{"id":"373acf03-4c43-4eed-80dc-ccf565c94787","arxiv_id":"2608.08324","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In biased q-voter models, persistence probability saturates when the dynamics reach consensus and decays exponentially when they do not, yielding phase diagrams that reproduce the fixed point structure.","lead":"This paper computes the persistence probability, the chance that an agent keeps its initial opinion, in two biased q-voter opinion dynamics models. It finds that persistence saturates exactly when the dynamics reach a consensus state and decays exponentially otherwise, so the persistence phase diagram mirrors the models' fixed point structure.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Phase boundary at c* rests on an unverified N→∞ assumption; the paper reports no system sizes, run counts, or error bars, so finite-size smearing near the separatrix cannot be assessed.","rationale":"The mean-field derivation is internally sound: in the N→∞ limit, Eq. (6) shows that P+(∞) is positive exactly when the deterministic trajectory approaches c=1, so the central correlation with consensus fixed points holds. The load-bearing weakness is the finite-size verification of the phase boundary, exactly as the reader's weakest_assumption states. The paper provides no system sizes, run counts, or error bars, and near the separatrix the analytical step function is smeared over a width of order N^{-1/2}. Additionally, P+(∞) vanishes continuously as ε↓→1/q, making the vertical boundary numerically undetectable without a threshold criterion. These are addressable limitations rather than fatal flaws, so the conditional verdict is unchanged.","tokens_in":17514,"tokens_out":16476,"duration_ms":154446,"concrete_test":"For DMSS q=3 at ε↑=0.1, ε↓=0.1 (bistable), compute c* from Eq. (26). Run Monte Carlo for N=10^3, 10^4, 10^5 with c0=c*+δ for δ=0.05, 0.01, 0.001 and at least 10^4 realizations each; measure the fraction of runs ending in positive consensus and the mean P+(∞). Compare with the analytical value from Eq. (28). If the fraction is not near 1 for δ≳N^{-1/2}, or the mean P+(∞) deviates from the analytical value by more than 1/√N, the boundary at c0=c* is only asymptotic, and the article must report N and error bars before the phase diagram can be accepted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section V identifies the persistence phase boundary for q=3 with the deterministic unstable fixed point c* (Eq. 57). This identification requires that, in the N→∞ limit, a system prepared at c0>c* always reaches positive consensus, so P+(∞) equals its positive saturation value and is zero for c0<c*. For any finite N, however, fluctuations can carry the density across the separatrix into the negative-consensus basin, so the average P+(∞) near c* is a convolution of the step function with a kernel of width ~N^{-1/2}; the apparent boundary shifts and smears. The paper states it verified numerically 'including values of c0 very close to c*', but reports no system size, number of runs, or error bars (Sections IV-V, Figs. 4-7), so this verification cannot be checked. The problem is compounded because P+(∞) also vanishes continuously as ε↓→1/q from below, making the vertical boundary at 1/q unobservable below any finite threshold in a simulation. Thus the sharp phase diagram is a thermodynamic-limit statement, and the simulation evidence as presented does not by itself establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies persistence probabilities P+(t) and P-(t), defined as the fractions of agents that have never changed opinion, in two biased q-voter models on fully connected networks: the DMSS generalized q-voter model with parameters (ε↑, ε↓) and the MS weighted-influence model with parameter p. In the N→∞ mean-field limit the authors write exact rate equations for P±, Eq. (3), integrate them using the deterministic evolution of the positive-opinion density c(t), and obtain closed-form persistence expressions for q=2, q=3, and q→∞ for both models. Monte Carlo data are compared with these expressions for selected parameters (Figs. 1-5), and exponential fits or saturation plateaus are used to construct 'persistence phase diagrams' in which P+(∞) is positive or zero. For the q=3 DMSS model the boundary is argued to be the deterministic unstable fixed point c*, Eq. (57), with a vertical boundary at ε↓=1/q; for the MS model the boundary is p=1/2. The central claim is that persistence saturation is strongly correlated with the existence of a stable consensus fixed point.","tokens_in":17683,"tokens_out":17657,"duration_ms":168337,"significance":"If the claims are correct, the paper provides a nontrivial example where a first-passage quantity (persistence) is controlled by the deterministic fixed-point structure of the mean-field dynamics, in contrast to voter-model-like persistence. The mean-field persistence equations are exact for the fully connected geometry, and the closed forms for q=2, 3, and ∞, including the reductions to the voter-model and original q-voter limits, are useful additions. The paper also makes falsifiable predictions for the phase boundary Eq. (57) and for the exponential decay rates. However, the numerical support for the phase diagram is currently under-reported, and the