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A balancing condition on transition rates equates annealed and quenched dynamics, maps heterogeneous binary-choice systems to homogeneous ones, and forbids oscillations.

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2026-07-10 20:58 UTC pith:NZSJMGH2

load-bearing objection Solid expansion of their 2025 PRL: clean mean-field derivation of the balancing condition, honest scope, useful cross-field illustrations; novelty is moderate but the math and taxonomy are worth having.

arxiv 2607.06803 v1 pith:NZSJMGH2 submitted 2026-07-07 physics.soc-ph nlin.AO

Unified Framework for Binary-Choice Dynamics: Analysis and Applications

Arkadiusz J\k{e}drzejewski , Jos\'e F. F. Mendes This is my paper
classification physics.soc-ph nlin.AO
keywords binary-choice dynamicsannealed disorderquenched disorderheterogeneityopinion dynamicsagent-based modelingcollective dynamicsbalancing condition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces a unified framework for binary-choice models in which agents switch between two states under two competing mechanisms selected by individual preferences. Preferences may be redrawn every step (annealed) or fixed once and for all (quenched), and may be identical for all agents (homogeneous) or drawn from a distribution (heterogeneous). The authors derive a simple algebraic constraint on the four transition-rate functions; whenever the constraint holds, the macroscopic equations collapse so that annealed and quenched dynamics become identical, any heterogeneous population behaves exactly like a homogeneous one whose preference equals the mean, and oscillations are impossible. The same condition explains why certain models of opinion dynamics, kinetic Ising systems and epidemic spreading are robust to modeling choices while others are highly sensitive. The result therefore supplies a practical test that tells a modeler when a simplified homogeneous description is rigorously sufficient.

Core claim

When the transition rates of the two mechanisms satisfy X_AB(a)+X_BA(a)=Y_AB(a)+Y_BA(a) for every density a, the high-dimensional quenched rate equations reduce to the one-dimensional annealed flow that depends only on the mean preference; consequently annealed and quenched dynamics coincide, every heterogeneous system maps onto a homogeneous one, and oscillatory solutions cannot appear.

What carries the argument

The balancing condition X_AB(a)+X_BA(a)=Y_AB(a)+Y_BA(a); it cancels the integral term that otherwise couples the full preference distribution into the macroscopic dynamics, collapsing both annealed and quenched descriptions onto a single mean-field equation.

Load-bearing premise

All derivations assume a well-mixed population on a complete graph with random sequential updates driven by exactly two mechanisms.

What would settle it

Construct a binary-choice model that obeys the balancing condition yet, when placed on a sparse network of low average degree, produces measurably different annealed versus quenched stationary densities or sustained oscillations.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any model satisfying the condition is completely insensitive to the shape of the preference distribution and to whether preferences are fixed or redrawn.
  • Heterogeneous populations under the condition reduce exactly to homogeneous systems whose single preference equals the distribution mean.
  • Oscillatory trajectories are rigorously excluded once the condition holds.
  • Phase diagrams (continuous versus discontinuous transitions, epidemic thresholds) become independent of disorder type for balanced models, while unbalanced models can change transition order under quenched heterogeneity.
  • Model builders can test the four rate functions once and immediately know whether a simple mean-field homogeneous description is exact.

Where Pith is reading between the lines

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

  • The balancing identity can be used as a design criterion: choose functional forms that satisfy it when robustness to heterogeneity is desired, or deliberately violate it when distribution-dependent effects are the scientific target.
  • On networks the condition almost certainly acquires degree-dependent corrections; deriving the network version would extend the same unifying power beyond well-mixed populations.
  • Multi-state or multi-mechanism generalizations should admit analogous higher-order balancing relations whose satisfaction again collapses annealed and quenched dynamics.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The manuscript develops a unified mean-field framework for binary-choice dynamics in which each agent updates its state under one of two competing mechanisms (X or Y) selected according to an individual preference p drawn from a distribution φ(p). It classifies existing models by homogeneity versus heterogeneity of φ(p) and by annealed versus quenched treatment of the preferences, then derives a balancing condition on the transition rates, X_AB(a)+X_BA(a)=Y_AB(a)+Y_BA(a) for all a∈[0,1]. When the condition holds, the annealed and quenched macroscopic equations coincide, any heterogeneous population maps onto a homogeneous one with preference equal to the mean of φ(p), and the resulting one-dimensional flow cannot oscillate. The three consequences are illustrated with nonlinear voter models (with and without anticonformity), kinetic Ising models under competing Glauber or Metropolis dynamics, and the SIS epidemic model; limitations (networks, multi-state extensions, multi-flip updates) are discussed openly.

