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REVIEW 4 major objections 4 minor 55 references

Modeling biases in binary decision-making within the generalized nonlinear q-voter model

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A generalized q-voter model with asymmetric flip probabilities produces two new results: for influence groups of size q>3, a phase D exists in which a fully adopted and a partially adopted state are simultaneously stable; and for q>=3 in…

desk verdict A useful two-parameter q-voter extension with a genuinely new phase and exit-probability plateau, but Eq. (22) is arithmetically wrong and contradicts the paper's own figures; the qualitative results survive. read the letter →

arxiv 2502.10172 v2 pith:GFYFZIXY submitted 2025-02-14 physics.soc-ph cond-mat.stat-mechphysics.comp-ph

Maciej Doniec , Pratik Mullick , Parongama Sen , Katarzyna Sznajd-Weron This is my paper · ORCID
classification physics.soc-phcond-mat.stat-mechphysics.comp-ph
keywords q-votermodelopiniondynamicsbinarydecision-makingasymmetricbiasexitprobabilityphasecoexistencecompletegraphcollectiveadoption
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a generalized q-voter model in which, when the q-person influence group is not unanimous, an individual's probability of switching opinion depends on its current state: unadopted individuals adopt with probability $\varepsilon_\uparrow$ and adopted individuals give up with probability $\varepsilon_\downarrow$. On a complete graph, the model is analyzed through a rate equation and Monte Carlo simulation, and it reduces to the original q-voter model when the two probabilities are equal and to a recently studied mass-media variant when $\varepsilon_\downarrow=0$. The central claim is that this asymmetry creates two previously unseen behaviors. First, for influence groups larger than three ($q>3$), a phase D exists in which a fully adopted state and a partially adopted state are simultaneously stable. Second, for $q\ge 3$ in small systems, the exit probability has a plateau, meaning that increasing initial support for an opinion does not increase the probability of final full adoption across a wide range of starting conditions; the paper argues this matters for real small groups such as organizations.

What carries the argument

The central object is the rate equation (Eq. 11) built from per-step transition probabilities $\gamma_+$ and $\gamma_-$, each decomposed into unanimity and non-unanimity parts with asymmetric weights $\varepsilon_\uparrow$ and $\varepsilon_\downarrow$. The phase diagram is obtained from the fixed points of this equation and their linear stability, with the decisive thresholds $\varepsilon=1/q$ and $\varepsilon=(q-1)/(2q-2)$. The exit probability is computed by representing the system as a Markov chain on $N+1$ concentration states, using transition rates taken from the infinite-$N$ expressions in Eq. (10), and solving the fundamental-matrix equation; for $q=2$ this yields the closed form in Eq. (27).

What would settle it

Recompute the exit probability using transition probabilities obtained from the exact finite-$N$ combinatorial expressions (Eqs. 6-9) instead of the $N\to\infty$ rates in Eq. (10), for $q=3$ and $N=64$; if the plateau in $E(c_0)$ changes shape, moves, or disappears, the plateau claim as stated would fail. A second check is to run the same model on a sparse random graph instead of a complete graph and see whether phase D survives.

Watch

Extended reading notes

Core claim

On a complete graph, the model's aggregate dynamics are governed by the rate equation $dc/dt = \gamma_+ - \gamma_-$, where the upward and downward transition probabilities are each sums of a unanimity contribution and a non-unanimity contribution weighted by $\varepsilon_\uparrow$ and $\varepsilon_\downarrow$. Analyzing the fixed points and their linear stability reveals five phases, labeled A through E. For $q>3$, a new phase D appears: when $\varepsilon_\uparrow > 1/q > \varepsilon_\downarrow$, the fully adopted state and a partially adopted state are both stable, while the unadopted state and another partially adopted state are unstable; the mirror situation holds for $\varepsilon_\downarrow > 1/q > \varepsilon_\uparrow$. In addition, for $q\ge 3$, the exit probability $E(c_0)$ in small systems takes a unique form with a wide plateau: over a broad range of initial concentrations $c_0$, the probability of eventually reaching full adoption is nearly constant, so a larger initial fraction of adopters does not improve the chance of full adoption. For $q=2$, the exit probability is given in closed form and reduces to the linear voter-model result $E(c_0)=c_0$ at the symmetric point $\varepsilon_\uparrow=\varepsilon_\downarrow=1/2$.

