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Recent advances in opinion propagation dynamics: A 2020 Survey
Hossein Noorazar
TL;DR
Opinion dynamics research needs models that explain how local interactions produce system-level outcomes relevant to decisions, markets, networks, and online information spread. This survey synthesizes established models and recent extensions, including stubbornness, bias, repulsion, power evolution, noise, and private opinions. It concludes by identifying new questions while noting inconsistent terminology and limits of existing interaction assumptions.
Problem
The field lacks convergent terminology and still needs improved models of human and technological interaction dynamics relevant to online misinformation and other applications.
Method
The paper surveys well-known continuous- and discrete-opinion models and reviews extensions incorporating stubbornness, bias, repulsive behavior, power evolution, interrelated topics, noise, and expressed and private opinions.
Results
The survey covers DeGroot, FJ, DW, HK, Galam, Sznajd, and voter models, together with extensions and newly posed questions for future exploration.
Takeaways & Limitations
Opinion dynamics provides models for studying consensus, polarization, fragmentation, opinion manipulation, and potential online applications such as fake-news detection.
Takeaways & Limitations
The literature has no convergent terminology, and existing models do not fully represent asynchronous human interaction or always-valid assumptions about repulsive behavior.
Abstract
from arXiv · showhide
Opinion dynamics have attracted the interest of researchers from different fields. Local interactions among individuals create interesting dynamics for the system as a whole. Such dynamics are important from a variety of perspectives. Group decision making, successful marketing, and constructing networks (in which consensus can be reached or prevented) are a few examples of existing or potential applications. The invention of the Internet has made the opinion fusion faster, unilateral, and on a whole different scale. Spread of fake news, propaganda, and election interferences have made it clear there is an essential need to know more about these dynamics. The emergence of new ideas in the field has accelerated over the last few years. In the first quarter of 2020, at least 50 research papers have emerged, either peer-reviewed and published or on pre-print outlets such as arXiv. In this paper, we summarize these ground-breaking ideas and their fascinating extensions and introduce newly surfaced concepts.
1 Introduction
Opinion dynamics examines how networked agents propagate and update opinions through local interactions. This survey introduces core concepts and newly developed modeling ideas across established and emerging applications.
- Opinion dynamics studies opinion propagation in networks through interactions among agents.
- The field includes foundational work from Asch, French, Abelson, DeGroot, Lehrer, and Latané.
- Models represent opinions as binary or continuous variables depending on the topic and interaction purpose.
- Agents begin with initial opinions, interact through an update rule, and continue until a termination criterion such as steady state.
- The survey introduces major concepts and newly developed ideas without pursuing exhaustive detail.
2 Preliminaries
The paper formalizes opinion networks, opinion spaces, agent states, and equilibrium outcomes. It also notes that terminology varies across the literature and establishes definitions for stubbornness and closed-mindedness.
- The agent network G contains N agents, with adjacency matrix A and row-stochastic influence matrix W describing connections and influence levels.
- Opinion space O contains possible opinions such as {0, 1}, {1, 2, . . . m}, or [0, 1].
- At equilibrium, opinion dynamics may converge to consensus, polarization with two clusters, or fragmentation with more than two clusters.
- The literature lacks convergent terminology for agents that retain opinions, using labels such as leader, media, stubborn, or inflexible agent.
- Fully-stubborn agents never change opinions, partially-stubborn agents retain initial opinions, and closed-minded agents have smaller confidence radii.
3 Milestones
The survey reviews continuous and discrete opinion models, beginning with DeGrootian and bounded-confidence frameworks before covering discrete models. It connects these models to convergence, consensus, cluster formation, and extensions studied across varied network settings.
- 3 Milestones: The survey organizes foundational models by opinion space, covering continuous models first and discrete models afterward.
- 3.1.1 DeGrootian models: The DeGroot model iteratively averages opinions through a row-stochastic matrix, with connected networks converging to consensus under a stated matrix condition.
- 3.1.1 DeGrootian models: The FJ model extends DeGroot by adding stubborn agents whose susceptibility determines how much they respond to influence.