manuscript does not specify the limiting procedure needed to define P+(∞) in a finite system.","major_comments":[{"comment":"The paper does not specify the limiting procedure behind the statement that P+(∞)=0 in phases without a stable consensus. Equation (3) is an N→∞ mean-field equation, and the decay to zero follows by letting t→∞ after N→∞. In any finite N the q-voter dynamics is absorbed into one of the consensus states with probability one, so an agent of the initial majority type that survives to consensus contributes a positive P+(∞) with nonzero probability; hence the true infinite-time limit in finite N is not zero. The numerical observation of 'decay to zero' in non-consensus phases is therefore a finite-time transient whose duration diverges with N. The manuscript should state the order of limits and report the system sizes and observation times; otherwise the phase diagram in Section V is not well defined.","section":"Sections III and V; Eq. (3)"},{"comment":"The claimed numerical verification of the boundary c0=c* is not backed by quantitative information. The manuscript states that P+(∞) 'remains finite for all initial conditions satisfying c0>c*, including values of c0 very close to c*', but it does not report N, the number of independent runs, error bars, or the criterion used to distinguish saturation from a slow exponential decay. In fact, for c0>c*, the integral in Eq. (6) has a logarithmic divergence as c0 approaches c* from above (since ω_{+→-}(c*)>0 and ċ(c)∼(c-c*)), so P+(∞)∼(c0-c*)^{ω(c*)/K} tends to zero continuously. Thus near the theoretical boundary the saturation value is arbitrarily small and may fall below the numerical resolution; a finite-size scaling analysis is required to substantiate the boundary. The same issue applies to the vertical boundary at ε↓=1/q and to the MS model near p=1/2.","section":"Section V, Eqs. (28) and (57)"},{"comment":"For q=5, no closed-form persistence expressions are available, and the persistence phase diagram is based entirely on Monte Carlo heatmaps and on numerical solution of the deterministic rate equation for c. As with the q=3 case, no system size, number of runs, or error bars are given, and the heatmap color scale is not described. The q=5 phase boundary is therefore not quantitatively established; the figure should either be supplemented with simulation details and a threshold criterion, or presented as an illustrative scan rather than a determined phase boundary.","section":"Section V, Fig. 7"}],"minor_comments":[{"comment":"The MS model description contains the typo 'unanibous' and the duplicated 'the the'; please correct.","section":"Section II"},{"comment":"'Exolicitly' should be 'Explicitly'.","section":"Section III.A.1"},{"comment":"'two basin of attraction' should be 'two basins of attraction'.","section":"Section III.B.1"},{"comment":"The formula should be written with parentheses: ε↑ = (3c0 ε↓ + 1 − 2c0)/(3 − 3c0); as printed, the fraction is ambiguous.","section":"Section V, Eq. (57)"},{"comment":"Please define the time unit used in the Monte Carlo simulations (single agent update vs full sweep), since Eq. (3) is a continuous-time rate equation and the comparison with simulation data depends on this convention.","section":"Section III.A and Figures 1-3"},{"comment":"Add colorbars and state how P+(t→∞) is estimated from finite-time data, for example by giving a plateau criterion or a fitting threshold.","section":"Figures 6 and 7"},{"comment":"The parenthetical remark that the voter-model exponents in Ref. [14] contain 'probably an oversight' should be either substantiated with a derivation or removed.","section":"Section III.A.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely useful paper. It does something new—persistence probability in q-voter models—and mostly earns its claims. The central result, that P±(t) saturates exactly when the mean-field dynamics reach consensus and decays otherwise, is credible and well supported. I spot-checked several special-case reductions (q=2 DMSS symmetric limit, ε↑+ε↓=1, ε↓=0; MS p=1/2) and they reduce correctly. The closed-form mean-field expressions for q=2, 3 and q→∞ are new, and the q→∞ DMSS limit is explicitly flagged as not reported in the earlier model paper, which I appreciate. The paper is also honest about what is simulation-only: q=5 phase diagram, and the phase boundary identification with the unstable fixed point.\n\nWhere it's soft: the paper never reports system size, run count, or error bars. For a fully connected network this isn't fatal—the mean-field rate equations are exact in the N→∞ limit and the analytic curves match the simulation data in the figures—but it does undermine the sharpness of the claimed phase diagram. The boundary c0 = c* (Eq. 57) is a step function in the thermodynamic limit; for finite N, fluctuations will smear it over a width ~N^{-1/2}. The paper says it verified numerics 'including values of c0 very close to c*', but without N that verification cannot be checked. That is a genuine gap, but it is a gap in reporting, not a flaw in the physics. The stress-test worry about the vertical boundary at ε↓=1/q is similar: P+(∞) vanishes continuously as you approach that line, so the boundary is a mean-field statement, which the paper implicitly acknowledges. Minor point: the 'approximate exponential' fits in Figure 4 are quoted with slopes but no discussion of residuals or fit quality; harmless but slightly sloppy.