Load-bearing premise

The exit-probability plateau is derived by inserting infinite-system transition rates into a Markov chain with only $N+1$ states, so the plateau's shape and location rest on the assumption that these rates stay accurate at $N=64$ without finite-size corrections.

Editorial extensions

If this is right

  • When $q>3$ and $\varepsilon_\uparrow>1/q>\varepsilon_\downarrow$, the final state depends on the initial concentration: only initial adoptions above a critical value lead to the fully adopted state, while lower initial support settles into a partially adopted state.
  • For $q\ge 3$ and small $N$, the exit-probability plateau implies that campaigns or interventions aimed at increasing initial adoption will not improve the odds of full consensus until they push the initial concentration past the plateau edge.
  • Because phase D is absent for $q=2$ and $q=3$, the model predicts that the size of the influence group, not just the strength of bias, qualitatively changes collective outcomes.
  • The generalized model contains the original q-voter model ($\varepsilon_\uparrow=\varepsilon_\downarrow$) and the mass-media model ($\varepsilon_\downarrow=0$) as special cases, so its phase diagram maps those earlier results onto a common parameter plane.
  • For $q=2$, the closed-form exit probability Eq. (27) recovers the linear voter behavior $E(c_0)=c_0$ at the symmetric point $\varepsilon_\uparrow=\varepsilon_\downarrow=1/2$ and gives quantitative predictions for all other asymmetries.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension (editorial): the plateau implies a measurable prediction for small-group experiments: in groups of about sixty people whose discussion panels have three or more members, varying initial support within the plateau range should leave the probability of unanimous adoption nearly unchanged.
  • Extension (editorial): the coexistence in phase D suggests a hysteresis-like dependence on initial conditions; a testable corollary is that the same group can end at full adoption or partial adoption depending only on initial concentration, so bimodal final outcomes should be observable in repeated runs with identical parameters.
  • Extension (editorial): the asymmetric flip probabilities can be read as a cost-benefit asymmetry for adoption; an economic analogue would predict that a product with a stronger pull can still fail to take over a market unless early adopters exceed the critical initial share.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a generalized nonlinear q-voter model on a complete graph in which, when the influence group is not unanimous, the probability of a target agent changing state depends on its current state, with parameters ε↑ and ε↓. The authors derive a mean-field rate equation (Eq. (11)), analyze fixed points and their stability, construct a phase diagram with five phases A–E, and compute exit probabilities for small systems using both a Markov chain and Monte Carlo simulations. Two main findings are reported: (1) for q>3, a phase D appears in which a fully adopted and a partially adopted state are simultaneously stable, and (2) for q≥3, the exit probability exhibits a plateau, meaning that larger initial support does not necessarily increase the probability of full adoption.

Significance. The model generalizes the original q-voter model and a recent mass-media variant, and the reported phase D and exit-probability plateau are novel and socially relevant. The paper contains analytic rate equations, linear stability analysis, phase diagrams, and Monte Carlo validation, with source code claimed to be available. However, the central stability threshold for the symmetric case is printed incorrectly, creating an internal inconsistency in the phase-diagram derivation; the qualitative conclusions appear recoverable with the correct formula, but the analytical scaffolding requires repair.

