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Opinion dynamics: models, extensions and external effects

Alina Sîrbu, Vittorio Loreto, Vito D. P. Servedio, Francesca Tria

arXiv:1605.06326v1physics.soc-phcs.SI

TL;DR

Opinion formation is shaped by peer interactions and external information, but their roles remain difficult to clarify across diverse models. This review classifies recent opinion-dynamics methods by information source and opinion representation, finding that models under different assumptions often produce similar results consistent with observed social behaviours.

  • Problem

    Understanding how peer interactions and external information shape opinion formation matters because opinions drive behaviour and collective responses to social and environmental challenges.

  • Method

    The paper reviews recent opinion-dynamics models, classifying them by external information and by discrete or continuous one- or multidimensional opinion representations.

  • Results

    Across different assumptions, many reviewed models produce similar results that also agree with observed behaviours in social systems.

  • Takeaways & Limitations

    Opinion-dynamics models provide a framework for examining disagreement, peer interaction, and external information in social modelling.

  • Takeaways & Limitations

    Application of opinion-dynamics models to real data remains very scarce, despite increasingly available behavioural data.

Abstract

from arXiv · show

Recently, social phenomena have received a lot of attention not only from social scientists, but also from physicists, mathematicians and computer scientists, in the emerging interdisciplinary field of complex system science. Opinion dynamics is one of the processes studied, since opinions are the drivers of human behaviour, and play a crucial role in many global challenges that our complex world and societies are facing: global financial crises, global pandemics, growth of cities, urbanisation and migration patterns, and last but not least important, climate change and environmental sustainability and protection. Opinion formation is a complex process affected by the interplay of different elements, including the individual predisposition, the influence of positive and negative peer interaction (social networks playing a crucial role in this respect), the information each individual is exposed to, and many others. Several models inspired from those in use in physics have been developed to encompass many of these elements, and to allow for the identification of the mechanisms involved in the opinion formation process and the understanding of their role, with the practical aim of simulating opinion formation and spreading under various conditions. These modelling schemes range from binary simple models such as the voter model, to multi-dimensional continuous approaches. Here, we provide a review of recent methods, focusing on models employing both peer interaction and external information, and emphasising the role that less studied mechanisms, such as disagreement, has in driving the opinion dynamics. [...]

1 Introduction

The section frames opinion dynamics as a complex interdisciplinary process linking individual opinions, social interactions, external information, and collective human behaviour. It introduces a review that classifies models by external information and opinion representation, while noting that it is not exhaustive.

  • Motivation: Opinions are internal states that drive human actions, making their formation and evolution central to explaining human choices.Opinion formation is shaped by interacting forces, including social context and other influences.
  • Motivation: Individual actions collectively affect local environmental conditions and global challenges such as climate change and resource use.Understanding how citizens’ environmental awareness can be enhanced is therefore important for sustainability.
  • Modeling approach: Opinion dynamics models apply tools from physics, mathematics, and computer science to represent connected agents with discrete or continuous opinions and rules for opinion change.These models seek to explain social processes through interactions among a finite number of agents.
  • Review structure: The review classifies models according to whether they include external information, then further divides them by the effective form of opinion.Section 2 excludes external information, whereas Section 3 models it as an immutable agent participating in the dynamics.
  • Scope: The review is not intended to be exhaustive, despite efforts to include as many contributions as possible.This defines the scope limitation of the survey.

2 Existing models of opinion dynamics and extensions

The Ising model is an early, popular agent-based framework for opinion dynamics, representing binary opinions as spins shaped by peer interactions and external information. Because this abstraction can oversimplify individual positions and interactions, many alternative models and developments have since been designed.

  • Ising model: The Ising model adapts a physics model to opinion dynamics by treating individuals as agents that communicate and update opinions according to fixed rules.Agents may interact pairwise or in groups, with connections defined by an underlying graph topology.
  • Ising model: Each Ising-model agent holds one of two opinions, represented by an up or down spin corresponding to a choice between two options.Spin couplings encode peer interactions, while the magnetic field represents external information.
  • Extensions: The Ising model can be too simple to capture the complexity of individual positions and interactions, motivating many alternative models and recent developments.The section presents selected models and extensions from the last decade, building on earlier reviews.

