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Journal ArticleDOI

Consensus and Polarization of Binary Opinions in Structurally Balanced Networks

TL;DR: By applying a fluid limit theorem for jump Markov processes, a system of differential equations for the density functions of opinions for large networks is derived and it is shown that the equilibrium points corresponding to consensus and polarization are the only stable equilibrium points.
Abstract: In this paper, we propose a new model for binary opinion dynamics in a (fully connected) structurally balanced network. In a structurally balanced network, agents are classified into two clusters and two agents in the same cluster (resp. different clusters) are connected with a positive (resp. negative) edge. Initially, every agent is assigned with one of the two opinions randomly. In every time slot, three agents are randomly selected to have their opinions updated. If the three agents belong to the same cluster, the majority rule (MR) is used to update their opinions. On the other hand, if the three agents belong to two different clusters, with probability $p$ , a consensus is reached by the MR, and with probability $1-p$ , a polarization (in line with the signs of the three edges) is reached. The probability $p$ , called the rationality probability, plays a significant role for measuring how rational the agents in a network behave when they encounter different opinions. By applying a fluid limit theorem for jump Markov processes, we derive a system of differential equations for the density functions of opinions for large networks. We show that the equilibrium points corresponding to consensus and polarization are the only stable equilibrium points. All other equilibrium points are all unstable. As such, as time goes on, the network eventually reaches a consensus or a polarization, depending on the rationality probability and the initial state of the network.
Citations
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Journal ArticleDOI
TL;DR: A review of the framework and formulation of opinion dynamics as well as some basic models, extensions, and applications are presented and several open problems are proposed for future research.

253 citations

Journal ArticleDOI
TL;DR: In this article, a modified Hegselmann-Krause model with heterogeneous population is investigated, where agents in different/same subpopulation have different bounded confidence and stubbornness, and it is shown that increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster.
Abstract: In opinion dynamics with continuous opinion, bounded confidence is a critical parameter. Agents can interact with each other only when the opinion difference between them is less than the bounded confidence. Larger bounded confidence always leads to fewer opinion clusters. Stubbornness characterizing the insistence of an agent on her own opinion is thought to only affect the transition time. In this work, a modified Hegselmann–Krause model with heterogeneous population is investigated, where agents in different/same subpopulation have different/same bounded confidence and stubbornness. We find that, due to the interaction among subpopulations, increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster. We also find that increasing the bounded confidence of a subpopulation leads to fewer clusters and a larger largest cluster provided that all the others have large bounded confidence. While one subpopulation is with a small bounded confidence, there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the largest size of the largest cluster.

41 citations

Journal ArticleDOI
TL;DR: A model of pair and triadic interactions of fully connected agents, where each agent has G underlying attributes, and the similarity between agents in attribute space is used to determine the link weight between them, reveals that many real-world social systems would have a subcritical number of attributes necessary to achieve structural balance.
Abstract: Homophily between agents and structural balance in connected triads of agents are complementary mechanisms thought to shape social groups leading to, for instance, consensus or polarization. To capture both processes in a unified manner, we propose a model of pair and triadic interactions. We consider $N$ fully connected agents, where each agent has $G$ underlying attributes, and the similarity between agents in attribute space (i.e., homophily) is used to determine the link weight between them. For structural balance we use a triad-updating rule where only one attribute of one agent is changed intentionally in each update, but this also leads to accidental changes in link weights and even link polarities. The link weight dynamics in the limit of large $G$ is described by a Fokker-Planck equation from which the conditions for a phase transition to a fully balanced state with all links positive can be obtained. This ``paradise state'' of global cooperation is, however, difficult to achieve requiring $GgO({N}^{2})$ and $pg0.5$, where the parameter $p$ captures a willingness for consensus. Allowing edge weights to be a consequence of attributes naturally captures homophily and reveals that many real-world social systems would have a subcritical number of attributes necessary to achieve structural balance.

32 citations

Journal ArticleDOI
TL;DR: This work presents proof of credit (PoC), a fair blockchain protocol based on the PoS blockchain protocol, and explains the method by which the PoC protocol resists the double-spending attacks and the selfish mining attacks.
Abstract: Proof of work and proof of stake (PoS) are commonly used in the current permissionless blockchains. These consensus protocols can be abstracted into a random process of selecting a node for accounting in a blockchain ledger. However, they are generally faced with resource consumption and vulnerability issues. We present proof of credit (PoC), a fair blockchain protocol based on the PoC blockchain protocol. It is a special PoS protocol where the credit is a special kind of stake quantifying whether the node’s activity is beneficial to the system. Any nodes cannot change their credits arbitrarily. We demonstrate that our PoC protocol satisfies the security properties, including common prefix, chain quality, and chain growth, under the assumption that the total credit the honest held is majority. In addition, we propose a self-audit mechanism and a hybrid incentive mechanism to enhance the security and stability. Finally, we explain the method by which the PoC protocol resists the double-spending attacks and the selfish mining attacks.