major comments (4)
  1. [Section IV.B, Eq. (22)] The stability threshold for the c=1/2 fixed point on the diagonal ε↑=ε↓=ε is printed as ε > (q−1)/(2q−2) = 1/2. Substituting c=1/2 and ε↑=ε↓=ε into Eq. (19) gives d/dc(dc/dt) = (q−1)2^{1−q} − 2ε(1−2^{1−q}), so the threshold is ε_c = (q−1)/[2(2^{q−1}−1)]. The printed expression makes the subsequent inequality (q−1)/(2q−2) < 1/q fail for every q≥2, which would eliminate phases B and D for all q and contradict the paper's headline claim and Fig. 3(a). This equation and the surrounding text must be corrected; with the correct ε_c the qualitative phase diagram is restored.
  2. [Section IV.C, phase diagram discussion] The paragraph beginning 'Let us first focus on Fig. 3(a)' states that increasing ε↑ to (q−1)/(2q−2) along the diagonal leads to phase B. Since (q−1)/(2q−2)=1/2, this value exceeds 1/q for all q≥2, so the statement is inconsistent with the phase diagram and with the previously stated boundary ε↑=1/q. The same paragraph's explanation that phases B and D are absent for q=2,3 because (q−1)/(2q−2)<1/q fails is also not the correct condition: with the corrected threshold, the absence follows from ε_c=1/q (with equality) for q=2,3, not from a violation of the printed inequality. The text should be revised to use the correct threshold and to explain the q-dependence consistently.
  3. [Section IV.A, Eq. (11)] The factor containing the asymmetry is written as (ε↑+ε↓)(ε↑/(ε↑+ε↑)−c). The denominator ε↑+ε↑ should read ε↑+ε↓; otherwise the equation is inconsistent with Eq. (13) and with the stated reduction to the original q-voter model when ε↑=ε↓. This typo should be corrected throughout the derivation.
  4. [Section IV.D, Markov chain construction] The transition probabilities in the (N+1)-state Markov chain are taken directly from Eq. (10), which are the infinite-N deterministic rates, rather than the exact finite-N transition probabilities given by Eqs. (6)–(9). The authors should either use the exact finite-N rates or justify why O(1/N) corrections are negligible for N=64. The Monte Carlo simulations agreeing with the Markov chain provide empirical support, but this is a load-bearing approximation for the claimed plateau and should be explicitly acknowledged and tested.
minor comments (4)
  1. [Figure 5 caption] The phrase 'a plateau region ... covering the range from ε↑ = 0.4 to ε↑ = 0.6' is confusing: the plateau is a feature of E(c0) as a function of c0 for fixed ε↑, not a range of ε↑ values.
  2. [Section IV.C, phase E description] The text 'ε↑ = ε↑ > 1/q (symmetric case)' should read 'ε↑ = ε↓ > 1/q (symmetric case)'.
  3. [Section V, first paragraph] The sentence 'a new phase D appears for q > 3, in which two stable and two unstable stable steady states exist' contains the redundant phrase 'unstable stable'; it should read 'two stable and two unstable steady states'.
  4. [Data availability statement] The data availability statement mentions a public GitHub repository but does not provide a URL or repository identifier; please add the link.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase diagram and exit probability are direct mathematical consequences of the model's transition probabilities; the only self-citation is a q=2 limiting-case cross-check, not load-bearing.

full rationale

The derivation is self-contained: the paper starts from elementary update probabilities (Eqs. (6)-(9)), passes to the N→∞ rates (Eq. (10)), and obtains the mean-field rate equation (Eq. (11)); all fixed-point and stability statements (Eqs. (19)-(22)) are computed from that equation rather than imposed to force the claimed phases. The exit probability is obtained either from the Fokker-Planck equation (Eq. (23)) or from a Markov chain whose transition probabilities are the same γ± of Eq. (10), and the analytical curves are compared with Monte Carlo simulations rather than fitted. The limiting reductions to the original q-voter model [13] and the media-influenced model [16] provide independent external cross-checks. The only self-citation, [48] (Mullick and Sen, including the corresponding author), concerns the q=2 special case with ε↑+ε↓=1, which is not the regime supporting the headline q>3 phase-D claim or the q≥3 plateau; it is a consistency check, not a load-bearing premise. The two headline results are not independent of one another—the plateau is a finite-size reflection of the stable partially adopted state—but that is a derivational relationship, not circular reasoning. The arithmetic inconsistency in Eq. (22) (the printed threshold (q−1)/(2q−2)=1/2 contradicts Fig. 2, whereas direct differentiation of Eq. (19) gives (q−1)/[2(2^{q−1}−1)]) is a correctness defect, not a circularity, and the qualitative q>3 conclusion survives the corrected threshold.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model is a mean-field Markovian binary dynamics with two asymmetry parameters; all assumptions are standard for the q-voter family. The only non-standard element is the use of infinite-N rates in the finite-N Markov chain, which is validated by Monte Carlo matching but not explicitly flagged.

