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arxiv: 2601.18895 · v4 · pith:GGER52VGnew · submitted 2026-01-26 · ⚛️ physics.soc-ph

Competition between private and expressed opinions in binary choice: the α-EPO q-voter model

Barbara Kami\'nska , Barbara Nowak , Arkadiusz Lipiecki , Katarzyna Sznajd-Weron This is my paper

Pith reviewed 2026-05-16 10:39 UTC · model grok-4.3

classification ⚛️ physics.soc-ph
keywords opinion dynamicsq-voter modelexpressed-private opinionsself-anticonformityphase transitionspair approximationsocial networksasynchronous updating
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The pith

The α-EPO q-voter model shows that self-anticonformity makes collective agreement robust to the probability of updating private versus expressed opinions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an asynchronous update rule in which each agent flips its private opinion with probability α or its expressed opinion with probability 1-α. This removes the arbitrary order between thinking and acting that appeared in earlier EPO variants. Mean-field theory and a new pair approximation, both validated on networks, demonstrate that the effect of α on the agreement-disagreement transition depends sharply on whether self-anticonformity is allowed: with self-anticonformity the threshold stays fixed, while without it α moves the threshold and can switch the transition between continuous and discontinuous. The pair approximation further reveals that low average degree k opens an extra regime in which both α and k jointly control hysteresis width for influence groups larger than three.

Core claim

In the α-EPO q-voter model an agent updates its private opinion with probability α or its expressed opinion with complementary probability 1-α. When self-anticonformity is present the location of the agreement-disagreement threshold is insensitive to α; when self-anticonformity is absent, α shifts the threshold and, for q=3 in mean-field theory, changes the transition from continuous to discontinuous. The pair approximation extends this dependence to larger q in the low-connectivity regime, where both α and average degree k control the width of the hysteresis loop.

What carries the argument

The parameter α that sets the probability an agent updates its private opinion rather than its expressed opinion in each asynchronous step.

If this is right

  • With self-anticonformity present, the agreement threshold remains fixed for any value of α.
  • Without self-anticonformity, raising α lowers the critical fraction of disagreeing agents needed for a transition to disagreement.
  • For q=3 the mean-field transition changes from continuous to discontinuous as α increases.
  • In low-degree networks the pair approximation predicts that both α and k control the width of the bistable region for q>3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The robustness induced by self-anticonformity suggests that internal consistency checks can shield collective outcomes from changes in how often people voice private views.
  • Real groups with strong community structure may require higher-order approximations to capture how α and local clustering interact.
  • The low-connectivity regime identified by the pair approximation could be tested by rewiring experiments that vary average degree while holding α fixed.
  • α offers a continuous dial between internal reflection and social expression that may map onto measurable differences in how often people revise private beliefs versus public statements.

Load-bearing premise

The mean-field and pair approximations assume the network is either complete or has a single well-defined average degree and that correlations beyond pairs can be ignored.

What would settle it

Simulate the model on a real organizational network with measured community structure and check whether the observed change in consensus threshold with α matches the pair-approximation prediction only when self-anticonformity is turned off.

Figures

Figures reproduced from arXiv: 2601.18895 by Arkadiusz Lipiecki, Barbara Kami\'nska, Barbara Nowak, Katarzyna Sznajd-Weron.

Figure 1
Figure 1. Figure 1: FIG. 1. Visualization of a single update in the models with and [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Visualization of a single update in the models with and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

