Arnold cat map, the Ulam method and time reversal

We study the properties of the Arnold cat map on a torus with several periodic sections using the Ulam method. This approach generates a Markov chain with the Ulam matrix approximant. We study numerically the spectrum and eigenstates of this matrix showing their relation with the Fokker-Planck relaxation and the Kolmogorov-Sinai entropy. We show that, in the frame of the Ulam method, the time reversal property of the map is preserved only on a short Ulam time which grows only logarithmically with the matrix size. Parallels with the evolution in a regime of quantum chaos are also discussed.     

Non-consensus Opinion Models on Complex Networks

Social dynamic opinion models have been widely studied to understand how interactions among individuals cause opinions to evolve. Most opinion models that utilize spin interaction models usually produce a consensus steady state in which only one opinion exists. Because in reality different opinions usually coexist, we focus on non-consensus opinion models in which above a certain threshold two opinions coexist in a stable relationship. We revisit and extend the non-consensus opinion (NCO) model introduced by Shao et al. (Phys. Rev. Lett. 103:01870, 2009). The NCO model in random networks displays a second order phase transition that belongs to regular mean field percolation and is characterized by the appearance (above a certain threshold) of a large spanning cluster of the minority opinion. We generalize the NCO model by adding a weight factor W to each individual’s original opinion when determining their future opinion (NCOW model). We find that as W increases the minority opinion holders tend to form stable clusters with a smaller initial minority fraction than in the NCO model. We also revisit another non-consensus opinion model based on the NCO model, the inflexible contrarian opinion (ICO) model (Li et al. in Phys. Rev. E 84:066101, 2011), which introduces inflexible contrarians to model the competition between two opinions in a steady state. Inflexible contrarians are individuals that never change their original opinion but may influence the opinions of others. To place the inflexible contrarians in the ICO model we use two different strategies, random placement and one in which high-degree nodes are targeted. The inflexible contrarians effectively decrease the size of the largest rival-opinion cluster in both strategies, but the effect is more pronounced under the targeted method. All of the above models have previously been explored in terms of a single network, but human communities are usually interconnected, not isolated. Because opinions propagate not only within single networks but also between networks, and because the rules of opinion formation within a network may differ from those between networks, we study here the opinion dynamics in coupled networks. Each network represents a social group or community and the interdependent links joining individuals from different networks may be social ties that are unusually strong, e.g., married couples. We apply the non-consensus opinion (NCO) rule on each individual network and the global majority rule on interdependent pairs such that two interdependent agents with different opinions will, due to the influence of mass media, follow the majority opinion of the entire population. The opinion interactions within each network and the interdependent links across networks interlace periodically until a steady state is reached. We find that the interdependent links effectively force the system from a second order phase transition, which is characteristic of the NCO model on a single network, to a hybrid phase transition, i.e., a mix of second-order and abrupt jump-like transitions that ultimately becomes, as we increase the percentage of interdependent agents, a pure abrupt transition. We conclude that for the NCO model on coupled networks, interactions through interdependent links could push the non-consensus opinion model to a consensus opinion model, which mimics the reality that increased mass communication causes people to hold opinions that are increasingly similar. We also find that the effect of interdependent links is more pronounced in interdependent scale free networks than in interdependent Erd?s Rényi networks.     

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.     

Poincaré recurrences and Ulam method for the Chirikov standard map

We study numerically the statistics of Poincaré recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows us to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincaré exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincaré recurrences.     

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.     

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.     

Poisson statistics of PageRank probabilities of Twitter and Wikipedia networks

We use the methods of quantum chaos and Random Matrix Theory for analysis of statistical fluctuations of PageRank probabilities in directed networks. In this approach the effective energy levels are given by a logarithm of PageRank probability at a given node. After the standard energy level unfolding procedure we establish that the nearest spacing distribution of PageRank probabilities is described by the Poisson law typical for integrable quantum systems. Our studies are done for the Twitter network and three networks of Wikipedia editions in English, French and German. We argue that due to absence of level repulsion the PageRank order of nearby nodes can be easily interchanged. The obtained Poisson law implies that the nearby PageRank probabilities fluctuate as random independent variables.     

An almost linear stochastic map related to the particle system models of social sciences

We propose a stochastic map model of economic dynamics. In the past decade, an array of observations in economics has been investigated in the econophysics literature, a major example being the universal features of inequality in terms of income and wealth. Another area of enquiry is the formation of opinion in a society. The model proposed attempts to produce positively skewed distributions and power law distributions as has been observed in the real data of income and wealth. Also, it shows a non-trivial phase transition in the opinion of a society (opinion formation). A number of physical models also generate similar results. In particular, kinetic exchange models have been successful especially in this regard. Therefore, we compare the results obtained from these two approaches and discuss a number of new features and drawbacks of this model.     

Renormalizing Sznajd model on complex networks taking into account the effects of growth mechanisms

We present a renormalization approach to solve the Sznajd opinion formation model on complex networks. For the case of two opinions, we present an expression of the probability of reaching consensus for a given opinion as a function of the initial fraction of agents with that opinion. The calculations reproduce the sharp transition of the model on a fixed network, as well as the recently observed smooth function for the model when simulated on a growing complex networks.     

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 selfaffirmation,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-affirmation 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.     

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.     