assumptions (5)
  • domain assumption Complete graph / all-to-all interactions
    The model is analyzed on a complete graph (Section III), which justifies the mean-field rate equation in Section IV.A.
  • domain assumption Random sequential update with q-panel selected without replacement
    The transition probabilities in Eqs. (6)-(9) assume uniform random selection of q distinct neighbors; the products reflect sampling without replacement.
  • standard math Infinite-N continuum limit for the rate equation
    Eq. (4) is obtained by taking N→∞ in the discrete-time rate equation; standard in sociophysics.
  • domain assumption Finite-N Markov chain uses the N→∞ transition rates
    In Section IV.D the Markov chain transition probabilities are taken from Eq. (10), which are the infinite-N limits of Eqs. (6)-(9), without explicit finite-size corrections.
  • standard math Diffusion approximation for the exit probability
    Eq. (16) is a Fokker-Planck approximation; the closed-form exit probability Eq. (27) is based on it, though Fig. 5 uses the exact Markov chain for q≥3.

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Cite this review

Pith. "Pith review of Modeling biases in binary decision-making within the generalized nonlinear q-voter model." pith.science (2026). https://pith.science/paper/GFYFZIXY

@misc{pith2026250210172,
  author       = {Pith},
  title        = {Pith review of: Modeling biases in binary decision-making within the generalized nonlinear q-voter model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFYFZIXY}},
  note         = {Machine review of arXiv:2502.10172}
}
abstract

Collective decision-making is a process by which a group of individuals determines a shared outcome that shapes societal dynamics; from innovation diffusion to organizational choices. A common approach to model these processes is using binary dynamics, where the choices are reduced to two alternatives. One of the most popular models in this context is the $q$-voter model, which assumes that opinion changes are driven by peer pressure from a unanimous group. However, real-world decisions are also shaped by prior personal choices and external influences, such as mass media, which introduce biases that can favor certain options over others. To address this, we propose a generalized $q$-voter model that incorporates these biases. In our model, when the influence group is not unanimous, the probability that an individual changes its opinion depends on its current state, breaking the symmetry between opinions. In limiting cases, our model recovers both the original $q$-voter model and several recently introduced modifications of the $q$-voter model, while extending the framework to capture a broader range of scenarios. We analyze the model on a complete graph using analytical methods and Monte Carlo simulations. Our results highlight two key findings: (1) for larger influence groups ($q>3$), a phase emerges where both adopted and partially adopted states coexist, (2) in small systems, greater initial support for an opinion does not necessarily increase its likelihood of widespread adoption, as reflected in the unique form of the exit probability. These results point to one of the key issues in social science, the importance of group size in collective action.

Figures

Figures reproduced from arXiv: 2502.10172 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of our model illustrating possible scenar [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The temporal behavior of the concentration [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Phase diagrams for (a) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Concentration of adopted agents in stationary states for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Exit probability [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Hirshleifer , author Y

    author author D. Hirshleifer , author Y. Levi , author B. Lourie , \ and\ author S. H. \ Teoh ,\ title title Decision fatigue and heuristic analyst forecasts , \ @noop journal journal Journal of Financial Economics \ volume 133 ,\ pages 83 – 98 ( year 2019 ) NoStop

    work page 2019

  2. [2]

    Sweller ,\ title title Cognitive load during problem solving: Effects on learning , \ @noop journal journal Cognitive Science \ volume 12 ,\ pages 257--285 ( year 1988 ) NoStop

    author author J. Sweller ,\ title title Cognitive load during problem solving: Effects on learning , \ @noop journal journal Cognitive Science \ volume 12 ,\ pages 257--285 ( year 1988 ) NoStop

    work page 1988

  3. [3]

    Ball , author B

    author author S. Ball , author B. Katz , author F. Li , \ and\ author A. Smith ,\ title title The effect of cognitive load on economic decision-making: a replication attempt , \ @noop journal journal Journal of Economic Behavior and Organization \ volume 210 ,\ pages 226 – 242 ( year 2023 ) NoStop

    work page 2023

  4. [4]