Significance. The central derivation (Secs. 6.1–6.3) is algebraically transparent, free of free parameters, and yields three immediately usable, falsifiable consequences that resolve a long-standing model-dependent puzzle in the literature. The framework cleanly unifies models from statistical physics, opinion dynamics and epidemiology without imposing functional forms on the rates, and the numerical illustrations match the analytic predictions. Open data and the explicit fixed-point formulae further enhance reproducibility. Under the stated well-mixed, two-mechanism hypotheses the result is a genuine advance that will be cited across disciplines.

minor comments (4)
  1. [Sections 1, 4.1, 9] Several typographical errors remain: “we preset here” (Sec. 1) should be “we present here”; “literate review”, “corse manner” and “tradition rates” (Sec. 9) should be “literature review”, “coarse manner” and “transition rates”; “indulging” (Sec. 4.1) should be “including”.
  2. [Figure 5] Figure 5 caption mixes panel labels: the text refers to (d) and (e) for the r eq1 case while the figure layout places those trajectories in (e) and (f). Correct the cross-references.
  3. [Section 7.3] In the SIS illustration the continuum of ODEs for continuous φ(p) is integrated numerically; a one-sentence statement of the integrator (or a pointer to the open repository) would improve reproducibility.
  4. [Section 6.2] The phrase “septate rate equation” (Sec. 6.2) is presumably a typographical slip for “separate rate equation”.

Circularity Check

1 steps flagged

No significant circularity: balancing condition and three consequences are re-derived from first-principles mean-field ODEs; prior Letter is cited only for framework introduction.

specific steps
  1. self citation load bearing [Introduction (paragraph after abstract) and Sec. 1]
    "In a recent Letter [11], we use this structure to formulate a unified framework that captures a broad class of models. … Although the core framework was introduced in Ref. [11], we present here the step-by-step mathematical derivations for completeness."

    The framework itself is introduced by self-citation. However, the citation is not load-bearing for the three claimed consequences: every rate equation, the balancing condition (Eq. 16), and the equivalence statements are re-derived in full in Sec. 6 from the elementary definitions given in Sec. 2. The self-citation therefore contributes only a minor historical pointer and does not force the results.

full rationale

The load-bearing derivation (Sec. 6) starts from the model definitions (two mechanisms X/Y chosen with preference p, well-mixed sequential updates) and obtains the annealed ODE (Eq. 5, depending only on mean p-bar via linearity and law of total probability) and the quenched system (Eqs. 9–12, involving the integral of p a_p). Setting the coefficient of the extra integral term to zero immediately yields the balancing condition (Eq. 16); under it the ODEs collapse to the identical one-dimensional flow (Eq. 17), fixed points coincide (Eq. 18), heterogeneous systems map to homogeneous ones, and oscillations are impossible. All algebraic steps are self-contained and parameter-free. The self-citation to the authors’ prior Letter [11] merely notes that the framework was first proposed there; the present paper explicitly re-derives every equation “for completeness.” No fitted parameters, no uniqueness theorems, no ansatz smuggling, and no renaming of external empirical patterns occur. Illustrations (nonlinear voter, competing-temperature Ising, SIS) simply instantiate the already-derived condition. Scope limitations (networks, multi-state, multi-flip) are stated openly in Sec. 8 and do not affect the internal derivation. Hence the central claim does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The paper is a mean-field theory of binary-state Markov processes. No numerical parameters are fitted; the only free objects are the four arbitrary rate functions X_AB, X_BA, Y_AB, Y_BA and the preference distribution ϕ(p). All claims rest on standard well-mixed rate-equation assumptions plus the modeling choice of exactly two competing mechanisms.

axioms (4)
  • domain assumption Population is well-mixed (complete graph); every agent interacts with the global density a only.
    Stated in Sec. 2; used to close the rate equations (1) and (9). Explicitly listed as a limitation in Sec. 8.1.
  • domain assumption Agents update asynchronously (random sequential updating); only one agent flips at a time.
    Sec. 2; multi-flip and synchronous schemes are excluded (Sec. 8.3).
  • ad hoc to paper Exactly two competing mechanisms; preference p selects between them.
    Core modeling choice of the framework (Sec. 2). Multi-mechanism extensions are noted as future work (Sec. 8.4).
  • domain assumption Preference distribution ϕ(p) is time-independent and identical for annealed and quenched ensembles.
    Used throughout Secs. 6.1–6.2 to interchange integrals and expectations.
invented entities (1)
  • Balancing condition X_AB(a)+X_BA(a)=Y_AB(a)+Y_BA(a) independent evidence
    purpose: Algebraic constraint that forces annealed–quenched equivalence and collapse of heterogeneity.
    Derived in Sec. 6.3 from the general rate equations; not postulated a priori but obtained as the condition under which the last term of Eq. (12) vanishes.