- 3.1.2 Bounded confidence models: Bounded-confidence models make agents ignore opinions that are too distant; DW is pairwise, whereas HK is synchronous.
- 3.1.2 Bounded confidence models: The DW model updates interacting agents only within a confidence radius, with learning rate µ usually in (0, 0.5] to avoid crossover.
- 3.1.2 Bounded confidence models: r = 0.5 is reported as the critical homogeneous DW confidence radius above which agents reach consensus, while equilibrium clusters are approximately 1/2r.
- 3 Milestones: The survey also reviews multidimensional HK, binary Galam dynamics, Sznajd dynamics on lattices, and voter-model variants across network topologies.
4 Milestones’ extensions
The survey reviews extensions that add stubbornness or opinion bias to established models, including variants spanning DeGroot, DW, HK, Galam, and voter dynamics.
- Extensions: The survey presents these developments as extensions of established opinion-dynamics models rather than as a single unified framework.The reviewed variants alter assumptions about persistence, interaction probabilities, and social similarity.
- Stubborn agents: Stubborn-agent extensions examine fixed, partially fixed, media, leader, and contrarian influences across DeGroot, DW, HK, Galam, and voter models.Several studies analyze how these agents affect influence, convergence, consensus, and minority-majority outcomes.
- Biased agents: Biased-agent models encode homophily by making agents more likely to interact with others holding similar opinions.Extensions modify DeGroot, DW, and HK dynamics, including social similarity based on attributes such as age and education.
4.3 Opinion manipulation
Opinion manipulation research considers how network structure, media, strategic agents, stubborn agents, and competing controllers can move or prevent movement toward particular outcomes.
- Goals of manipulation: Manipulation studies seek either to drive networks toward consensus or predetermined opinions, or to prevent consensus.The FJ model is identified as lacking prior consensus-directed manipulation, while recent work studies adversarial prevention of consensus.
- Model-specific interventions: Interventions include adding a minimum number of network edges, using media accounts, deploying strategic or stubborn agents, and selecting influence-maximization strategies.These approaches are studied in DeGroot, DW, HK, and voter models.
- Media influence: Media-based influence can be tuned through the number of media accounts, their followers, content quality, and heterogeneous confidence radii.The cited studies examine conditions under which media impact is maximized.
- Competing forces: Competing controllers may face budget constraints, with Nash control strategies determining how each seeks to maximize its followers.Other work models influencers rewiring networks to maximize their impact.
- Power evolution: DeGroot consensus on a connected network is a weighted average of initial opinions, motivating models that study how social power evolves.These final-state properties provide the context for power-evolution extensions.
- Power evolution: Power evolution links sequential opinion discussions to changing influence weights, with later topics governed by outcomes of earlier topics.The DeGroot–Friedkin model and related extensions study social power, centrality, equilibria, and democratic or autocratic structures.
- Power evolution: In the FJ model, power-evolution research examines equilibria and democracy, and reports that autocracy cannot be achieved when stubborn agents are present.Stubbornness is treated as a source of social power in FJ and DeGroot models.
4.5 Repulsive behavior
Repulsive interactions extend opinion models beyond attraction and indifference, allowing polarization, fragmentation, and antagonistic relationship structures.
- Motivation: Sensitive topics can produce repulsive behavior, which the survey associates with polarization or fragmentation.Earlier models discussed in the survey support attraction or indifference only.
- DeGroot model: A repulsive DeGroot extension uses one entrenchment parameter to capture both bias and backfire effects and supports polarization.Polarization was not supported by the original DeGroot model.
- DW model: A potential-function formulation can recover the DW model or the Jager–Amblard model, depending on the potential selected.This formulation is based on minimizing interaction energy between agents.
- DW model: With confidence radius r = 0.3, different potential functions support attraction and indifference, or add repulsion to modified DW dynamics.The three behaviors are explicitly illustrated through the surveyed potential functions.
- Signed graphs: Signed-graph and balance-theory models represent friendship and antagonism through positive and negative edge signs, respectively.Related HK extensions implement attraction, indifference, and repulsion and can lead to consensus, bipartite consensus, and fragmentation.