    Jusup , author P

    author author M. Jusup , author P. Holme , author K. Kanazawa , author M. Takayasu , author I. Romić , author Z. Wang , author S. Geček , author T. Lipić , author B. Podobnik , author L. Wang , author W. Luo , author T. Klanjšček , author J. Fan , author S. Boccaletti , \ and\ author M. Perc ,\ title title Social physics , \ @noop journal journal Physics ...

    work page 2022

  5. [5]

    Chica , author M

    author author M. Chica , author M. Perc , \ and\ author F. C. \ Santos ,\ title title Success-driven opinion formation determines social tensions , \ @noop journal journal iScience \ volume 27 ,\ pages 109254 ( year 2024 ) NoStop

    work page 2024

  6. [6]

    Cheng , author S

    author author S. Cheng , author S. Chen , \ and\ author Z. Shi ,\ title title Opposing sequential biases in direction and time reproduction: Influences of task relevance and working memory , \ @noop journal journal British Journal of Psychology \ volume 115 ,\ pages 825 – 842 ( year 2024 ) NoStop

    work page 2024

  7. [7]

    author author R. S. S. \ Kramer \ and\ author L. R. \ Pustelnik ,\ title title Sequential effects in facial attractiveness judgments: Separating perceptual and response biases , \ @noop journal journal Visual Cognition \ volume 29 ,\ pages 679 – 688 ( year 2021 ) NoStop

    work page 2021

  8. [8]

    Frey \ and\ author A

    author author V. Frey \ and\ author A. van de Rijt ,\ title title Social influence undermines the wisdom of the crowd in sequential decision making , \ @noop journal journal Management Science \ volume 67 ,\ pages 4273--4286 ( year 2021 ) NoStop

    work page 2021

Show all 55 references
  1. [9]

    Grabisch \ and\ author A

    author author M. Grabisch \ and\ author A. Rusinowska ,\ title title A survey on nonstrategic models of opinion dynamics , \ @noop journal journal Games \ volume 11 ,\ pages 1 – 28 ( year 2020 ) NoStop

    2020

  2. [10]

    Olsson \ and\ author M

    author author H. Olsson \ and\ author M. Galesic ,\ title title Analogies for modeling belief dynamics , \ @noop journal journal Trends in Cognitive Sciences \ volume 28 ,\ pages 907--923 ( year 2024 ) NoStop

    2024

  3. [11]

    author author J. P. \ Gleeson ,\ title title Binary-state dynamics on complex networks: Pair approximation and beyond , \ @noop journal journal Phys. Rev. X \ volume 3 ,\ pages 021004 ( year 2013 ) NoStop

    2013

  4. [12]

    J e drzejewski \ and\ author K

    author author A. J e drzejewski \ and\ author K. Sznajd-Weron ,\ title title Statistical physics of opinion formation: Is it a spoof? \ @noop journal journal Comptes Rendus Physique \ volume 20 ,\ pages 244--261 ( year 2019 ) NoStop

    2019

  5. [13]

    Castellano , author M

    author author C. Castellano , author M. A. \ Mu\ noz , \ and\ author R. Pastor-Satorras ,\ title title Nonlinear q -voter model , \ @noop journal journal Phys. Rev. E \ volume 80 ,\ pages 041129 ( year 2009 ) NoStop

    2009

  6. [14]

    Peralta , author A

    author author A. Peralta , author A. Carro , author M. San Miguel , \ and\ author R. Toral ,\ title title Analytical and numerical study of the non-linear noisy voter model on complex networks , \ @noop journal journal Chaos \ volume 28 ,\ pages 075516 ( year 2018 ) NoStop

    2018

  7. [15]

    author author L. S. \ Ramirez , author F. Vazquez , author M. San Miguel , \ and\ author T. Galla ,\ title title Ordering dynamics of nonlinear voter models , \ @noop journal journal Phys. Rev. E \ volume 109 ,\ pages 034307 ( year 2024 ) NoStop

    2024

  8. [16]

    Muslim , author R

    author author R. Muslim , author R. A. \ NQZ , \ and\ author M. A. \ Khalif ,\ title title Mass media and its impact on opinion dynamics of the nonlinear q-voter model , \ @noop journal journal Physica A: Statistical Mechanics and its Applications \ volume 633 ,\ pages 129358 ...