pith-pipeline@v1.1.0-grok45 · 28122 in / 2351 out tokens · 26449 ms · 2026-07-10T20:58:45.999000+00:00 · methodology

0 comments
read the original abstract

We demonstrate how the unified framework for binary-choice dynamics can be used to study the role of annealed and quenched disorders in homogeneous and heterogeneous systems. The framework defines the structure of interactions between agents without imposing their functional forms. Such a high level of generality allows us to connect many different models across disciplines and find universal rules that apply to all of them. Within this framework, agents update their states under the influence of two competing mechanisms chosen according to individual preferences. We review the literature to classify existing models as homogeneous or heterogeneous based on their preference distribution, and we discuss the role of annealed (changing) and quenched (fixed) disorders in modeling these preferences. Using the framework, we derive a constraint on the transition rates. When a model meets this condition, three major things happen: annealed and quenched dynamics become equivalent, any heterogeneous system can be mapped into a homogeneous one, and oscillations cannot emerge. We illustrate these consequences using models from statistical physics, opinion dynamics, and disease spreading. Finally, we discuss the framework limitations and its potential further developments.

Figures

Figures reproduced from arXiv: 2607.06803 by Arkadiusz J\k{e}drzejewski, Jos\'e F. F. Mendes.

Figure 1
Figure 1. Figure 1: Agent i chooses between two options, A and B, under the influence of either mechanism X or Y . The mechanisms are defined by transition probabilities (or rates), which specify the likelihood of switching from A to B (AB) and from B to A (BA). These functions depend on the current fraction of agents choosing option A in the system, denoted by a, and they may take arbitrary forms. Agent i follows mechanism X… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Annealed dynamics: At every time step [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the preference distributions with the same mean preference [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For homogeneous systems, the annealed and quenched dynamics are trivially equivalent since only a single value of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of different variants of the nonlinear voter model with anticonfomrity for four preference distributions [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Phase diagrams for the model with q1 = 4, q2 = 3, T2 = 1 under the quenched dynamics. Each column corresponds to a different distribution of p with the same mean p¯ = 0.5, from left to right: degenerate distribution, logit￾normal distribution (with location and squared scale parameters: µ = 0 and σ 2 = 4), and Bernoulli distribution. Note that for the degenerate distribution, annealed and quenched dynamics… view at source ↗
Figure 7
Figure 7. Figure 7: Time evolution (a,b) and stable fixed points (c) for the SIS model with [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

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Reference graph

Works this paper leans on

109 extracted references · 109 canonical work pages · 2 internal anchors

  1. [1]

    Starnini, F

    M. Starnini, F. Baumann, T. Galla, D. Garcia, G. Iñiguez, M. Karsai, J. Lorenz, K. Sznajd-Weron, Opiniondynamics: Statisticalphysicsandbeyond, Rev.Mod.Phys.(2026)–doi:10.1103/j1zg-ddqv

    work page internal anchor Pith review doi:10.1103/j1zg-ddqv 2026

  2. [2]

    J. P. Gleeson, Binary-state dynamics on complex networks: Pair approximation and beyond, Phys. Rev. X 3 (2013) 021004.doi:10.1103/PhysRevX.3.021004. 18

    work page doi:10.1103/physrevx.3.021004 2013

  3. [3]

    P. L. Garrido, A. Labarta, J. Marro, Stationary nonequilibrium states in the Ising model with locally competing temperatures, J. Stat. Phys. 49 (3) (1987) 551–568.doi:10.1007/BF01009348

    work page doi:10.1007/bf01009348 1987

  4. [4]

    T. Tomé, M. J. de Oliveira, M. A. Santos, Non-equilibrium Ising model with competing Glauber dynamics, J. Phys. A 24 (15) (1991) 3677.doi:10.1088/0305-4470/24/15/033

    work page doi:10.1088/0305-4470/24/15/033 1991

  5. [5]

    Tamayo, F

    P. Tamayo, F. J. Alexander, R. Gupta, Two-temperature nonequilibrium Ising models: Critical be- havior and universality, Phys. Rev. E 50 (1994) 3474–3484.doi:10.1103/PhysRevE.50.3474

    work page doi:10.1103/physreve.50.3474 1994

  6. [6]

    M. J. Keeling, K. T. Eames, Networks and epidemic models, J. R. Soc. Interface. 2 (4) (2005) 295–307. doi:10.1098/rsif.2005.0051

    work page doi:10.1098/rsif.2005.0051 2005

  7. [7]

    K. M. Page, M. A. Nowak, Unifying evolutionary dynamics, J. Theor. Biol. 219 (1) (2002) 93–98. doi:10.1006/jtbi.2002.3112

    work page doi:10.1006/jtbi.2002.3112 2002

  8. [8]