2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model

The voter model describes binary-opinion agents updating through random peer copying, producing consensus in finite systems but dimension-dependent behavior in infinite systems. Its extensions show that network adaptation, nonlinear herding, zealots, popularity bias, additional opinions, and negative interactions can substantially alter consensus, disorder, and metastability.

  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: The voter model assigns each of N agents one of two opinions, s = ±1, and updates an agent by copying a randomly selected neighbour on an underlying graph.It is among the simplest opinion-dynamics models and originated in studies of species competition before being applied to electoral competition.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: In two-dimensional lattices, same-opinion domains grow while interfaces remain very rough, reflecting the absence of a direct majority rule and a lack of surface tension.Neighbour influence is felt through peer interaction rather than direct majority selection.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: Finite d-dimensional hyper-cubic systems reach one of two consensus states, with the selected outcome determined by the population’s initial state.The consensus states are all agents holding s = 1 or all holding s = −1.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: Temperature-dependent voter dynamics models financial-market opinion change by linking noise to agents’ nervousness and feedback from market imbalance.The system passes through long-lived striped configurations or shorter mean-field-like metastable states.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: Power-law intervals between interactions slow convergence compared with the original interaction timing, with the largest slowdown on rings and little or no difference on complete graphs.Regular random graphs show an intermediate effect.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: Adaptive similarity-based rewiring permits consensus in finite systems but allows infinitely persistent metastable states in infinite systems.Links between agents with different opinions are replaced by connections to agents sharing the same opinion.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: Nonlinear herding recovers the original voter model at α = 1, approaches majority-rule behavior for large α, and minimizes convergence time at moderate α.Very low α produces slowly forming large clusters, whereas very large α makes large clusters slow to merge.
  • 2.1 One-dimensional models · 2.1.1 Discrete opinions · The voter model: Extensions with contrarians, zealots, popularity bias, centrist opinions, strategic voting, negative interactions, and co-evolving networks generate coexistence, disorder, altered consensus, or metastability.Popularity bias yields consensus in T ∼[ln N]^2, while zealots can prevent consensus even in small proportions; strategic voting produces persistent two-party competition with a minority third party.

The majority rule (MR) model

The majority rule model updates randomly selected groups to their internal majority, with group size and tie-breaking bias shaping consensus. Extensions apply this framework to public debates, hypergraphs, coexistence of opinions, and tax-evasion dynamics.

  • The majority rule (MR) model: Agents with discrete opinions ±1 interact on a complete graph, and each randomly selected group adopts its internal majority opinion.The group size may be fixed or sampled from a specified distribution.
  • The majority rule (MR) model: For odd group sizes, the consensus threshold is pc(r) = 1/2; for even sizes, pc < 1/2 because the favored opinion can prevail from an initial minority.Consensus requires a number of updates per agent scaling like log N.
  • The majority rule (MR) model: In public-debate extensions, inflexible agents can drive outcomes when scientific data are insufficient, while collective beliefs strongly influence debate results.Acquiring inflexible agents is described as a winning strategy, including through overstated or exaggerated statements.
  • The majority rule (MR) model: On hypergraphs, spatial majority rule converges to a majority of +1 for even hyperedge size and clusters for odd size, even with infinitely many hyperedges.Ties result in adoption of opinion +1.
  • The majority rule (MR) model: Related extensions introduce self-opinion effects and can produce stable coexistence of the two competing opinions across several network types.The cited networks include coupled networks.
  • The majority rule (MR) model: In majority-vote tax-evasion models, agents respond probabilistically to neighbors’ average opinions and may be forced honest temporarily after punishment.The audit punishment occurs with probability p and lasts for k population updates; square-lattice, Barabási–Albert, and Honisch–Stauffer topologies are analyzed.

Social impact and the Sznajd model

The section presents social impact theory as a basis for opinion-formation models and describes the Sznajd model, in which agreeing groups influence neighbouring agents. It reviews extensions incorporating noise, disagreement, independence, reputation, additional opinions, and alternative network structures.

