{"id":"7c4d5f11-0354-413f-a5a8-95577ae8db3a","arxiv_id":"2501.11863","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A network model of opinion spreading with nonlinear (asymmetric) perception shows first-order explosive transitions and hysteresis in opinion polarization, explained by a mean-field bifurcation analysis.","lead":"This paper builds a mathematical model of opinion change on social networks where people perceive others' opinions unevenly, and shows this asymmetry can make the whole network flip explosively to a new opinion and then abruptly back. It matters because it suggests a mechanism, with hysteresis, behind sudden polarization and depolarization in online and offline communities.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The mean-field proxy that turns asymmetric perception into a subcritical-bifurcation explanation drops correlation terms the SM itself admits control the critical point; on broad-degree networks these terms are not small, so the analytical mechanism is not established.","rationale":"I read the paper in good faith: the full numerical integration of Eq. (2) does show explosive transitions and hysteresis for alpha > 1 on the tested modular ER and BA networks, so the qualitative phenomenon is real within the model. The weak point is the analytical explanation: the one-dimensional normal form Eq. (3) is obtained by discarding correlation terms that are asserted, not shown, to be small, and the SM itself contains the admission that the critical point depends on one of the discarded sums. Because the model's own xi ~ ki assumption makes delta x and delta k proportional, the neglected term is linear in <x> and scales with Var(k)/<k>, so it can be large exactly where the linear term of the proxy vanishes. The second discarded sum can renormalize the quadratic coefficient that determines whether the bifurcation is subcritical or supercritical. This matches the reader's weakest assumption. I do not recommend rejection: a corrected closure might preserve the qualitative conclusion, and the direct simulations stand on their own. The verdict should remain conditional, with the condition being a quantitative justification of the truncation or a clear limitation of the claim to narrow-degree networks.","tokens_in":15132,"tokens_out":13926,"duration_ms":149078,"concrete_test":"Evaluate the dropped correlation terms R(t) from SM Eq. (6) along a numerical trajectory of Eq. (2) on the 3-module BA network (m=5, d=2, alpha=1.2) immediately below the backward critical point: compute R = beta/N[(1-2<x>+2 alpha <x>(1-<x>)) sum_j delta x_j delta k_j - (1+2 alpha <x>) sum_{i,j} A_ij delta x_i delta x_j] and compare |R| to the retained terms beta<k>(alpha-1)<x>^2 - alpha beta<k><x>^3 at the same <x>. If |R| is comparable to or larger than the retained quadratic term, Eq. (3) is not a faithful reduction and the subcritical classification is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (3) is the only analytical route to the central \"alpha > 1 implies subcritical\" claim. In the SM, the reduction to (3) drops the correlation terms R = beta/N [ (1 - 2<x> + alpha d <x>^{d-1}(1-<x>)) sum_j delta x_j delta k_j - (1 + alpha d <x>^{d-1}) sum_{i,j} A_ij delta x_i delta x_j ] on the assertion that deviations are small and P(k) is narrow and symmetric. This assertion is not checked. Under the model's own ansatz x_i ~ k_i, delta x_i is approximately (<x>/<k>) delta k_i, so the first sum is about N <x> Var(k)/<k>: it is first order in <x>, not a higher-order correction. Near the claimed critical point gamma = beta <k>, the retained linear term (beta<k> - gamma)<x> vanishes, so this dropped term can dominate and shift the critical point; the SM explicitly concedes that the difference in critical points \"depends on sum_j delta x_j delta k_j\". The second sum has no sign-definite cancellation and can renormalize the coefficient of <x>^2, which is exactly the coefficient deciding subcritical versus supercritical. Thus the normal-form mechanism is not a controlled consequence of Eq. (2). The direct simulations on dense ER and BA networks do show jumps and hysteresis, so the phenomenon is not refuted; what is unsupported is the paper's stronger claim that the mean-field derivation establishes nonlinear perception as the minimal universal mechanism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an SIS-type opinion dynamics model in which asymmetric perception is encoded by a nonlinear incidence term, β(1−xi)Σ_j A_ij x_j (1 + α