    Byrka, A

    K. Byrka, A. J˛ edrzejewski, K. Sznajd-Weron, R. Weron, Difficulty is critical: The importance of social factors in modeling diffusion of green products and practices, Renew. Sustain. Energy Rev. 62 (2016) 723–735.doi:10.1016/j.rser.2016.04.063

    work page doi:10.1016/j.rser.2016.04.063 2016

  9. [9]

    J˛ edrzejewski, K

    A. J˛ edrzejewski, K. Sznajd-Weron, Statistical Physics Of Opinion Formation: Is it a SPOOF?, C. R. Phys. 20 (4) (2019) 244–261.doi:10.1016/j.crhy.2019.05.002

    work page doi:10.1016/j.crhy.2019.05.002 2019

  10. [10]

    V. C. Yang, M. Galesic, H. McGuinness, A. Harutyunyan, Dynamical system model predicts when so- cial learners impair collective performance, Proc. Natl. Acad. Sci. U.S.A. 118 (35) (2021) e2106292118. doi:10.1073/pnas.2106292118

    work page doi:10.1073/pnas.2106292118 2021

  11. [11]

    J˛ edrzejewski, J

    A. J˛ edrzejewski, J. F. F. Mendes, When does population diversity matter? A unified framework for binary-choice dynamics, Phys. Rev. Lett. 135 (2025) 217401.doi:10.1103/4db2-7dpd

    work page doi:10.1103/4db2-7dpd 2025

  12. [12]

    Gross, U

    T. Gross, U. Feudel, Generalized models as a universal approach to the analysis of nonlinear dynamical systems, Phys. Rev. E 73 (2006) 016205.doi:10.1103/PhysRevE.73.016205

    work page doi:10.1103/physreve.73.016205 2006

  13. [13]

    J. C. Massing, T. Gross, Generalized structural kinetic modeling: A survey and guide, Front. Mol. Biosci. 9 (2022) 825052.doi:10.3389/fmolb.2022.825052

    work page doi:10.3389/fmolb.2022.825052 2022

  14. [14]

    Kuehn, S

    C. Kuehn, S. Siegmund, T. Gross, Dynamical analysis of evolution equations in generalized models, IMA J. Appl. Math. 78 (5) (2013) 1051–1077.doi:10.1093/imamat/hxs008

    work page doi:10.1093/imamat/hxs008 2013

  15. [15]

    Galam, Local dynamics vs

    S. Galam, Local dynamics vs. social mechanisms: A unifying frame, Europhys. Lett. 70 (6) (2005) 705.doi:10.1209/epl/i2004-10526-5

    work page doi:10.1209/epl/i2004-10526-5 2005

  16. [16]

    Galam, Opinion dynamics and unifying principles: A global unifying frame, Entropy 24 (9) (2022)

    S. Galam, Opinion dynamics and unifying principles: A global unifying frame, Entropy 24 (9) (2022). doi:10.3390/e24091201

    work page doi:10.3390/e24091201 2022

  17. [17]

    J˛ edrzejewski, L

    A. J˛ edrzejewski, L. Hernández, Symmetric conformity functions make decision-making processes independent of the distribution of learning strategies, Phys. Rev. Res. 6 (2024) 033093.doi: 10.1103/PhysRevResearch.6.033093

    work page doi:10.1103/physrevresearch.6.033093 2024

  18. [18]

    Sznajd-Weron, J

    K. Sznajd-Weron, J. Szwabiński, R. Weron, Is the person-situation debate important for agent-based modeling and vice-versa?, PLOS ONE 9 (11) (2014) 1–7.doi:10.1371/journal.pone.0112203

    work page doi:10.1371/journal.pone.0112203 2014

  19. [19]

    J˛ edrzejewski, K

    A. J˛ edrzejewski, K. Sznajd-Weron, Nonlinearq-voter model from the quenched perspective, Chaos 30 (1) (2020) 013150.doi:10.1063/1.5134684

    work page doi:10.1063/1.5134684 2020

  20. [20]

    J˛ edrzejewski, K

    A. J˛ edrzejewski, K. Sznajd-Weron, Person-situation debate revisited: Phase transitions with quenched and annealed disorders, Entropy 19 (8) (2017) 415.doi:10.3390/e19080415. 19

    work page doi:10.3390/e19080415 2017

  21. [21]

    Pradhan, P

    A. Pradhan, P. Mullick, P. Sen, Analyzing contrarian behavior using nonlinear biasedq-voter model, Phys. Rev. E 113 (2026) 034301.doi:10.1103/dwpm-66dj

    work page doi:10.1103/dwpm-66dj 2026

  22. [22]