People often express opinions that differ from their privately held views, a phenomenon known in economy as preference falsification. Expressed-private opinion (EPO) models capture this by assigning each agent two dynamical variables: a private (internal) and an expressed (external) opinion. Within the nonlinear $q$-voter model, two EPO variants have been studied so far: with and without self-anticonformity. In both formulations, agents update private and expressed binary opinions, one after another and at the same rate, which has led to two update schemes studied previously: AT (act then think), in which an agent first updates its expressed and then its private opinion, and TA (think then act), in which the order is reversed. To eliminate this ad hoc distinction and quantify the interplay between private and expressed opinions, we introduce the $\alpha$-EPO $q$-voter model with asynchronous updating -- in each elementary step, an agent updates its private opinion with probability $\alpha$ or its expressed opinion with complementary probability $1-\alpha$. We derive mean-field theory and, for the first time for EPO $q$-voter dynamics, a pair approximation, and validate them with Monte Carlo simulations on artificial and real organizational networks. Comparing the two model variants, we show that the collective outcome controlled by $\alpha$ strongly depends on self-anticonformity: with self-anticonformity the results are robust to $\alpha$, whereas without it $\alpha$ shifts the agreement-disagreement threshold and can change the type of phase transition. In the mean-field limit this change occurs only for $q=3$, but the pair approximation reveals an additional low-connectivity regime in which both $\alpha$ and the average degree $k$ control the emergence and width of hysteresis also for larger influence groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the α-EPO q-voter model, which replaces ad-hoc AT/TA update orders with asynchronous updating in which each agent updates its private opinion with probability α (or expressed opinion with probability 1-α). Mean-field and pair-approximation equations are derived for the two variants (with and without self-anticonformity), compared to Monte Carlo simulations on Erdős–Rényi, scale-free, and real organizational networks, and used to show that α-dependence of agreement-disagreement thresholds and phase-transition type is robust when self-anticonformity is present but sensitive without it; the pair approximation additionally identifies a low-connectivity regime in which both α and average degree k control hysteresis width for q>3.

Significance. If the derivations and network validations hold, the work unifies prior EPO q-voter formulations, isolates the modulating role of self-anticonformity, and demonstrates that pair-level closure captures connectivity-dependent effects missed by mean-field theory. The explicit control parameter α and the falsifiable predictions for hysteresis width versus k and α constitute a clear advance for modeling preference falsification on structured networks.

major comments (2)
  1. [Pair approximation] Pair-approximation section: the moment closure assumes that three-body and higher correlations (including between private and expressed opinions of neighboring agents) remain negligible; on real organizational networks with community structure this assumption can break, potentially shifting the reported low-k regime in which both α and k control hysteresis width for the no-self-anticonformity variant.
  2. [Results] Monte Carlo validation (results section): quantitative agreement between pair-approximation curves and simulation transition points is asserted for the no-self-anticonformity case, yet no error bars on critical α or hysteresis widths are reported, nor is the deviation between PA and MC quantified as a function of k; this weakens the claim that α can change the type of phase transition in the low-connectivity regime.
minor comments (2)
  1. [Introduction] The abstract states that the pair approximation is derived 'for the first time for EPO q-voter dynamics'; the introduction should explicitly contrast the new closure with any earlier pair-level treatments of related EPO models.
  2. [Figures] Figure captions for the real-network panels should state the measured average degree k and the precise definition of the order parameter used to locate the transition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate the suggestions where appropriate.

read point-by-point responses

  1. Referee: Pair-approximation section: the moment closure assumes that three-body and higher correlations (including between private and expressed opinions of neighboring agents) remain negligible; on real organizational networks with community structure this assumption can break, potentially shifting the reported low-k regime in which both α and k control hysteresis width for the no-self-anticonformity variant.

    Authors: We agree that the pair approximation relies on neglecting higher-order correlations, an assumption that may not hold on networks with pronounced community structure. In the revised manuscript we have added an explicit discussion of this limitation in the pair-approximation section, noting its potential influence on the low-k regime for the no-self-anticonformity variant. At the same time, the Monte Carlo simulations performed on real organizational networks (which exhibit community structure) continue to show qualitative agreement with the pair-approximation predictions, supporting the robustness of the reported trends. revision: partial

  2. Referee: Monte Carlo validation (results section): quantitative agreement between pair-approximation curves and simulation transition points is asserted for the no-self-anticonformity case, yet no error bars on critical α or hysteresis widths are reported, nor is the deviation between PA and MC quantified as a function of k; this weakens the claim that α can change the type of phase transition in the low-connectivity regime.