28 citations


Cites background from "Consensus and Polarization of Binar..."

  • ...Consensus agreement is the core of the blockchain protocol [8], [9]....

    [...]

Journal ArticleDOI
TL;DR: Experiments show that OOD is superior to the improvement of the original Hegselmann–Krause (HK) model, HK-13 and HK-17, with respect to qualitative predictions of the evolution trend of an event.
Abstract: Predicting and utilizing the evolution trend of hot topics is critical for contingency management and decision-making purposes of government bodies and enterprises. This paper proposes a model named online opinion dynamics (OODs) where any node in a social network has its unique confidence threshold and influence radius. The nodes in the OOD are mainly affected by their neighbors and are also randomly influenced by unfamiliar nodes. In the traditional opinion model, however, each node is affected by all other nodes, including its friends. Furthermore, many traditional opinion evolution approaches are reviewed to see if all nodes (participants) can eventually reach a consensus. On the contrary, OOD is more focused on such details as concluding the overall trend of events and evaluating the support level of each participant through numerical simulation. Experiments show that OOD is superior to the improvement of the original Hegselmann–Krause (HK) model, HK-13 and HK-17, with respect to qualitative predictions of the evolution trend of an event. The quantitative predictions of the HK model cannot be used to make decisions, whereas the results of the OOD model are proved to be acceptable.

27 citations

References
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Journal ArticleDOI
TL;DR: In this article, a wide list of topics ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading are reviewed and connections between these problems and other, more traditional, topics of statistical physics are highlighted.
Abstract: Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

3,840 citations


"Consensus and Polarization of Binar..." refers methods in this paper

  • ...[1] for more extensions of the MR model....

    [...]

Journal ArticleDOI
Fritz Heider1
TL;DR: A comparison of attitudes and cognitive Organization in the context of war and post-war Europe shows marked differences in the attitudes of men and women towards one another and towards Europe in general.
Abstract: (1946). Attitudes and Cognitive Organization. The Journal of Psychology: Vol. 21, No. 1, pp. 107-112.

3,204 citations


"Consensus and Polarization of Binar..." refers background in this paper

  • ...We refer the reader to [9]–[11] for references on structurally balanced networks....

    [...]

Book
19 Jul 2010
TL;DR: In this article, an introductory undergraduate textbook takes an interdisciplinary look at economics, sociology, computing and information science, and applied mathematics to understand networks and behavior, addressing fundamental questions about how the social, economic, and technological worlds are connected.
Abstract: Over the past decade there has been a growing public fascination with the complex connectedness of modern society. This connectedness is found in many incarnations: in the rapid growth of the Internet, in the ease with which global communication takes place, and in the ability of news and information as well as epidemics and financial crises to spread with surprising speed and intensity. These are phenomena that involve networks, incentives, and the aggregate behavior of groups of people; they are based on the links that connect us and the ways in which our decisions can have subtle consequences for others. This introductory undergraduate textbook takes an interdisciplinary look at economics, sociology, computing and information science, and applied mathematics to understand networks and behavior. It describes the emerging field of study that is growing at the interface of these areas, addressing fundamental questions about how the social, economic, and technological worlds are connected.

2,650 citations

Book
16 Jun 2006
TL;DR: McCarty et al. as mentioned in this paper examined the relationship of polarization, wealth disparity, immigration, and other forces, characterizing it as a dance of give and take and back and forth causality.
Abstract: The idea of America as politically polarized--that there is an unbridgeable divide between right and left, red and blue states--has become a cliche. What commentators miss, however, is that increasing polarization in recent decades has been closely accompanied by fundamental social and economic changes--most notably, a parallel rise in income inequality. In Polarized America, Nolan McCarty, Keith Poole, and Howard Rosenthal examine the relationships of polarization, wealth disparity, immigration, and other forces, characterizing it as a dance of give and take and back and forth causality.Using NOMINATE (a quantitative procedure that, like interest group ratings, scores politicians on the basis of their roll call voting records) to measure polarization in Congress and public opinion, census data and Federal Election Commission finance records to measure polarization among the public, the authors find that polarization and income inequality fell in tandem from 1913 to 1957 and rose together dramatically from 1977 on; they trace a parallel rise in immigration beginning in the 1970s. They show that Republicans have moved right, away from redistributive policies that would reduce income inequality. Immigration, meanwhile, has facilitated the move to the right: non-citizens, a larger share of the population and disproportionately poor, cannot vote; thus there is less political pressure from the bottom for redistribution than there is from the top against it. In "the choreography of American politics" inequality feeds directly into political polarization, and polarization in turn creates policies that further increase inequality.

1,894 citations