  • Social impact theory: Social impact theory models an individual’s susceptibility to a group through the group’s number, distance, and strength, using cellular automata for opinion dynamics.Distance is represented through a decreasing function, while individual strength is incorporated through a scaling function in the updating rule.
  • Social impact theory: The social-impact model produces spatially localized opinion clusters, including minority clusters supported by strong individuals across several network topologies.The reported topologies include fully connected, hierarchical, strongly diluted, and Euclidean networks.
  • Sznajd model: The one-dimensional Sznajd model assumes that neighbouring agents sharing an opinion influence their neighbours more strongly than a single agent, while disagreement prevents influence.The model accounts for proximity through lattice neighbours but does not include individual strength as a social-impact factor.
  • Sznajd extensions: Sznajd-model extensions add two-dimensional plaquette agreement, a centrist or indifferent option, and social temperature that probabilistically reverses the original updating rule.The social-temperature formulation applies the original rule with probability p and the opposite outcome with probability 1 −p, producing disagreement and potentially disordered states.
  • Sznajd extensions: Further variants study anti-conformist reactions, agent independence and flexibility, reputation-based influence, random-graph clustering, and competition with Voter dynamics.Independence favors coexistence of both opinions, while enhanced-clustering random graphs prevent full consensus; reputation dynamics can eliminate the phase transition for p < pc ∼0.69.

The q-voter model

The q-voter model generalizes discrete opinion dynamics through group influence and probabilistic flipping when sampled neighbours disagree. Its non-conformity and anti-conformity variants exhibit distinct phase-transition behaviour as group size q changes.

  • Model formulation: The q-voter model places N individuals with opinions ±1 on a fully connected network, where an agreeing group of q neighbours influences one randomly selected neighbour.If the group disagrees, the selected agent flips with probability ϵ; voter and Sznajd models are special cases of this model and its extensions.
  • Non-conformity and anti-conformity: Non-conformity makes some agents flip with probability p regardless of the group’s opinion, whereas anti-conformity makes them adopt the opposite opinion with probability p.These two dynamics appear similar but produce important differences.
  • Non-conformity and anti-conformity: For anti-conformism, the critical value pc for the order-disorder phase transition increases with q, while for non-conformism, pc decreases with q.The comparison directly distinguishes how group size affects the phase transition under the two dynamics.

Other approaches · 2.1.2 Continuous opinions · Deffuant-Weisbuch

The Deffuant-Weisbuch model represents opinions continuously and explains convergence into opinion clusters through bounded-confidence interactions. Its extensions show that initial conditions, noise, disagreement, heterogeneous confidence, network structure, and bias can substantially alter consensus, pluralism, and final-opinion distributions.

  • Other approaches: Binary opinions on interdependent party networks were analyzed by minimizing a Hamiltonian counting conflicting connections, and the most connected network wins the
  • 2.1.2 Continuous opinions: In the Deffuant-Weisbuch model, agents hold opinions x_i ∈ [−1, 1] and interact only when |x_i − x_j| < d, moving closer according to convergence parameter µ.The population size is N, and d is the bounded-confidence parameter.
  • Deffuant-Weisbuch: The population converges to one or more clusters, with the approximate cluster count determined by c ≈⌊1
  • Deffuant-Weisbuch: Parameters µ and N determine convergence speed and final-opinion distribution width, while small extreme clusters are a characteristic model outcome.
  • Deffuant-Weisbuch: Extensions incorporate disagreement through partial contrarians, heterogeneous or adaptive confidence thresholds, noise, and opinion drift, broadening the model beyond agreement dynamics.Partial contrarians can change opinions in the opposite direction from differently minded individuals.
  • Deffuant-Weisbuch: Segregated initial conditions hinder consensus, whereas initial cohesion produces one cluster; noise can partially remove this effect.Initial condition and noise (‘free will’) strongly affect the number of clusters.
  • Deffuant-Weisbuch: Opinion-dependent segregation produces one large cluster when α is very small or β is very large, alongside small extreme clusters, whereas pluralism is conserved only when extremist clusters are conneExtremists interact only with similar individuals, while moderated individuals have a wider interaction range.
  • Deffuant-Weisbuch: Directed scale-free networks yield more final opinions than undirected networks at high d and fewer at low d; other extensions impose bias, hierarchical interactions, adaptive networks, or public-goods coupling.The bias and hierarchical interaction structure are described as approaches aimed at consensus, while public-goods coupling uses opinion values as strategy probabilities.