\n\nOverall, I don't see a load-bearing error. The derivation of the persistence equations is independent and correct, and the agreement between theory and simulation for q=2,3 is strong. The missing simulation details are exactly what a referee should ask for, and they are trivially addressable. This paper deserves peer review, not desk rejection.\n\nSummary: worth citing if you work on q-voter models or persistence in binary opinion dynamics. A serious referee should engage and ask for the simulation parameters, but the science is honest and the results are useful.","headline":"Solid, workmanlike addition to q-voter literature: first persistence calculations with closed-form mean-field results, but missing simulation details keep it from being a clean accept.","tokens_in":18274,"tokens_out":1517,"would_cite":true,"duration_ms":16370,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C05","82C31","91D30"],"pacs":["05.40.-a","89.65.-s"],"model":"deepseek-v4-flash","headline":"The persistence probability saturates exactly when a biased q-voter model reaches consensus, and decays to zero otherwise.","keywords":["persistence probability","q-voter model","opinion dynamics","mean-field theory","phase diagram","consensus","biased dynamics","Monte Carlo simulation"],"falsifier":"Simulate the DMSS model at $q=3$ with large $N$, fix $\\epsilon_\\downarrow < 1/3$, and choose $c_0$ just below the theoretical boundary $\\epsilon_\\uparrow = (3c_0 \\epsilon_\\downarrow + 1 - 2c_0)/(3 - 3c_0)$ from Eq. (57); if a sizable fraction of runs still reach positive consensus and $P_+(\\infty)$ stays positive, the claimed boundary is not the correct one.","tokens_in":17261,"feed_emoji":"🗳️","tokens_out":5204,"duration_ms":42933,"temperature":0.7,"pith_summary":"This paper studies persistence — the probability that an agent has kept its initial opinion up to time t — in two biased generalizations of the q-voter model, the DMSS model and the MS model. It shows that the asymptotic persistence probability for a given opinion is non-zero precisely when the mean-field density dynamics converge to the corresponding consensus fixed point, and decays to zero otherwise. The persistence phase diagram therefore coincides with the fixed-point phase diagram of the opinion dynamics. This is not what one would expect from other models: in the two-dimensional voter model, persistence decays even though consensus is reached.","feed_headline":"Persistence saturates exactly when q-voter models reach consensus","feed_subtitle":"In two biased variants, non-zero persistence marks the consensus basin; elsewhere it dies exponentially.","key_machinery":"The central object is the coupled system of mean-field rate equations for the density $c(t)$ of positive opinion and the persistence probabilities $P_\\pm(t)$, closed through the per-agent flip rates $\\omega_{+\\to-}(c)$ and $\\omega_{-\\to+}(c)$. The key identity is the change of variables $dP_\\pm/dc = -\\omega(c) P_\\pm / \\dot{c}(c)$, whose integration from the initial density $c_0$ to $c(t)$ yields $P_\\pm(t)$ in closed form when the density equation is solvable ($q=2,3,\\infty$). This identity converts persistence, a history-dependent quantity, into a quadrature over the deterministic density trajectory, which is what ties saturation to fixed points.","core_discovery":"On the paper's own terms, the central discovery is that in both the DMSS and MS biased q-voter models, $P_+(t\\to\\infty)>0$ exactly when the stable fixed point of the density equation is the positive consensus $c=1$, and $P_+(t\\to\\infty)=0$ otherwise, with the complementary statement for $P_-$. The same dichotomy holds for all $q$ studied — $q=2$, $q=3$, and the $q\\to\\infty$ limit — with closed-form persistence expressions obtained for these cases. In the DMSS model the boundary is initial-condition dependent and is given by the unstable fixed point $c^*$ when both flip rates are below $1/q$; in the MS model the boundary is simply $p=1/2$, independent of initial density. The authors present heatmaps of $P_+(\\infty)$ in $(\\epsilon_\\uparrow,\\epsilon_\\downarrow)$ space and compare them with equilibrium phase diagrams based on consensus formation, finding close correspondence that improves as $c_0$ increases.","pith_inferences":["The paper establishes a diagnostic: in any binary-opinion dynamics whose density equation is separable, a persistence phase diagram can be constructed directly from the fixed points, provided the quadrature for the persistence probability is integrable — this may extend to other q-voter variants, including ones with contrarians.","Near the boundary the sharp phase transition is the fragile part: if finite-size fluctuations allow a system prepared just below $c^*$ to occasionally reach the positive consensus, the thermodynamic-limit boundary would be rounded in finite systems; a finite-size scaling study of $P_+(\\infty)$ near $c_0=c^*$ would settle that.","The persistence saturation value is given by the integral of the flip rates along the trajectory, so it could serve as a parameter-free probe of the basin structure in models where the fixed points are known but