    Authors: We acknowledge that the original submission lacked error bars and a quantitative assessment of deviations. In the revised version we have added error bars to the critical α values and hysteresis widths in the relevant figures. We have also included a supplementary analysis that quantifies the relative deviation between pair-approximation and Monte Carlo results as a function of average degree k for the no-self-anticonformity case. These additions strengthen the evidence that α can alter the phase-transition type in the low-connectivity regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity: α introduced as explicit parameter; thresholds emerge from derived equations

full rationale

The paper defines the α-EPO model by introducing α as a new asynchronous update probability (private vs. expressed opinion) that replaces the prior AT/TA distinction. Mean-field and pair-approximation equations are obtained directly from the microscopic stochastic rules; the reported thresholds, hysteresis widths, and phase-transition changes are solutions to those closed equations rather than inputs. Monte Carlo simulations on artificial and real networks serve as independent validation. Self-citations to earlier EPO q-voter variants supply context for the two model families but do not carry the load of the central claim, which follows from the new α-controlled dynamics. No parameters are fitted to the same observables that are later 'predicted,' and no uniqueness theorem or ansatz is smuggled in via self-reference.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard binary-opinion update rules plus the new α probability; no new particles or forces are postulated. The pair approximation itself is an additional closure assumption.

free parameters (2)
  • α
    Probability that an agent updates its private opinion in a given step; controls the relative speed of private versus expressed dynamics.
  • q
    Size of the influence group; taken from the original q-voter model.
axioms (2)
  • domain assumption Agents are updated asynchronously with probability α for private opinion and 1-α for expressed opinion.
    This replaces the earlier discrete AT/TA schemes and is the central modeling choice.
  • standard math Pair approximation closes the hierarchy of moment equations by neglecting correlations beyond nearest neighbors.
    Standard closure used in network opinion models; invoked to obtain analytic results beyond mean-field.

pith-pipeline@v0.9.0 · 5653 in / 1542 out tokens · 41724 ms · 2026-05-16T10:39:07.161180+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    act then think

    and Kami ´nska and Sznajd-Weron [3] considered their models only on a complete graph, corresponding to a fully connected society in which everyone knows everyone. For large social systems, this assumption is clearly unrealistic. Here, we move beyond the complete graph by studyingα- EPOq-voter models on networks using Monte Carlo (MC) simulations and pair ...

    work page

  2. [2]

    We use two types of initial conditions: ordered,c S(0) =c σ (0) =1, and random, wherec S(0) = cσ (0) =1/2

    We assume that agents are initially in harmony, i.e., ∀i;S i(0) =σ i(0). We use two types of initial conditions: ordered,c S(0) =c σ (0) =1, and random, wherec S(0) = cσ (0) =1/2. After initialization, we equilibrate the system for 3000 MCS, and then perform time averaging over the subsequent 2000 MCS. The simulation codes are publicly available at github...

    work page 2000

  3. [3]

    Model with self-anticonformity Below we simplify the mean-field equations for the sta- tionary states for the model with self-anticonformity. c↑↑ =c σ (1−α) [1−(1−p)(1−c S)q] +c ↑↑α[1−p/2−(1−p)(1−c S)q] +c ↑↓α p/2+ (1−p)c q S ,(14) c↑↓ = (1−c σ )(1−α)(1−p)c q S +c ↑↑α[p/2+ (1−p)(1−c S)q] +c ↑↓α 1−p/2−(1−p)c q S ,(15) c↓↑ =c σ (1−α)(1−p)(1−c S)q +c ↓↑α[1−p...

    work page

  4. [4]

    1−(1−p) k↑↑ +k ↑↓ !(k−q)! k↑↑ +k ↑↓ −q !k! 1k↑↑+k↑↓⩾q # , (38) f ↓↑→↓↓ (κ) =α(1−p) k↓↑ +k ↓↓ !(k−q)! k↓↑ +k ↓↓ −q !k! 1k↓↑+k↓↓⩾q + 1 2 αp,(39) f ↓↑→↑↑ (κ) = (1−α) ×

    Model without self-anticonformity As stated above, for the model without self- anticonformity, the system is reduced analogously, we simply remove the framed terms, which yields: c↑↑ =c σ [1−(1−p)(1−c S)q] (1−α) +c ↑↑α[1−p/2] +c ↑↓α p/2+ (1−p)c q S ,(22) c↑↓ = (1−c σ )(1−α)(1−p)c q S +c ↑↑αp/2+c ↑↓α 1−p/2−(1−p)c q S ,(23) c↓↑ =c σ (1−α)(1−p)(1−c S)q +c ↓↑...

    work page 2023

  5. [5]

    Kuran, Preference Falsification, Policy Continuity and Col- lective Conservatism, The Economic Journal97, 642 (1987)

    T. Kuran, Preference Falsification, Policy Continuity and Col- lective Conservatism, The Economic Journal97, 642 (1987)

    work page 1987

  6. [6]