    2024

  9. [17]

    author author R. A. \ Holley \ and\ author T. M. \ Liggett ,\ title title Ergodic theorems for weakly interacting infinite systems and the voter model , \ @noop journal journal The Annals of Probability \ volume 3 ,\ pages 643--663 ( year 1975 ) NoStop

    1975

  10. [18]

    author author T. M. \ Liggett \ and\ author T. M. \ Liggett ,\ @noop title Interacting particle systems ,\ Vol. volume 2 \ ( publisher Springer ,\ year 1985 ) NoStop

    1985

  11. [19]

    author author S. Redner ,\ title title Reality-inspired voter models: A mini-review; [modèles d'électeurs inspirés de la réalité: une mini-revue] , \ @noop journal journal Comptes Rendus Physique \ volume 20 ,\ pages 275--292 ( year 2019 ) NoStop

    2019

  12. [20]

    Mobilia ,\ title title Does a single zealot affect an infinite group of voters? \ @noop journal journal Phys

    author author M. Mobilia ,\ title title Does a single zealot affect an infinite group of voters? \ @noop journal journal Phys. Rev. Lett. \ volume 91 ,\ pages 028701 ( year 2003 ) NoStop

    2003

  13. [21]

    author author I. Ferreira ,\ title title The probability of survival for the biased voter model in a random environment , \ @noop journal journal Stochastic processes and their applications \ volume 34 ,\ pages 25--38 ( year 1990 ) NoStop

    1990

  14. [22]

    Czaplicka , author C

    author author A. Czaplicka , author C. Charalambous , author R. Toral , \ and\ author M. San Miguel ,\ title title Biased-voter model: How persuasive a small group can be? \ @noop journal journal Chaos, Solitons & Fractals \ volume 161 ,\ pages 112363 ( year 2022 ) NoStop

    2022

  15. [23]

    Durocher , author P

    author author L. Durocher , author P. Karras , author A. Pavlogiannis , \ and\ author J. Tkadlec ,\ title title Invasion dynamics in the biased voter process , \ @noop journal journal arXiv preprint arXiv:2201.08207 \ ( year 2022 ) NoStop

    2022 arXiv

  16. [24]

    Masuda , author N

    author author N. Masuda , author N. Gibert , \ and\ author S. Redner ,\ title title Heterogeneous voter models , \ @noop journal journal Phys. Rev. E \ volume 82 ,\ pages 010103 ( year 2010 ) NoStop

    2010

  17. [25]

    Llabr\'es , author M

    author author J. Llabr\'es , author M. San Miguel , \ and\ author R. Toral ,\ title title Partisan voter model: Stochastic description and noise-induced transitions , \ @noop journal journal Phys. Rev. E \ volume 108 ,\ pages 054106 ( year 2023 ) NoStop

    2023

  18. [26]

    author author B. L. \ Granovsky \ and\ author N. Madras ,\ title title The noisy voter model , \ @noop journal journal Stochastic Processes and their Applications \ volume 55 ,\ pages 23--43 ( year 1995 ) NoStop

    1995

  19. [27]

    Sood \ and\ author S

    author author V. Sood \ and\ author S. Redner ,\ title title Voter model on heterogeneous graphs , \ @noop journal journal Phys. Rev. Lett. \ volume 94 ,\ pages 178701 ( year 2005 ) NoStop

    2005

  20. [28]

    Sood , author T

    author author V. Sood , author T. Antal , \ and\ author S. Redner ,\ title title Voter models on heterogeneous networks , \ @noop journal journal Phys. Rev. E \ volume 77 ,\ pages 041121 ( year 2008 ) NoStop

    2008

  21. [29]

    Galam ,\ title title Sociophysics: A review of galam models , \ @noop journal journal International Journal of Modern Physics C \ volume 19 ,\ pages 409--440 ( year 2008 ) NoStop

    author author S. Galam ,\ title title Sociophysics: A review of galam models , \ @noop journal journal International Journal of Modern Physics C \ volume 19 ,\ pages 409--440 ( year 2008 ) NoStop

