{"id":"1c39f1ce-f2f9-4a8d-aa5d-ae2939abe729","arxiv_id":"2502.10172","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A generalized q-voter model with asymmetric state-dependent flips exhibits a new coexistence phase for peer groups larger than three and a plateau in the exit probability for small systems.","lead":"This paper extends the q-voter model of opinion dynamics by adding two different flip probabilities for supporters and opponents when a peer group is not unanimous, and analyzes the resulting phases. It finds a new regime for large peer groups where full and partial adoption coexist, and a surprising small-group effect where more initial support does not raise the chance of full adoption.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (22) sets the c=1/2 stability threshold at (q-1)/(2q-2)=1/2 for all q, making the paper's inequality for phase B (ε_c<1/q) false for every q≥2 and contradicting its own q>3 phase-D claim; the threshold from Eq. (19) is actually (q-1)/[2(2^{q-1}-1)].","rationale":"I focused on the analytical stability criterion because the paper's phase diagram, and hence the claimed new phase D, is derived from it. The reader's concern about finite-size corrections in the Markov chain is reasonable, but the authors already compare the infinite-N transition-rate approximation to Monte Carlo simulations at N=64 and show qualitative agreement; that concern is not the weakest link. The Eq. (22) error is a demonstrable internal inconsistency: the stated threshold is constant 1/2, making the subsequent inequality impossible for any q≥2, in direct conflict with the paper's own figures and central claim. No ad hominem is intended; this is a checkable algebraic defect. The correct threshold computed from the paper's own Eq. (19) still yields phase B/D for q>3, so the central scientific claim is not refuted, but the manuscript must correct Eq. (22), the black-dot description, and the inequality argument in Section IV.C. Thus the verdict remains CONDITIONAL with an added required fix. I disagree with the reader's identification of the weakest assumption: the finite-size issue is secondary compared to the explicit algebraic error in the stability analysis.","tokens_in":16421,"tokens_out":14636,"duration_ms":129838,"concrete_test":"Evaluate Eq. (19) at c=1/2 with ε↑=ε↓=ε, set the derivative to zero, and solve for ε; confirm whether the threshold equals (q-1)/(2q-2) or (q-1)/[2(2^{q-1}-1)]. If the latter, recompute the q=5 phase boundaries in Fig. 3(a) with the corrected threshold and verify that phase B occupies the diagonal interval (q-1)/[2(2^{q-1}-1)] < ε < 1/q; if the phase diagram matches the corrected threshold, the paper's Eq. (22) must be revised but the phase-D claim stands.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that a new phase D appears for q>3 rests on the existence of phases B and D, which require a window of ε where the central partially-adopted fixed point is stable while at least one absorbing state is stable. On the diagonal ε↑=ε↓=ε, the text (Section IV.B and IV.C) states that c=1/2 becomes stable for ε > (q-1)/(2q-2), and that phases B and D are absent for q=2,3 because (q-1)/(2q-2) < 1/q fails. But (q-1)/(2q-2)=1/2 for every q≠1, so this inequality fails for every q≥2; under Eq. (22) there would be no phase B or D for any group size, directly contradicting the headline result. Direct differentiation of Eq. (19) at c=1/2, ε↑=ε↓=ε gives d/dc(dc/dt) = (q-1)2^{1-q} - 2ε(1-2^{1-q}), so the correct stability threshold is ε_c = (q-1)/[2(2^{q-1}-1)] (e.g., q=4: 3/14; q=5: 4/30 ≈ 0.133). With this correction, ε_c < 1/q holds for q≥4, restoring phases B and D for q>3. However, the paper's printed Eq. (22) and the surrounding inequality are arithmetically wrong and internally inconsistent with Fig. 2(e), where q=5 and ε=0.21 (<0.5) already shows a stable partially adopted state. The analytical scaffolding of the phase diagram therefore needs correction, even though the qualitative central claim appears to survive.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16840,"tokens_out":10912,"duration_ms":87306,"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":[{"comment":"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.","section":"Section IV.B, Eq. (22)"},{"comment":"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.","section":"Section IV.C, phase diagram discussion"},{"comment":"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.","section":"Section IV.A, Eq. (11)"},{"comment":"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.","section":"Section IV.D, Markov chain construction"}],"minor_comments":[{"comment":"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.","section":"Figure 5 caption"},{"comment":"The text 'ε↑ = ε↑ > 1/q (symmetric case)' should read 'ε↑ = ε↓ > 1/q (symmetric case)'.","section":"Section IV.C, phase E description"},{"comment":"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'.","section":"Section V, first paragraph"},{"comment":"The data availability statement mentions a public GitHub repository but does not provide a URL or repository identifier; please add the link.","section":"Data availability statement"}],"recommendation":"major_revision","confidential_remarks":"The error in Eq. (22) appears to be a typographical slip rather than a fundamental conceptual failure, because the correct threshold follows immediately from the paper's own Eq. (19) and the simulation and figure results are consistent with the corrected value. The qualitative phase D and exit-probability plateau claims are plausible and supported by the Monte Carlo data, so the paper is worth serious revision rather than rejection. The authors should be asked to correct the threshold, revisit all statements that depend on it, and clarify the finite-size approximation in the Markov chain calculation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends the q-voter model by making the non-unanimous flip probability depend on the target agent's current state, giving two asymmetric parameters ε↑ and ε↓. That is a natural and useful generalization, and the limiting cases (ε↑=ε↓ recovers the original q-voter; ε↓=0 recovers the mass-media model) provide good external cross-checks. The new phase D for q>3, where an adopted and a partially adopted state are simultaneously stable, and the plateau in the exit probability for q≥3 in small systems are genuinely not present in the cited prior work, including the closest model by Mullick and Sen. The mean-field rate equation and the Markov-chain exit-probability calculation are standard and clearly presented, and the Monte Carlo simulations support the qualitative phase diagram.