    J. Dong, J. Hu, Y . Zhao, and Y . Peng, Opinion formation analysis for Expressed and Private Opinions (EPOs) mod- els: Reasoning private opinions from behaviors in group decision-making systems, Expert Systems with Applications 236, 121292 (2024)

    work page 2024

  7. [7]

    Kami ´nska and K

    B. Kami ´nska and K. Sznajd-Weron, Impact of cognitive dis- sonance on social hysteresis: Insights from the expressed and private opinions model, Expert Systems with Applications 273, 126851 (2025)

    work page 2025

  8. [8]

    T. C. Schelling, Hockey helmets, concealed weapons, and day- light saving: A study of binary choices with externalities, Journal of Conflict Resolution17, 381 (1973)

    work page 1973

  9. [9]

    Castellano, M

    C. Castellano, M. A. Muñoz, and R. Pastor-Satorras, Nonlin- ear q-voter model, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics80(2009)

    work page 2009

  10. [10]

    Nyczka, K

    P. Nyczka, K. Sznajd-Weron, and J. Cisło, Phase transitions in the q -voter model with two types of stochastic driving, Phys- ical Review E86, 011105 (2012)

    work page 2012

  11. [11]

    Mobilia, Nonlinear q-voter model with inflexible zealots, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics92(2015)

    M. Mobilia, Nonlinear q-voter model with inflexible zealots, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics92(2015). 11 FIG. 5.Stationary concentration of positive expressed opinionsc S as a function of the independence probabilitypforα=0.1,q=3. Monte Carlo results on empirical networks [29] are compared with simulations on Watts–Stro...

    work page 2015

  12. [12]

    Gradowski and A

    T. Gradowski and A. Krawiecki, Pair approximation for the q- voter model with independence on multiplex networks, Phys- ical Review E102(2020)

    work page 2020

  13. [13]

    Masuda and S

    N. Masuda and S. Redner, Can partisan voting lead to truth?, Journal of Statistical Mechanics: Theory and Experiment 2011, L02002 (2011)

    work page 2011

  14. [14]

    Gaisbauer, E

    F. Gaisbauer, E. Olbrich, and S. Banisch, Dynamics of opinion expression, Physical Review E102, 042303 (2020)

    work page 2020

  15. [15]

    Manfredi, A

    R. Manfredi, A. Guazzini, C. A. Roos, T. Postmes, and N. Koudenburg, Private-Public Opinion Discrepancy, PLOS ONE15, e0242148 (2020)

    work page 2020

  16. [16]

    Llabrés, M

    J. Llabrés, M. San Miguel, and R. Toral, Partisan voter model: Stochastic description and noise-induced transitions, Physical Review E108, 054106 (2023)

    work page 2023

  17. [17]

    Banisch and E

    S. Banisch and E. Olbrich, Opinion polarization by learning from social feedback, The Journal of Mathematical Sociology 43, 76 (2019)

    work page 2019

  18. [18]

    M. Ye, Y . Qin, A. Govaert, B. D. Anderson, and M. Cao, An influence network model to study discrepancies in expressed and private opinions, Automatica107, 371 (2019)

    work page 2019

  19. [19]

    F. J. León-Medina, J. Tena-Sánchez, and F. J. Miguel, Fak- ers becoming believers: how opinion dynamics are shaped by preference falsification, impression management and coher- ence heuristics, Quality & Quantity54, 385 (2020)

    work page 2020

  20. [20]

    Jacob and S

    D. Jacob and S. Banisch, Polarization in Social Media: A Vir- tual Worlds-Based Approach, Journal of Artificial Societies and Social Simulation26, 11 (2023)

    work page 2023

  21. [21]

    Y . Peng, Y . Zhao, J. Dong, and J. Hu, Adaptive opinion dy- namics over community networks when agents cannot express opinions freely, Neurocomputing618, 129123 (2025)

    work page 2025

  22. [22]

    Piras, S

    S. Piras, S. Righi, M. Setti, N. Koseoglu, M. J. Grainger, G. B. Stewart, and M. Vittuari, From social interactions to private environmental behaviours: The case of consumer food waste, Resources, Conservation and Recycling176, 105952 (2022)

    work page 2022

  23. [23]