4.6 Noisy models
The survey covers noise as a way to model behavioral variability, uncertainty, population change, and external influences, while also introducing models of interrelated topics.
- Noise models: Noise is used to represent uniqueness tendencies, agent birth or death, internal thoughts, external sources, and uncertainty in opinions.Examples include interval-valued opinions and noisy interaction probabilities.
- DW model: In DW dynamics, interaction noise replaces a sharp confidence threshold with a probability dependent on opinion difference.Another noise mechanism models an agent changing its opinion as a proxy for death and birth.
- Other models: Noise studies in HK, Galam, and voter models examine consensus formation, environmental effects, phase transitions, convergence time, contrarians, and zealots.The surveyed work spans homogeneous and heterogeneous HK systems, mobile-agent Galam dynamics, and complex-network voter models.
- Sznajd model: Appropriate random noise is predicted to increase consensus in a modified synchronous Sznajd model.The result is stated as depending on the amount of noise.
- Interrelated topics: Opinion changes on one topic can affect another, but interrelated-topic models remain limited in the surveyed literature.Examples include health affecting exercise and diet.
- Interrelated topics: Interrelated-topic models include multidimensional and sequentially dependent FJ dynamics, pairwise bounded-confidence DW interactions, and coupled-topic opinion updates.In sequential dependence, topic s affects the dynamics of topic s + 1; in coupled models, discussing one topic can update another.
4.8 Expressed vs. private opinions
The survey examines models in which agents’ expressed opinions can differ from their private beliefs, including co-evolutionary and manipulative extensions.
- Social pressure can make expressed opinions diverge from agents’ internally held beliefs.The survey traces this distinction to work by Nowak and Latané and earlier work by Latané.
- Expressed vs. private opinions in FJ: The FJ model has been extended to study the co-evolution and convergence of expressed and private opinions.
- Expressed vs. private opinions in voter model: Several studies develop expressed–private opinion models within the voter model.
- More on EPO: Other proposed models examine private and expressed opinions without belonging to the field’s well-known model families.
- An extension inspired by the DW model uses informed agents who pretend their opinions resemble those of other agents.
5 Last words
The survey closes by highlighting additional opinion-dynamics models, convergence analyses, engineering applications, physical approaches, and evolving susceptibility to persuasion.
- Additional models include interactions triggered by large opinion differences, including social-pressure and beyond-confidence-interval interactions.
- Synchronous bounded-confidence models can update from randomly selected neighbors, placing them between DW pairwise interaction and synchronous HK interaction.
- Further work studies convergence under inertia, heterogeneous confidence radii, and other HK-model variations.
- Opinion dynamics have also been applied to anomaly detection and engineering voting processes.
- Physics-inspired research uses kinetic-theory and mean-field approaches while incorporating leaders and noise, with applications including economics.
- Other extensions model changing susceptibility to persuasion and use edge weights to represent interaction frequency.
6 New questions
The survey identifies open questions about realism, universality, and applications of opinion-dynamics models, especially where human behavior exceeds simplified assumptions.
- Human interactions do not generally involve all neighbors simultaneously, unlike the DeGroot model, and even computers face physical interaction limits.
- Balanced-graph assumptions for repulsive behavior rely on social principles that are not always true.
- Existing models are not universal because they target particular traits or deliberately create specific dynamics.
- The authors propose narrowing the gap between theoretical-model simplicity and the complexity of human behavior.
- Potential applications include detecting fake-news resources and identifying individuals susceptible to online terrorist recruitment.
- Future work could apply opinion dynamics to flag computers or processors sending erroneous or corrupted bus messages.
7 Conclusions
The paper reviews established opinion-dynamics models across continuous and discrete opinion spaces, surveys extensions with additional mechanisms, and poses questions for future exploration.
- The review covers continuous-opinion models including DeGroot, FJ, DW, and HK, alongside discrete Galam, Sznajd, and voter models.
- Selected extensions add stubbornness, bias, repulsive behavior, power evolution, interrelated topics, noise, and expressed and private opinions.
- The authors conclude by posing new questions for future explorations.