    Przybyła, K

    P. Przybyła, K. Sznajd-Weron, R. Weron, Diffusion of innovation within an agent-based model: Spin- sons, independence and advertising, Adv. Complex Syst. 17 (01) (2014) 1450004.doi:10.1142/ S0219525914500040

    work page 2014

  23. [23]

    Y. Liu, X. Chen, Evolutionary dynamics of collective decision-making with local social influence on static and dynamic networks, Inf. Fusion. (2026) 104568doi:10.1016/j.inffus.2026.104568

    work page doi:10.1016/j.inffus.2026.104568 2026

  24. [24]

    Weron, A

    T. Weron, A. Kowalska-Pyzalska, R. Weron, The role of educational trainings in the diffusion of smart metering platforms: An agent-based modeling approach, Physica A 505 (2018) 591–600.doi:https: //doi.org/10.1016/j.physa.2018.03.086

    work page doi:10.1016/j.physa.2018.03.086 2018

  25. [25]

    Kowalska-Pyzalska, K

    A. Kowalska-Pyzalska, K. Maciejowska, K. Suszczyński, K. Sznajd-Weron, R. Weron, Turning green: Agent-based modeling of the adoption of dynamic electricity tariffs, Energy Policy 72 (2014) 164–174. doi:https://doi.org/10.1016/j.enpol.2014.04.021

    work page doi:10.1016/j.enpol.2014.04.021 2014

  26. [26]

    Brauer, Compartmental Models in Epidemiology, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008, pp

    F. Brauer, Compartmental Models in Epidemiology, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008, pp. 19–79.doi:10.1007/978-3-540-78911-6_2

    work page doi:10.1007/978-3-540-78911-6_2 2008

  27. [27]

    Barrat, M

    A. Barrat, M. Barthelemy, A. Vespignani, Dynamical processes on complex networks, Cambridge university press, 2008

    work page 2008

  28. [29]

    T. J. H. Morgan, L. E. Rendell, M. Ehn, W. Hoppitt, K. N. Laland, The evolutionary basis of human social learning, Proc. Royal Soc. B 279 (1729) (2012) 653–662.doi:10.1098/rspb.2011.1172

    work page doi:10.1098/rspb.2011.1172 2012

  29. [30]

    Hoppitt, K

    W. Hoppitt, K. N. Laland, Social Learning: An Introduction to Mechanisms, Methods, and Models, Princeton University Press, 2013. URLhttp://www.jstor.org/stable/j.ctt2jc8mh

    work page 2013

  30. [31]

    M. B. Donnellan, R. E. Lucas, W. Fleeson, Introduction to personality and assessment at age 40: Reflections on the legacy of the person–situation debate and the future of person–situation integration, J. Res. Pers. 43 (2) (2009) 117–119.doi:10.1016/j.jrp.2009.02.010

    work page doi:10.1016/j.jrp.2009.02.010 2009

  31. [32]

    Rendell, L

    L. Rendell, L. Fogarty, W. J. Hoppitt, T. J. Morgan, M. M. Webster, K. N. Laland, Cognitive culture: Theoretical and empirical insights into social learning strategies, Trends Cogn. Sci. 15 (2) (2011) 68–76. doi:10.1016/j.tics.2010.12.002

    work page doi:10.1016/j.tics.2010.12.002 2011

  32. [33]

    McElreath, A

    R. McElreath, A. V. Bell, C. Efferson, M. Lubell, P. J. Richerson, T. Waring, Beyond existence and aiming outside the laboratory: estimating frequency-dependent and pay-off-biased social learning strategies, Philos. Trans. R. Soc. B 363 (1509) (2008) 3515–3528.doi:10.1098/rstb.2008.0131

    work page doi:10.1098/rstb.2008.0131 2008

  33. [34]

    D. L. Stein, C. M. Newman, Spin Glasses and Complexity, Princeton University Press, 2013. URLhttp://www.jstor.org/stable/j.ctt12f4hf

    work page 2013

  34. [35]

    Galam, S

    S. Galam, S. Moscovici, Towards a theory of collective phenomena: Consensus and attitude changes in groups, Eur. J. Soc. Psychol. 21 (1) (1991) 49–74.doi:10.1002/ejsp.2420210105

    work page doi:10.1002/ejsp.2420210105 1991

  35. [36]

    Galam, Rational group decision making: A random field Ising model atT= 0, Physica A 238 (1) (1997) 66–80.doi:10.1016/S0378-4371(96)00456-6

    S. Galam, Rational group decision making: A random field Ising model atT= 0, Physica A 238 (1) (1997) 66–80.doi:10.1016/S0378-4371(96)00456-6. 20

    work page doi:10.1016/s0378-4371(96)00456-6 1997

  36. [37]

