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REVIEW 3 major objections 7 minor 30 references

Persistence probability based dynamics and phase diagrams in biased q-voter models

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The persistence probability saturates exactly when a biased q-voter model reaches consensus, and decays to zero otherwise.

desk verdict 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. read the letter →

arxiv 2608.08324 v1 pith:H3OF75VI submitted 2026-08-08 cond-mat.stat-mech

Amit Pradhan , Pratik Mullick , Parongama Sen This is my paper · ORCID
classification cond-mat.stat-mech MSC 82C0582C3191D30 PACS 05.40.-a89.65.-s
keywords persistenceprobabilityq-votermodelopiniondynamicsmean-fieldtheoryphasediagramconsensusbiasedMonteCarlosimulation
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 7 minor

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.

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 (3)
  1. [Sections III and V; Eq. (3)] 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.
  2. [Section V, Eqs. (28) and (57)] 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.
  3. [Section V, Fig. 7] 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.
minor comments (7)
  1. [Section II] The MS model description contains the typo 'unanibous' and the duplicated 'the the'; please correct.
  2. [Section III.A.1] 'Exolicitly' should be 'Explicitly'.
  3. [Section III.B.1] 'two basin of attraction' should be 'two basins of attraction'.
  4. [Section V, Eq. (57)] The formula should be written with parentheses: ε↑ = (3c0 ε↓ + 1 − 2c0)/(3 − 3c0); as printed, the fraction is ambiguous.
  5. [Section III.A and Figures 1-3] 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.
  6. [Figures 6 and 7] Add colorbars and state how P+(t→∞) is estimated from finite-time data, for example by giving a plateau criterion or a fitting threshold.
  7. [Section III.A.1] The parenthetical remark that the voter-model exponents in Ref. [14] contain 'probably an oversight' should be either substantiated with a derivation or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the persistence probabilities are derived from the model's flip rates, and the fixed-point correlation is a computed consequence rather than an input-output reversal.

full rationale

The persistence derivation is not circular. The paper starts from per-agent flip rates (DMSS: Eqs. (7), (24), (43)-(44); MS: Eqs. (19), (34), (52)) taken as model definitions from Refs. [19,20], constructs the persistence rate equations (3), and solves them via Eq. (6) to obtain closed forms (11), (22), (28)-(29), (40)-(41), (47), and (56). The asymptotic behavior (saturation vs. exponential decay) is determined by where the deterministic density c(t) flows, i.e., by the fixed points of the same rate equation, so the agreement between the persistence phase diagram and the fixed-point phase diagram is a mathematical consequence of the solved equations, not an assumption built into the derivation. The self-citations to Refs. [19,20] supply the rate equations and fixed-point stability analysis; these are parameter-free model definitions that do not themselves contain persistence results, and therefore constitute independent support rather than a circular chain. The q=3 persistence phase boundary, Eq. (57), is obtained directly from the known unstable fixed point c* of Eq. (26), and the numerical heatmaps in Figs. 6 and 7 are comparisons against that analytic line, not fits of it. The only substantive weakness is that the numerical verification in Section V ('We have verified numerically that P+(t->infinity) remains finite for all initial conditions satisfying c0 > c*, including values of c0 very close to c*') reports no system size, run count, or error bars, so finite-size smearing near the separatrix is not quantified; however, that is a verification gap, not circular reasoning.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard mean-field persistence formalism (Eq. 3) and on the previously published rate equations for the two models. No new entities or fitted parameters are introduced; the model parameters ε↑, ε↓, p, q, c0 are inputs from the model definitions. The derived persistence expressions are new, but their phase boundaries echo the fixed point structure taken from Refs. [19] and [20].

assumptions (4)
  • domain assumption The mean-field rate equations for c(t) from Refs. [19] and [20] are correct for the fully connected network in the N→∞ limit.
    Used in Section III to derive persistence probabilities, e.g., Eqs. (8), (25), (37), (45), and (53).
  • domain assumption The persistence probabilities satisfy dP±/dt = -ω_{±→∓}(c(t)) P±, i.e., the flip rate of an agent that has never flipped depends only on the global density c(t).
    Exact on fully connected networks in the thermodynamic limit; stated in Eq. (3) and used throughout Section III.
  • domain assumption For q→∞, the unanimous panel probabilities c^q and (1-c)^q vanish for 0<c<1, yielding constant or rational flip rates.
    Used to derive Eqs. (45)-(47) and (52)-(56) in Section III C; valid for fixed interior c, but requires care near the boundaries.
  • domain assumption The persistence phase boundary for q=3 is given by the unstable fixed point c* of the density equation, i.e., P+(∞)>0 iff c0>c*.
    Introduced in Section V, Eq. (57), and verified numerically; this is a nontrivial mapping between persistence saturation and density dynamics.

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Cite this review

Pith. "Pith review of Persistence probability based dynamics and phase diagrams in biased q-voter models." pith.science (2026). https://pith.science/paper/H3OF75VI

@misc{pith2026260808324,
  author       = {Pith},
  title        = {Pith review of: Persistence probability based dynamics and phase diagrams in biased q-voter models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H3OF75VI}},
  note         = {Machine review of arXiv:2608.08324}
}
read the original abstract

Persistence probability in opinion dynamics models estimates the tendency of the agents not to change their initial opinion till the present time. Here we consider two nonlinear q-voter models with binary opinions, where the dynamics are governed by a biased choice when the q panel is not unanimous. The models are studied for different parameter ranges corresponding to the known stationary states. Mean field theory and numerical simulations are used to compute the persistence probability for the two types of opinion separately. The long time behavior in general is either a saturation or a decay that can be approximated by an exponential form, depending on the chosen parameters. Based on this, phase diagrams in the parameter space are presented for both the models. The regions in the phase diagrams indicate a strong correlation with the behavior of fixed points in the corresponding models, which is non-trivial as far as the persistence probability is concerned.

Figures

Figures reproduced from arXiv: 2608.08324 by the authors.

Figure 1
Figure 1. FIG. 1: Time evolution of the persistence probabilities (a) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Time evolution of the persistence probabilities (a) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Persistence probability [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Persistence probabilities [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Heatmaps showing the variation of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: (a) A heatmap showing the variation of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Asymptotic positive persistence probability [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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    The MS model In the weighted influence q voter model [20], a mixed panel favors the positive opinion with relative weight p and the negative opinion with weight 1 − p. In the mean field picture, the finite q transition probabilities are bi- nomial sums, but for large q as q → ∞, as 0 < c < 1, the unanimous terms vanish, giving simple forms for the flip rates:...

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