the full dynamics are not analytically solvable.","The paper notes that the $q\\to\\infty$ limit of the DMSS model was not reported in the original model paper; the exponential persistence decay in this limit suggests that for large finite $q$ the saturation plateau shrinks as the mixed fixed point destabilizes consensus, a crossover that could be quantified."],"forward_implications":["In the DMSS model, $P_+(\\infty)>0$ for all initial densities $c_0$ above the unstable fixed point $c^*$, and for $q=3$ the boundary line $\\epsilon_\\uparrow = (3c_0\\epsilon_\\downarrow + 1 - 2c_0)/(3 - 3c_0)$ is exact within mean-field theory.","In the MS model, the persistence phase boundary is $p=1/2$ for every $q$ and initial density: for $p>1/2$ the positive persistence saturates and the negative decays, and vice versa for $p<1/2$.","In non-consensus phases the persistence decays approximately exponentially, $P_\\pm(t) \\approx \\alpha + \\beta e^{-\\gamma t}$, with rates $\\gamma$ that vary with parameters in the DMSS model but are nearly universal in the MS model.","For $q\\to\\infty$ the unanimous-panel terms vanish and both models have only a mixed fixed point; the paper gives $P_+(t)=c_0 e^{-\\epsilon_\\downarrow t}$ and $P_-(t)=(1-c_0)e^{-\\epsilon_\\uparrow t}$ for the DMSS model.","The saturation value of persistence is not universal: it depends on the initial density $c_0$ and on the model parameters, even within a consensus phase."],"supporting_citations":[{"why":"Supplies the DMSS model, its flip rates, and the fixed-point structure used to construct the persistence phase diagram.","marker":"[19]"},{"why":"Supplies the MS model and its mean-field density equation, from which the persistence quadrature and the $p=1/2$ boundary follow.","marker":"[20]"},{"why":"Defines the original q-voter model and its unanimous-panel update rule, the baseline from which both biased models deviate.","marker":"[18]"},{"why":"Provides the voter-model persistence behavior against which the biased models are compared, including the exponential mean-field decay.","marker":"[14]"},{"why":"Gives the $\\epsilon_\\downarrow=0$ special case of the DMSS model used to test the closed-form persistence expressions.","marker":"[22]"}],"fun_headline_variants":["q-voter persistence maps consensus basins via phase diagrams","Persistence probability divides q-voter parameter space into decay and saturation","For biased q-voter models, persistence saturates only at consensus fixed points","Persistence reveals phase boundaries in two biased q-voter models","Saturation of persistence marks consensus in DMSS and MS q-voter models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharp location of the persistence phase boundary assumes that, in a finite system, the dynamics always follow the deterministic mean-field trajectory, so a system starting just above the unstable fixed point never escapes to the negative-consensus basin.","fun_headline_variants_meta":{"raw":{"variants":["q-voter persistence maps consensus basins via phase diagrams","Persistence probability divides q-voter parameter space into decay and saturation","For biased q-voter models, persistence saturates only at consensus fixed points","Persistence reveals phase boundaries in two biased q-voter models","Saturation of persistence marks consensus in DMSS and MS q-voter models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00049,"raw_usage":{"total_tokens":2389,"prompt_tokens":900,"completion_tokens":1489,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":1407}},"tokens_in":516,"tokens_out":1489,"duration_ms":10986,"temperature":1.0,"reasoning_tokens":1407,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:08:44.023766+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the DMSS model at $q=3$ with large $N$, fix $\\epsilon_\\downarrow < 1/3$, and choose $c_0$ just below the theoretical boundary $\\epsilon_\\uparrow = (3c_0 \\epsilon_\\downarrow + 1 - 2c_0)/(3 - 3c_0)$ from Eq. (57); if a sizable fraction of runs still reach positive consensus and $P_+(\\infty)$ stays positive, the claimed boundary is not the correct one.","supporting_citations":[{"cited_title":"This equation is not invariant under any non-trivial symmetry transformation, including c → 1−c, since such a transformation changes the coefficients of the equation","cited_arxiv_id":null,"evidence_quote":"Supplies the DMSS model, its flip rates, and the fixed-point structure used to construct the persistence phase diagram."},{"cited_title":"Jusup, P","cited_arxiv_id":null,"evidence_quote":"Supplies the MS model and its mean-field density equation, from which the persistence quadrature and the $p=1/2$ boundary follow."},{"cited_title":"Castellano, S","cited_arxiv_id":null,"evidence_quote":"Defines the original q-voter model and its unanimous-panel update rule, the baseline from which both biased models deviate."},{"cited_title":"Ben-Naim, P","cited_arxiv_id":null,"evidence_quote":"Provides the voter-model persistence behavior against which the biased models are compared, including the exponential mean-field decay."},{"cited_title":"Ben-Naim, L","cited_arxiv_id":null,"evidence_quote":"Gives the $\\epsilon_\\downarrow=0$ special case of the DMSS model used to test the closed-form persistence expressions."}],"review_version":1}