    Efferson, R

    C. Efferson, R. Lalive, P. J. Richerson, R. McElreath, M. Lubell, Conformists and mavericks: the empirics of frequency-dependent cultural transmission, Evol. Hum. Behav. 29 (1) (2008) 56–64.doi: 10.1016/j.evolhumbehav.2007.08.003

    work page doi:10.1016/j.evolhumbehav.2007.08.003 2008

  37. [38]

    J.M.Gonzalez-Miranda, P.L.Garido, J.Marro, J.L.Lebowitz, NonequilibriumphasediagramofIsing model with competing dynamics, Phys. Rev. Lett. 59 (1987) 1934–1937.doi:10.1103/PhysRevLett. 59.1934

    work page doi:10.1103/physrevlett 1987

  38. [39]

    T. Tomé, M. J. de Oliveira, Self-organization in a kinetic Ising model, Phys. Rev. A 40 (1989) 6643– 6646.doi:10.1103/PhysRevA.40.6643

    work page doi:10.1103/physreva.40.6643 1989

  39. [40]

    R. A. Dumer, M. Godoy, Metastable states in the Ising model with Glauber-Kawasaki competing dynamics, Phys. Rev. E 110 (2024) 024315.doi:10.1103/PhysRevE.110.024315

    work page doi:10.1103/physreve.110.024315 2024

  40. [41]

    Szolnoki, Phase transitions in the kinetic Ising model with competing dynamics, Phys

    A. Szolnoki, Phase transitions in the kinetic Ising model with competing dynamics, Phys. Rev. E 62 (2000) 7466–7469.doi:10.1103/PhysRevE.62.7466

    work page doi:10.1103/physreve.62.7466 2000

  41. [42]

    R. A. Dumer, M. Godoy, Nonequilibrium Ising model on a two-dimensional additive small-world network, Phys. Rev. E 107 (2023) 044115.doi:10.1103/PhysRevE.107.044115

    work page doi:10.1103/physreve.107.044115 2023

  42. [43]

    R. A. Dumer, D. R. da Costa, M. Godoy, Classical three-dimensional heisenberg model with competing dynamics, Phys. Rev. E 112 (2025) 044119.doi:10.1103/n6z3-1m6g

    work page doi:10.1103/n6z3-1m6g 2025

  43. [44]

    M. J. de Oliveira, Isotropic majority-vote model on a square lattice, J. Stat. Phys. 66 (1992) 273–281. doi:10.1007/BF01060069

    work page doi:10.1007/bf01060069 1992

  44. [45]

    Crochik, T

    L. Crochik, T. Tomé, Entropy production in the majority-vote model, Phys. Rev. E 72 (2005) 057103. doi:10.1103/PhysRevE.72.057103

    work page doi:10.1103/physreve.72.057103 2005

  45. [46]

    J. J. Schneider, The influence of contrarians and opportunists on the stability of a democracy in the Sznajd model, Int. J. Mod. Phys. C 15 (05) (2004) 659–674.doi:10.1142/S012918310400611X

    work page doi:10.1142/s012918310400611x 2004

  46. [47]

    M. S. de la Lama, J. M. López, H. S. Wio, Spontaneous emergence of contrarian-like behaviour in an opinion spreading model, EPL 72 (5) (2005) 851.doi:10.1209/epl/i2005-10299-3

    work page doi:10.1209/epl/i2005-10299-3 2005

  47. [48]

    Nowak, K

    B. Nowak, K. Sznajd-Weron, Homogeneous symmetrical threshold model with nonconformity: Inde- pendence versus anticonformity, Complexity 2019 (2019) 1.doi:10.1155/2019/5150825

    work page doi:10.1155/2019/5150825 2019

  48. [49]

    Nowak, M

    B. Nowak, M. Grabisch, K. Sznajd-Weron, Threshold model with anticonformity under random se- quential updating, Phys. Rev. E 105 (2022) 054314.doi:10.1103/PhysRevE.105.054314

    work page doi:10.1103/physreve.105.054314 2022

  49. [50]

    Nowak, K

    B. Nowak, K. Sznajd-Weron, Symmetrical threshold model with independence on random graphs, Phys. Rev. E 101 (2020) 052316.doi:10.1103/PhysRevE.101.052316

    work page doi:10.1103/physreve.101.052316 2020

  50. [51]

    Castellano, M

    C. Castellano, M. A. Muñoz, R. Pastor-Satorras, Nonlinearq-voter model, Phys. Rev. E 80 (2009) 041129.doi:10.1103/PhysRevE.80.041129

    work page doi:10.1103/physreve.80.041129 2009

  51. [52]