The Hegselmann-Krause (HK) model

The Hegselmann–Krause model uses bounded confidence and simultaneous averaging over all compatible neighbours, making it suited to group interactions such as formal meetings. Its dynamics converge polynomially, form opinion clusters, and exhibit analytically characterized separation properties.

  • The Hegselmann-Krause (HK) model: Each agent averages the opinions of all neighbours within its confidence interval [x_i − ϵ, x_i + ϵ].Unlike pairwise updating, agents interact with all compatible neighbours simultaneously.
  • The Hegselmann-Krause (HK) model: The HK model is suited to large-group interactions such as formal meetings, whereas Deffuant is suited to pairwise interaction in large populations.
  • The Hegselmann-Krause (HK) model: At least a quadratic number of steps is required, although the model is proven to converge in polynomial time.
  • The Hegselmann-Krause (HK) model: As ϵ increases, the final number of opinion clusters decreases; above a threshold ϵ_c, only one cluster can remain.Convergence to one cluster can nevertheless be very slow.
  • The Hegselmann-Krause (HK) model: For real opinions in [0,L] with ϵ = 1, the population always converges to clusters separated by distances larger than 1.Analytical lower bounds for inter-cluster distances were obtained for finite populations and a continuum of agents.

Other models

Other agent-based opinion models extend continuous dynamics through adaptive noise, stubbornness, biased assimilation, disagreement, kinetic exchange, and community connectivity. These mechanisms produce consensus alternatives including individualism, pluralism, polarization, persistent disagreement, symmetry breaking, opposite clusters, travelling waves, and incoherence.

  • Adaptive-noise models: Adaptive noise combined with individualization yields consensus, individualism, or preserved pluralism, depending on their respective levels.The model balances individualization against social integration in continuous opinions.
  • Social influence: Social influence on continuous opinions depends on the population’s initial condition in a model of the wisdom of crowds.Agents update through the average opinion of their peers, while aggregated opinions may be closer to truth than individual opinions.
  • Stubborn agents: Poissonian interactions with stubborn agents generate continuous opinion fluctuation and disagreement, so consensus is never reached.Stubborn agents do not change their opinions.
  • Biased assimilation: Adding biased assimilation to weighted-neighbour updating enables polarization, whereas the model without it does not produce polarization even with homophily.Biased assimilation reinforces or extremizes opinions when information is inconclusive.
  • Disagreement: Disagreement in coupled-oscillator opinion dynamics produces opposite clusters, travelling waves, or complete incoherence even when oscillators share the same frequency.Disagreeing oscillators are negatively coupled to the mean field.
  • Kinetic-exchange models: λ_c = 2/3 marks kinetic-exchange symmetry breaking: below it the average opinion remains 0, whereas above it the average opinion is non null.Extensions distinguish conviction from influence, with a symmetry-breaking boundary set by λ = 1 + µ^2.
  • Community connectivity: The maximum inter- to intra-community link ratio without consensus increases as intra-community connectivity increases in the information accumulation model.The model studies two initially differently opinionated communities connected by inter-community links.

2.1.3 Hybrid models · The CODA model

The CODA model combines continuous probabilities for opinion adherence with discrete public actions, Bayesian peer updates, and memory effects. Extensions show how migration, additional opinions, trust, observation range, and external information shape agreement, polarization, extremism, and adoption.