    2008

  22. [30]

    author author P. L. \ Krapivsky \ and\ author S. Redner ,\ title title Divergence and consensus in majority rule , \ @noop journal journal Phys. Rev. E \ volume 103 ,\ pages L060301 ( year 2021 ) NoStop

    2021

  23. [31]

    Nyczka , author K

    author author P. Nyczka , author K. Sznajd-Weron , \ and\ author J. Cis o ,\ title title Phase transitions in the q -voter model with two types of stochastic driving , \ @noop journal journal Phys. Rev. E \ volume 86 ,\ pages 011105 ( year 2012 ) NoStop

    2012

  24. [32]

    author author A. M. \ Timpanaro \ and\ author C. P. C. \ Prado ,\ title title Exit probability of the one-dimensional q -voter model: Analytical results and simulations for large networks , \ @noop journal journal Phys. Rev. E \ volume 89 ,\ pages 052808 ( year 2014 ) NoStop

    2014

  25. [33]

    author author M. A. \ Javarone \ and\ author T. Squartini ,\ title title Conformism-driven phases of opinion formation on heterogeneous networks: the q-voter model case , \ @noop journal journal Journal of Statistical Mechanics: Theory and Experiment \ volume 2015 ,\ pages P10...

    2015

  26. [34]

    Mobilia ,\ title title Nonlinear q-voter model with inflexible zealots , \ @noop journal journal Phys

    author author M. Mobilia ,\ title title Nonlinear q-voter model with inflexible zealots , \ @noop journal journal Phys. Rev. E \ volume 92 ,\ pages 012803 ( year 2015 ) NoStop

    2015

  27. [35]

    Byrka , author A

    author author K. Byrka , author A. Jedrzejewski , author K. Sznajd-Weron , \ and\ author R. Weron ,\ title title Difficulty is critical: The importance of social factors in modeling diffusion of green products and practices , \ @noop journal journal Renewable and Sustainable E...

    2016

  28. [36]

    Rinto Anugraha , author R

    author author N. Rinto Anugraha , author R. Muslim , author H. Henokh Lugo , author F. Nugroho , author I. S. \ Alam , \ and\ author M. A. \ Khalif ,\ title title Nonlinear q-voter model involving nonconformity on networks , \ @noop journal journal Physica D: Nonlinear Phenome...

    2025

  29. [37]

    Gradowski \ and\ author A

    author author T. Gradowski \ and\ author A. Krawiecki ,\ title title Pair approximation for the q-voter model with independence on multiplex networks , \ @noop journal journal Phys. Rev. E \ volume 102 ,\ pages 022314 ( year 2020 ) NoStop

    2020

  30. [38]

    Chmiel \ and\ author K

    author author A. Chmiel \ and\ author K. Sznajd-Weron ,\ title title Phase transitions in the q-voter model with noise on a duplex clique , \ @noop journal journal Phys. Rev. E \ volume 92 ,\ pages 052812 ( year 2015 ) NoStop

    2015

  31. [39]

    author author A. R. \ Vieira , author A. F. \ Peralta , author R. Toral , author M. S. \ Miguel , \ and\ author C. Anteneodo ,\ title title Pair approximation for the noisy threshold q-voter model , \ @noop journal journal Phys. Rev. E \ volume 101 ,\ pages 052131 ( year 2020 ) NoStop

    2020

  32. [40]

    Krawiecki \ and\ author T

    author author A. Krawiecki \ and\ author T. Gradowski ,\ title title q -voter model with independence on signed random graphs: Homogeneous approximations , \ @noop journal journal Phys. Rev. E \ volume 109 ,\ pages 014302 ( year 2024 ) NoStop

    2024

  33. [41]

    Fardela , author Z

    author author R. Fardela , author Z. Abdullah , \ and\ author R. Muslim ,\ title title Opinion evolution under mass media influence on the Barabasi–Albert network based on the q-voter model , \ @noop journal journal International Journal of Modern Physics C (IJMPC) \ volume ...

    2025

  34. [42]

    author author A. Lipiecki ,\ title title Strategic promotional campaigns for sustainable behaviors: Maximizing influence in competitive complex contagions , \ in\ @noop booktitle Computational Science -- ICCS 2024 ,\ editor edited by\ editor L. Franco , editor C. de Mulatier ,...