    A. F. Peralta, A. Carro, M. San Miguel, R. Toral, Analytical and numerical study of the non-linear noisy voter model on complex networks, Chaos 28 (7) (2018) 075516.doi:10.1063/1.5030112

    work page doi:10.1063/1.5030112 2018

  52. [53]

    Muslim, J

    R. Muslim, J. Kim, N. Oikawa, A. R. NQZ, Z. Akbar, Ordering-disordering dynamics of theq-voter model under random external bias, Phys. Rev. E 112 (2025) 034312.doi:10.1103/hcy5-pnwf

    work page doi:10.1103/hcy5-pnwf 2025

  53. [54]

    Nyczka, K

    P. Nyczka, K. Byrka, P. R. Nail, K. Sznajd-Weron, Conformity in numbers—does criticality in social responses exist?, PLOS ONE 13 (12) (2018) 1–18.doi:10.1371/journal.pone.0209620

    work page doi:10.1371/journal.pone.0209620 2018

  54. [55]

    A. R. Vieira, C. Anteneodo, Thresholdq-voter model, Phys. Rev. E 97 (2018) 052106.doi:10.1103/ PhysRevE.97.052106. 21

    work page 2018

  55. [56]

    A. R. Vieira, A. F. Peralta, R. Toral, M. S. Miguel, C. Anteneodo, Pair approximation for the noisy thresholdq-voter model, Phys. Rev. E 101 (2020) 052131.doi:10.1103/PhysRevE.101.052131

    work page doi:10.1103/physreve.101.052131 2020

  56. [57]

    Nyczka, K

    P. Nyczka, K. Sznajd-Weron, Anticonformity or independence?—insights from statistical physics, J. Stat. Phys. 151 (1) (2013) 174–202.doi:10.1007/s10955-013-0701-4

    work page doi:10.1007/s10955-013-0701-4 2013

  57. [58]

    Nyczka, K

    P. Nyczka, K. Sznajd-Weron, J. Cisło, Phase transitions in theq-voter model with two types of stochastic driving, Phys. Rev. E 86 (2012) 011105.doi:10.1103/PhysRevE.86.011105

    work page doi:10.1103/physreve.86.011105 2012

  58. [59]

    A. F. Peralta, M. Neri, J. Kertész, G. Iñiguez, Effect of algorithmic bias and network structure on coexistence, consensus, and polarization of opinions, Phys. Rev. E 104 (2021) 044312.doi:10.1103/ PhysRevE.104.044312

    work page 2021

  59. [60]

    Doniec, P

    M. Doniec, P. Mullick, P. Sen, K. Sznajd-Weron, Modeling biases in binary decision-making within the generalized nonlinearq-voter model, Chaos 35 (4) (2025) 043133.doi:10.1063/5.0266510

    work page doi:10.1063/5.0266510 2025

  60. [61]

    the hung elections scenario

    S. Galam, Contrarian deterministic effects on opinion dynamics: “the hung elections scenario”, Physica A 333 (2004) 453–460.doi:10.1016/j.physa.2003.10.041

    work page doi:10.1016/j.physa.2003.10.041 2004

  61. [62]

    Galam, Ratio-dependent contrarian activation in opinion dynamics, Entropy 28 (4) (2026).doi: 10.3390/e28040443

    S. Galam, Ratio-dependent contrarian activation in opinion dynamics, Entropy 28 (4) (2026).doi: 10.3390/e28040443

    work page doi:10.3390/e28040443 2026

  62. [63]

    Borghesi, S

    C. Borghesi, S. Galam, Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect, Phys. Rev. E 73 (2006) 066118.doi:10.1103/PhysRevE.73.066118

    work page doi:10.1103/physreve.73.066118 2006

  63. [64]

    A. L. Vilela, F. Moreira, A. J. de Souza, Majority-vote model with a bimodal distribution of noises, Physica A 391 (24) (2012) 6456–6462.doi:10.1016/j.physa.2012.07.068

    work page doi:10.1016/j.physa.2012.07.068 2012

  64. [65]

    A. L. Vilela, A. J. de Souza, Majority-vote model with a bimodal distribution of noises in small-world networks, Physica A 488 (2017) 216–223.doi:https://doi.org/10.1016/j.physa.2017.06.029

    work page doi:10.1016/j.physa.2017.06.029 2017

  65. [66]

    M. A. Javarone, Social influences in opinion dynamics: The role of conformity, Physica A 414 (2014) 19–30.doi:10.1016/j.physa.2014.07.018

    work page doi:10.1016/j.physa.2014.07.018 2014

  66. [67]

    Krawiecki, Spin-glass-like transition in the majority-vote model with anticonformists, Eur