  • The CODA model: CODA represents adherence continuously through p_i while deriving discrete opinions σ_i between +1 and −1 using a hard threshold.Individuals occupy a square lattice, with 1−p_i representing agreement to opinion −1.
  • The CODA model: Agents observe neighbours’ discrete opinions and update p_i Bayesianly, preserving discrete public dynamics while adding continuous change and memory.Agents do not jump directly between −1 and +1, but gradually change their opinions.
  • The CODA model: Migration in social networks reduces observed extremism and ultimately produces one cluster, while introducing a third opinion modifies the prevalence of mild opinions.The third opinion can be undecided when p_i is near 1/2 or represented through probabilities p_i, q_i, and r_i.
  • The CODA model: Trust extensions yield either agreement for higher trust or polarization as agents’ evolving estimates of others’ trustworthiness shape the dynamics.Agents hold arrays of probabilities describing whether others are trustworthy, and these probabilities evolve over time.
  • The CODA model: Agreement is reached faster than polarization, and clustered early adopters are no more likely than randomly distributed adopters to impose their opinion.The latter result follows from extending agents’ observation range and grouping neighbours with the same opinion.
  • The CODA model: In scientific-theory adoption, experimentalists receive information from Nature as well as peers, and a small fraction τ makes persuasion difficult.Experimentalists comprise fraction τ of the scientific world and can support competing theories with specified probabilities.

2.2 Multi-dimensional models

Multi-dimensional opinion models represent individuals through multiple discrete or continuous features, allowing homophily, social influence, bounded confidence, disagreement, and evolving relationships to shape collective outcomes. These extensions produce agreement, fragmentation, clustering, metastability, and empirically similar transient dynamics under different conditions.

  • Axelrod model: The Axelrod model represents each individual with F cultural features, each taking one of q discrete traits, and makes interaction more likely between similar individuals.Interactions increase similarity by copying a differing feature; agents sharing no traits cannot interact.
  • Axelrod model: Small q favors ordered absorbing states, whereas larger q favors frozen states with coexisting cultural regions; on regular lattices, the transition occurs at a critical qc depending on F.The final state is determined by how many traits individuals initially share, which controls interaction opportunities and cultural-domain formation.
  • Axelrod model: Analytical results show that F = q = 2 yields one majority cluster, while q > F produces fragmentation; in one dimension, fixation occurs when F ≤ cq, with e−c = c.Here, fixation means the system eventually stops changing.
  • Axelrod extensions: Extensions show that surface tension creates metastable states, cultural drift drives agreement at very small noise rates but fragmentation at large rates, and disagreement favors cultural fragmentation.Interaction noise has small effects on the phase transition but reduces relaxation times.
  • Axelrod extensions: Committed individuals and evolving social links extend Axelrod dynamics, while scale-free-network analyses find global fragmentation even when many individual features are shared.The network model introduces a fraction p of agents who do not change one issue’s opinion.
  • Continuous multidimensional models: Continuous vectorial models generate bounded-confidence clusters, self-reinforcing groups under high vanity, and affinity-based interactions that can occur despite distant opinions.In affinity models, opinions and affinities are updated during interaction, and affinities are interpreted as weighted social-network ties.

2.3 Modelling norms

Norm-compliance models extend opinion dynamics from expressed views to behaviour, incorporating social and external pressures. Agent-based approaches examine norm emergence, violation, cooperation, sanctions, group interactions, persistence, and collective participation.

  • Conceptual basis: Norm compliance concerns final behaviour rather than opinions alone, because social or external pressure can produce actions contrary to stated opinions.Norms are socially enforced rules and may also be enforced by law.
  • Agent-based approaches: Agent-based norm models commonly use game-theoretic strategies, with cooperation representing compliance and defection representing violation.Agents adjust strategies or strategy probabilities using utilities that include costs, punishments, rewards, and peer behaviour.
  • Group interactions: In-group similarity, out-group distance, and behavioural persistence shape norm evolution, with agents tending to resemble in-groups and resist behavioural change.The utility function includes in-group similarity, out-group difference, and the closeness between behaviour at times t and t + 1.
  • Evolutionary norm dynamics: Spatial interactions give moralists an advantage, allowing them to prevail and enabling the social norm to win.Moralists are cooperators who punish at a cost, while immoralists are defectors who punish.
  • Information, sanctions, and calibration: Norm-compliance research also models hidden violations, sanctions, social herding, and calibrated user-page interactions to explain compliance and norm emergence.These approaches include inspectors, concealment effort, neighbourhood cooperator fractions, and Wikipedia-based editing and sanction processes.
  • Collective participation: Weak self-reinforcement increases stability and produces larger long-run participation in collective behaviour.Participation incentives increase with the number of participants, making participation analogous to compliance with a norm.