The spreading of opposite opinions on online social networks with authoritative nodes

The study of opinion dynamics, such as spreading and controlling of rumors, has become an important issue on social networks. Numerous models have been devised to describe this process, including epidemic models and spin models, which mainly focus on how opinions spread and interact with each other, respectively. In this paper, we propose a model that combines the spreading stage and the interaction stage for opinions to illustrate the process of dispelling a rumor. Moreover, we set up authoritative nodes, which disseminate positive opinion to counterbalance the negative opinion prevailing on online social networking sites. With analysis of the relationship among positive opinion proportion, opinion strength and the density of authoritative nodes in networks with different topologies, we demonstrate that the positive opinion proportion grows with the density of authoritative nodes until the positive opinion prevails in the entire network. In particular, the relationship is linear in homogeneous topologies. Besides, it is also noteworthy that initial locations of the negative opinion source and authoritative nodes do not influence positive opinion proportion in homogeneous networks but have a significant impact on heterogeneous networks. The results are verified by numerical simulations and are helpful to understand the mechanism of two different opinions interacting with each other on online social networking sites.     

Sampling from complex networks with high community structures

In this paper, we propose a novel link-tracing sampling algorithm, based on the concepts from PageRank vectors, to sample from networks with high community structures. Our method has two phases; (1) Sampling the closest nodes to the initial nodes by approximating personalized PageRank vectors and (2) Jumping to a new community by using PageRank vectors and unknown neighbors. Empirical studies on several synthetic and real-world networks show that the proposed method improves the performance of network sampling compared to the popular link-based sampling methods in terms of accuracy and visited communities.     

Social bots and mass media manipulated public opinion through dual opinion climate

Opinion dynamics models based on the multi-agent method commonly assume that interactions between individuals in a social network result in changes in their opinions.However,formation of public opinion in a social network is a macroscopic statistical result of opinions of all expressive individuals(corresponding to silent individuals).Therefore,public opinion can be manipulated not only by changing individuals'opinions,but also by changing their states of expression(or silence)which can be interpreted as the phenomenon"spiral of silence"in social psychology.Based on this theory,we establish a"dual opinion climate"model,involving social bots and mass media through a multi-agent method,to describe mechanism for manipulation of public opinion in social networks.We find that both social bots(as local variables)and mass media(as a global variable)can interfere with the formation of public opinion,cause a significant superposition effect when they act in the same direction,and inhibit each other when they act in opposite directions.     

The map equation

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.     

The noisy Hegselmann-Krause model for opinion dynamics

In the model for continuous opinion dynamics introduced by Hegselmann and Krause, each individual moves to the average opinion of all individuals within an area of confidence. In this work we study the effects of noise in this system. With certain probability, individuals are given the opportunity to change spontaneously their opinion to another one selected randomly inside the opinion space with different rules. If the random jump does not occur, individuals interact through the Hegselmann-Krause’s rule. We analyze two cases, one where individuals can carry out opinion random jumps inside the whole opinion space, and other where they are allowed to perform jumps just inside a small interval centered around the current opinion. We found that these opinion random jumps change the model behavior inducing interesting phenomena. Using pattern formation techniques, we obtain approximate analytical results for critical conditions of opinion cluster formation. Finally, we compare the results of this work with the noisy version of the Deffuant et al. model [G. Deffuant, D. Neu, F. Amblard, G. Weisbuch, Adv. Compl. Syst. 3, 87 (2000)] for continuous-opinion dynamics.     

Attack Vulnerability of Network with Duplication-Divergence Mechanism

We study the attack vulnerability of network with duplication-divergence mechanism. Numerical results have shown that the duplication-divergence network with larger retention probability a is more robust against target attack relatively. Furthermore, duplication-divergence network is broken down more quickly than its counterpart BA network under target attack. Such result is consistent with the fact of WWW and Internet networks under target attack. So duplication-divergence model is a more realistic one for us to investigate the characteristics of the world wide web in future. We also observe that the exponent γ of degree distribution and average degree are important parameters of networks, reflecting the performance of networks under target attack. Our results are helpful to the research on the security of network.     

The strength of the minority

In this paper, we focus on the problem of opinion formation by introducing a simple model with a certain community structure. To understand the strength of the community, we took a particular interest in a special problem of how the opinion of a small but cohesive community could persist or even be finally accepted by the majority of the society. Both simulation and analysis has been done in the absence and presence of noise. In the noiseless environment, assuming the population of the community is fixed, if the cohesion of the community reaches a certain level, then the phase transition will occur in the evolution process that the community will never be assimilated even if it can assimilate the other nodes in the network, which depends on the population of the community. On the other hand, in the presence of noise, the process of opinion formation seems more complex that two transition behaviors occur outside and inside the community as the noise level increases. And the outcomes of the evolution may be completely opposite under different noise conditions.     

Community structure in the United Nations General Assembly

We study the community structure of networks representing voting on resolutions in the United Nations General Assembly. We construct networks from the voting records of the separate annual sessions between 1946 and 2008 in three different ways: (1) by considering voting similarities as weighted unipartite networks; (2) by considering voting similarities as weighted, signed unipartite networks; and (3) by examining signed bipartite networks in which countries are connected to resolutions. For each formulation, we detect communities by optimizing network modularity using an appropriate null model. We compare and contrast the results that we obtain for these three different network representations. We thereby illustrate the need to consider multiple resolution parameters and explore the effectiveness of each network representation for identifying voting groups amidst the large amount of agreement typical in General Assembly votes.