\n\nThe soft spot is real and load-bearing. Eq. (22) states that the c=1/2 state becomes stable for ε↑ > (q-1)/(2q-2), but (q-1)/(2q-2) = 1/2 for every q>1. The surrounding text claims this inequality is satisfied for q>3, which is false. Differentiating Eq. (19) directly gives the correct threshold ε_c = (q-1)/[2(2^{q-1}-1)], which is below 1/q for q≥4 and therefore restores phases B and D for q>3, consistent with Fig. 2(e). So the printed equation contradicts the paper's own figure, and the analytical scaffolding needs correction even though the central qualitative claim survives. This is a fixable error, but it must be fixed.\n\nTwo smaller points. The finite-size Markov chain uses transition rates taken from the infinite-N limit; that is a reasonable approximation but should be stated explicitly, and the authors should quantify the O(1/N) corrections for N=64. And the 'unique' exit probability claim should be tempered unless the authors compare with the biased voter models cited in their own introduction; a plateau might well appear there too.\n\nThe citation pattern looks fair, and the authors explicitly distinguish their results from the closest existing model. This paper is for the sociophysics community, especially people working on q-voter variants. It deserves a serious referee: the model is clean, the new phase and plateau are worth reporting, and the errors are correctable. I would send it to review and ask the authors to fix Eq. (22), the inequality discussion, and the finite-size caveat.","headline":"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.","tokens_in":17373,"tokens_out":3689,"would_cite":true,"duration_ms":33835,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["q-voter model","opinion dynamics","binary decision-making","asymmetric bias","exit probability","phase coexistence","complete graph","collective adoption"],"falsifier":"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.","tokens_in":16242,"feed_emoji":"🗳️","tokens_out":6857,"duration_ms":64189,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bigger influence groups make full and partial adoption coexist","feed_subtitle":"In small groups with q>=3, extra initial support stops raising the odds of full consensus.","key_machinery":"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).","core_discovery":"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 $\\varepsilon_\\uparrow=\\varepsilon_\\downarrow=1/2$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original q-voter model, recovered here at $\\varepsilon_\\uparrow=\\varepsilon_\\downarrow$, and the unanimity-based conformity mechanism that the generalized model extends.","marker":"[13]"},{"why":"Supplies the mass-media q-voter variant, recovered at $\\varepsilon_\\downarrow=0$, and the comparison showing that phase D is absent in that earlier model.","marker":"[16]"},{"why":"Provides the weighted-influence q-voter model with which the present model is identical for $q=2$, serving as the baseline for exit-probability comparison.","marker":"[48]"},{"why":"Supplies the rate-equation formalism with $\\gamma_+$ and $\\gamma_-$ decomposition used to build the infinite-size transition probabilities in Eq. (10).","marker":"[31]"},{"why":"Supplies the Markov-chain aggregation and fundamental-matrix method used to compute the exit probability for finite systems.","marker":"[52]"}],"fun_headline_variants":["Coexisting full and partial adoption emerges when q>3","In small systems, extra initial support doesn't improve adoption odds","Group size q>3 gives rise to stable mixed adoption","Exit probability plateau: initial support no longer helps for q≥3","Group size effects: coexistence for q>3 and plateau for small systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Coexisting full and partial adoption emerges when q>3","In small systems, extra initial support doesn't improve adoption odds","Group size q>3 gives rise to stable mixed adoption","Exit probability plateau: initial support no longer helps for q≥3","Group size effects: coexistence for q>3 and plateau for small systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001183,"raw_usage":{"total_tokens":4964,"prompt_tokens":1100,"completion_tokens":3864,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":716,"completion_tokens_details":{"reasoning_tokens":3777}},"tokens_in":716,"tokens_out":3864,"duration_ms":28852,"temperature":1.0,"reasoning_tokens":3777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T19:08:30.881773+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Castellano , author M","cited_arxiv_id":null,"evidence_quote":"Supplies the original q-voter model, recovered here at $\\varepsilon_\\uparrow=\\varepsilon_\\downarrow$, and the unanimity-based conformity mechanism that the generalized model extends."},{"cited_title":"Muslim , author R","cited_arxiv_id":null,"evidence_quote":"Supplies the mass-media q-voter variant, recovered at $\\varepsilon_\\downarrow=0$, and the comparison showing that phase D is absent in that earlier model."},{"cited_title":"Mullick \\ and\\ author P","cited_arxiv_id":null,"evidence_quote":"Provides the weighted-influence q-voter model with which the present model is identical for $q=2$, serving as the baseline for exit-probability comparison."},{"cited_title":"Nyczka , author K","cited_arxiv_id":null,"evidence_quote":"Supplies the rate-equation formalism with $\\gamma_+$ and $\\gamma_-$ decomposition used to build the infinite-size transition probabilities in Eq. (10)."},{"cited_title":"Banisch ,\\ @noop title Markov chain aggregation for agent-based models \\ ( publisher Springer ,\\ year 2015 ) NoStop","cited_arxiv_id":null,"evidence_quote":"Supplies the Markov-chain aggregation and fundamental-matrix method used to compute the exit probability for finite systems."}],"review_version":1}