3 Effect of external information on opinion dynamics

External information extends peer-only opinion models by introducing media, truth-oriented sources, or other fields that can alter consensus and clustering. Its effects depend on exposure frequency, information strength and accuracy, network structure, and whether external sources interact or disagree.

  • 3 Effect of external information on opinion dynamics: Peer-only models omit any external reservoir, whereas external-information models represent populations interacting with media, information sources, monitors, experts, or fields.This extension is motivated by cases where individual opinions are shaped by information beyond mutual peer interactions.
  • 3 Effect of external information on opinion dynamics: External influence can produce consensus or clustering, with outcomes depending on initial opinion density, media probability, exposure timing, confidence thresholds, and information strength.In the Sznajd model, final all-up or all-down states depend on initial up-spin density and p; in Deffuant-type models, timing and strength can generate additional or antagonistic clusters.
  • 3 Effect of external information on opinion dynamics: Extreme information requires sufficiently frequent exposure and continued peer interaction, while mild information may require longer intervals between re-exposures to achieve complete agreement.For mild information or large confidence, agreement occurs only when T exceeds a threshold; otherwise, substantial fractions can still cluster around the information value.
  • 3 Effect of external information on opinion dynamics: Strong or aggressive information can generate antagonistic clusters, leave individuals away from the source, or otherwise reduce adoption rather than improve alignment.Truth-seeking models similarly show that large α may leave more α = 0 individuals away from the truth, although all truth seekers converge to it analytically.
  • 3 Effect of external information on opinion dynamics: External sources need not remain fixed: endogenous fields and competing media can produce alignment, rejection, or large rejecting minorities, while repulsive links can favor consensus with external information.Media competition is modeled through source interactions and disagreement, and populations influenced by one another’s global fields can either align or reject the information.
  • 3 Effect of external information on opinion dynamics: Peer interaction can counter false media messages only when media reach remains limited, while disagreement-based models with multiple media sources can yield stable non-polarised clusters or full agreement.The reported threshold for escaping a false message is that media do not reach more than 60% of individuals; outcomes also vary with pI and message mildness.

4 Final remarks

The review highlights disagreement, independence, zealots, noise, network structure, and external information as important extensions of opinion-dynamics models. It also identifies limited treatment of changing, multiple, bidirectional information sources and new opportunities from behavioural data.

  • 4 Final remarks: The review spans opinion models from discrete one-dimensional to continuous multidimensional representations, emphasizing disagreement and external information.These ingredients model less-explored social mechanisms, including interaction with mass media.
  • 4 Final remarks: Disagreement, independence, and zealots can facilitate coexistence of multiple opinion states beyond attractive interactions.These mechanisms were introduced to improve models’ applicability to real settings.
  • 4 Final remarks: Low noise facilitates consensus, whereas high noise produces instability as disorder or fluctuating clusters in continuous and discrete models.The review discusses these effects for examples including Deffuant and Axelrod models.
  • 4 Final remarks: Most network topologies have limited effects on consensus time and final qualitative structure, but directed interactions can induce fragmentation.Evolving networks can still produce consensus or clustered states, while cluster quantities may change across topologies.
  • 4 Final remarks: External information generally causes fragmentation, while its modelling remains limited beyond Axelrod, Deffuant, and a few multidimensional continuous approaches.For other discrete models, an external field generally produces trivial consensus.
  • 4 Final remarks: Literature rarely models multiple changing information sources whose evolution receives population feedback, leaving bidirectional media–agent interaction unresolved.New communication technologies also make behavioural traces increasingly available through sensors, human computation, and gaming.
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