    A. Krawiecki, Spin-glass-like transition in the majority-vote model with anticonformists, Eur. Phys. J. B 91 (2018) 1–7.doi:10.1140/epjb/e2018-80551-9

    work page doi:10.1140/epjb/e2018-80551-9 2018

  67. [68]

    A. L. Vilela, C. Wang, K. P. Nelson, H. E. Stanley, Majority-vote model for financial markets, Physica A 515 (2019) 762–770.doi:https://doi.org/10.1016/j.physa.2018.10.007

    work page doi:10.1016/j.physa.2018.10.007 2019

  68. [69]

    M. F. B. Granha, A. L. M. Vilela, C. Wang, K. P. Nelson, H. E. Stanley, Opinion dynamics in financial markets via random networks, Proc. Natl. Acad. Sci. U.S.A. 119 (49) (2022) e2201573119. doi:10.1073/pnas.2201573119

    work page doi:10.1073/pnas.2201573119 2022

  69. [70]

    I. V. Oliveira, C. Wang, G. Dong, R. Du, C. E. Fiore, A. L. Vilela, H. E. Stanley, Entropy production on cooperative opinion dynamics, Chaos Solit. Fractals. 181 (2024) 114694.doi:https://doi.org/ 10.1016/j.chaos.2024.114694

    work page doi:10.1016/j.chaos.2024.114694 2024

  70. [71]

    J˛ edrzejewski, K

    A. J˛ edrzejewski, K. Sznajd-Weron, Pair approximation for theq-voter models with quenched disorder on networks, Phys. Rev. E 105 (2022) 064306.doi:10.1103/PhysRevE.105.064306

    work page doi:10.1103/physreve.105.064306 2022

  71. [72]

    M. A. Javarone, T. Squartini, Conformism-driven phases of opinion formation on heterogeneous networks: Theq-voter model case, J. Stat. Mech.: Theory Exp. 2015 (10) (2015) P10002.doi: 10.1088/1742-5468/2015/10/P10002

    work page doi:10.1088/1742-5468/2015/10/p10002 2015

  72. [73]

    Nowak, B

    B. Nowak, B. Stoń, K. Sznajd-Weron, Discontinuous phase transitions in the multi-state noisy q-voter model: Quenched vs. annealed disorder, Sci. Rep. 11 (1) (2021) 6098.doi:10.1038/ s41598-021-85361-9. 22

    work page 2021

  73. [74]

    Nowak, K

    B. Nowak, K. Sznajd-Weron, Switching from a continuous to a discontinuous phase transition under quenched disorder, Phys. Rev. E 106 (2022) 014125.doi:10.1103/PhysRevE.106.014125

    work page doi:10.1103/physreve.106.014125 2022

  74. [75]

    Grabisch, F

    M. Grabisch, F. Li, Anti-conformism in the threshold model of collective behavior, Dyn. Games Appl. 10 (2) (2020) 444–477.doi:10.1007/s13235-019-00332-0

    work page doi:10.1007/s13235-019-00332-0 2020

  75. [76]

    J. S. Juul, M. A. Porter, Hipsters on networks: How a minority group of individuals can lead to an antiestablishment majority, Phys. Rev. E 99 (2019) 022313.doi:10.1103/PhysRevE.99.022313

    work page doi:10.1103/physreve.99.022313 2019

  76. [77]

    Czaplicka, C

    A. Czaplicka, C. Charalambous, R. Toral, M. San Miguel, Biased-voter model: How persuasive a small group can be?, Chaos Solit. Fractals 161 (2022) 112363.doi:https://doi.org/10.1016/j.chaos. 2022.112363

    work page doi:10.1016/j.chaos 2022

  77. [78]

    Galam, F

    S. Galam, F. Jacobs, The role of inflexible minorities in the breaking of democratic opinion dynamics, Physica A 381 (2007) 366–376.doi:10.1016/j.physa.2007.03.034

    work page doi:10.1016/j.physa.2007.03.034 2007

  78. [79]

    Mobilia, Does a single zealot affect an infinite group of voters?, Phys

    M. Mobilia, Does a single zealot affect an infinite group of voters?, Phys. Rev. Lett. 91 (2003) 028701. doi:10.1103/PhysRevLett.91.028701

    work page doi:10.1103/physrevlett.91.028701 2003

  79. [80]

    Stauffer, J

    D. Stauffer, J. Sá Martins, Simulation of Galam’s contrarian opinions on percolative lattices, Physica A 334 (3) (2004) 558–565.doi:10.1016/j.physa.2003.12.003

    work page doi:10.1016/j.physa.2003.12.003 2004

  80. [82]

    S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, CRC press, Boca Raton, 2018

    work page 2018

Showing first 80 references.