    Festinger,A theory of cognitive dissonance(Stanford Uni- versity Press, 1957)

    L. Festinger,A theory of cognitive dissonance(Stanford Uni- versity Press, 1957)

    work page 1957

  24. [24]

    M. T. Gastner, B. Oborny, and M. Gulyás, Consensus time in a voter model with concealed and publicly expressed opin- ions, Journal of Statistical Mechanics: Theory and Experiment 2018, 063401 (2018)

    work page 2018

  25. [25]

    J˛ edrzejewski, G

    A. J˛ edrzejewski, G. Marcjasz, P. R. Nail, and K. Sznajd- Weron, Think then act or act then think?, PLOS ONE13, e0206166 (2018)

    work page 2018

  26. [26]

    Carro, R

    A. Carro, R. Toral, and M. San Miguel, The noisy voter model on complex networks, Scientific Reports6, 24775 (2016)

    work page 2016

  27. [27]

    Khalil, M

    N. Khalil, M. San Miguel, and R. Toral, Zealots in the mean- field noisy voter model, Physical Review E97, 012310 (2018)

    work page 2018

  28. [28]

    Khalil and R

    N. Khalil and R. Toral, The noisy voter model under the influ- ence of contrarians, Physica A: Statistical Mechanics and its 12 Applications515, 81 (2019)

    work page 2019

  29. [29]

    A. F. Peralta, A. Carro, M. S. Miguel, and R. Toral, Stochastic pair approximation treatment of the noisy voter model, New Journal of Physics20, 103045 (2018)

    work page 2018

  30. [30]

    A. R. Vieira, A. F. Peralta, R. Toral, M. S. Miguel, and C. An- teneodo, Pair approximation for the noisy threshold q -voter model, Physical Review E101, 052131 (2020)

    work page 2020

  31. [31]

    P. R. Nail, S. I. Di Domenico, and G. MacDonald, Proposal of a Double Diamond Model of Social Response, Review of General Psychology17, 1 (2013)

    work page 2013

  32. [32]

    Kroesen and C

    M. Kroesen and C. Chorus, The role of general and specific at- titudes in predicting travel behavior – A fatal dilemma?, Travel Behaviour and Society10, 33 (2018)

    work page 2018

  33. [33]

    Fire and R

    M. Fire and R. Puzis, Organization Mining Using Online Social Networks, Networks and Spatial Economics16, 545 (2016)

    work page 2016

  34. [34]

    Je ¸drzejewski, Pair approximation for the q-voter model with independence on complex networks, Physical Review E 95, 012307 (2017)

    A. Je ¸drzejewski, Pair approximation for the q-voter model with independence on complex networks, Physical Review E 95, 012307 (2017)

    work page 2017

  35. [35]

    J˛ edrzejewski and K

    A. J˛ edrzejewski and K. Sznajd-Weron, Pair approximation for the q -voter models with quenched disorder on networks, Physical Review E105, 064306 (2022)

    work page 2022

  36. [36]

    L. S. Ramirez, F. Vazquez, M. San Miguel, and T. Galla, Or- dering dynamics of nonlinear voter models, Phys. Rev. E109, 034307 (2024)

    work page 2024

  37. [37]

    Lipiecki and K

    A. Lipiecki and K. Sznajd-Weron, When heterogeneity drives hysteresis: Anticonformity in the multistate q -voter model on networks, Physical Review E112, 054316 (2025)

    work page 2025

  38. [38]

    D. T. Miller, A century of pluralistic ignorance: what we have learned about its origins, forms, and consequences, Frontiers in Social Psychology1(2023)

    work page 2023

  39. [39]

    Dvorak, U

    F. Dvorak, U. Fischbacher, and K. Schmelz, Strategic Con- formity or Anti-Conformity to Avoid Punishment and Attract Reward, The Economic Journal135, 556 (2024)

    work page 2024

  40. [40]

    Lipiecki and K

    A. Lipiecki and K. Sznajd-Weron, Depolarizing power of an- ticonformity, Expert Systems with Applications285, 127879 (2025)

    work page 2025

  41. [41]

    Gerstel, N

    H. Gerstel, N. Kreilkamp, M. Schmidt, and A. Wöhrmann, Mitigating escalation of commitment through error manage- ment climate and the devil’s advocate approach, Journal of Business Economics95, 975 (2025)

    work page 2025