    2024

  35. [43]

    author author A. R. \ Vieira \ and\ author C. Anteneodo ,\ title title Threshold q-voter model , \ @noop journal journal Phys. Rev. E \ volume 97 ,\ pages 052106 ( year 2018 ) NoStop

    2018

  36. [44]

    Nyczka , author K

    author author P. Nyczka , author K. Byrka , author P. R. \ Nail , \ and\ author K. Sznajd-Weron ,\ title title Conformity in numbers—does criticality in social responses exist? \ @noop journal journal PLOS ONE \ volume 13 ,\ pages 1--18 ( year 2018 ) NoStop

    2018

  37. [45]

    author author V. V. \ Vasconcelos , author S. A. \ Levin , \ and\ author F. L. \ Pinheiro ,\ title title Consensus and polarization in competing complex contagion processes , \ @noop journal journal Journal of The Royal Society Interface \ volume 16 ,\ pages 20190196 ( year 20...

    2019

  38. [46]

    Nowak \ and\ author K

    author author B. Nowak \ and\ author K. Sznajd-Weron ,\ title title Switching from a continuous to a discontinuous phase transition under quenched disorder , \ @noop journal journal Phys. Rev. E \ volume 106 ,\ pages 014125 ( year 2022 ) NoStop

    2022

  39. [47]

    Ramirez , author M

    author author L. Ramirez , author M. San Miguel , \ and\ author T. Galla ,\ title title Local and global ordering dynamics in multistate voter models , \ @noop journal journal Phys. Rev. E \ volume 106 ,\ pages 054307 ( year 2022 ) NoStop

    2022

  40. [48]

    Mullick \ and\ author P

    author author P. Mullick \ and\ author P. Sen ,\ title title Social influence and consensus building: Introducing a q-voter model with weighted influence , \ @noop journal journal PLOS ONE \ volume 20 ,\ pages 1--19 ( year 2025 ) NoStop

    2025

  41. [49]

    Civitarese ,\ title title External fields, independence, and disorder in q-voter models , \ @noop journal journal Phys

    author author J. Civitarese ,\ title title External fields, independence, and disorder in q-voter models , \ @noop journal journal Phys. Rev. E \ volume 103 ,\ pages 012303 ( year 2021 ) NoStop

    2021

  42. [50]

    Galam \ and\ author T

    author author S. Galam \ and\ author T. Cheon ,\ title title Asymmetric contrarians in opinion dynamics , \ @noop journal journal Entropy \ volume 22 ,\ pages 25 ( year 2020 ) NoStop

    2020

  43. [51]

    author author M. C. \ Gimenez , author L. Reinaudi , author S. Galam , \ and\ author F. Vazquez ,\ title title Contrarian majority rule model with external oscillating propaganda and individual inertias , \ @noop journal journal Entropy \ volume 25 ,\ pages 1402 ( year 2023 ) NoStop

    2023

  44. [52]

    Banisch ,\ @noop title Markov chain aggregation for agent-based models \ ( publisher Springer ,\ year 2015 ) NoStop

    author author S. Banisch ,\ @noop title Markov chain aggregation for agent-based models \ ( publisher Springer ,\ year 2015 ) NoStop

    2015

  45. [53]

    Esteban \ and\ author D

    author author J. Esteban \ and\ author D. Ray ,\ title title Collective action and the group size paradox , \ @noop journal journal American Political Science Review \ volume 95 ,\ pages 663 – 672 ( year 2001 ) NoStop

    2001

  46. [54]

    author author P. E. \ Oliver \ and\ author G. Marwell ,\ title title The paradox of group size in collective action: A theory of the critical mass. ii. \ @noop journal journal American Sociological Review \ volume 53 ,\ pages 1 -- 8 ( year 1988 ) NoStop

    1988

  47. [55]

    Galesic , author D

    author author M. Galesic , author D. Barkoczi , author A. M. \ Berdahl , author D. Biro , author G. Carbone , author I. Giannoccaro , author R. L. \ Goldstone , author C. Gonzalez , author A. Kandler , author A. B. \ Kao , author R. Kendal , author M. Kline , author E. Lee , a...

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.