x_j^{d−1}). The central claim is that for underestimating perception (d>1) with α>1, the model exhibits explosive, first-order polarization transitions and hysteresis, while overestimation (0<d<1) mitigates abrupt depolarization. To explain this analytically, the authors reduce the network dynamics to a one-dimensional mean-field equation, Eq. (3), which takes the form of a cubic normal form whose bifurcation type is controlled by α. Numerical simulations on a small modular network of three Erdős–Rényi modules show the predicted continuous-to-discontinuous transition and hysteresis, and the pattern of the polarized state is connected to the spectral properties of modular networks via perturbation theory. The Supplemental Material extends the comparison to scale-free and small-world networks, reporting qualitative agreement for scale-free networks and a quantitative error analysis for Watts–Strogatz networks.","tokens_in":15464,"tokens_out":6562,"duration_ms":64403,"significance":"If the central claim were rigorously established, the model would provide a simple and appealing mechanism for explosive opinion shifts and for the difficulty of reversing them, with falsifiable predictions (a first-order transition when α>1 and d>1). The paper includes concrete numerical simulations on modular, scale-free, and small-world networks, and it makes an honest attempt to connect the network-level phenomenology to a mean-field normal form. However, the analytical bridge—the derivation of Eq. (3)—rests on uncontrolled approximations that, under the model's own ansatz, are not guaranteed to be small. Because the sign of the quadratic coefficient in the normal form is precisely what determines whether the bifurcation is subcritical or supercritical, the paper's strongest analytical claim (that nonlinear perception is a minimal universal mechanism) is not yet supported. The numerical evidence for the phenomenon on specific networks is credible, but the general mechanism remains to be demonstrated.","major_comments":[{"comment":"The reduction to Eq. (7) drops the correlation terms (β/N)[(1−2⟨x⟩+αd⟨x⟩^{d−1}(1−⟨x⟩)) Σ_j δx_j δk_j − (1+αd⟨x⟩^{d−1}) Σ_{i,j} A_ij δx_i δx_j] with the assertion that deviations are small and the degree distribution is narrow and symmetric. This assertion is not checked and is in fact inconsistent with the model's own ansatz x_i ∼ k_i: under that ansatz δx_i ≈ (⟨x⟩/⟨k⟩)δk_i, so Σ_j δx_j δk_j ≈ N (⟨x⟩/⟨k⟩) Var(k), which is first order in ⟨x⟩ and does not vanish for a narrow distribution. Near the mean-field critical point γ = β⟨k⟩, the retained linear term (β⟨k⟩ − γ)⟨x⟩ vanishes, so this dropped term can dominate and shift the critical point; the SM itself concedes that the difference in critical points depends on Σ_j δx_j δk_j. Moreover, the term Σ_{i,j} A_ij δx_i δx_j renormalizes the coefficient of ⟨x⟩² in the normal form, which is exactly the coefficient that decides whether the bifurcation is subcritical or supercritical. Therefore Eq. (3) and the claim that α>1 implies a first-order transition are not a controlled consequence of the individual-based model, Eq. (2). The numerical simulations on the modular network support the phenomenon, but the analytical mechanism is not established.","section":"Supplemental Material, Section I, Eq. (6) and the paragraph that follows"},{"comment":"The paper's own quantitative comparison on small-world networks shows that the error between the individual-based simulation and the degree-based mean-field proxy can be large and generally grows with the rewiring probability p. For example, at k=4 the error is large across all p, and at moderate k the error increases with p before declining near p=1. This contradicts the impression given in the main text that Eq. (3) provides a general explanation for the observed transitions. The paper needs to either characterize the regime in which the approximation is quantitatively faithful (e.g., by verifying that the dropped correlation terms are indeed negligible) or substantially weaken the claim that the mean-field proxy explains the mechanism in a universal way.","section":"Supplemental Material, Section IV.B (Fig. 7)"}],"minor_comments":[{"comment":"The word 'Understimate' should be 'Underestimate'.","section":"Fig. 1 caption"},{"comment":"The sentence beginning 'When all terms ⟨x⟩ are considered' should be reworded to 'When all terms involving ⟨x⟩ are considered', since the current phrasing is unclear.","section":"Supplemental Material, Section I, after Eq. (6)"},{"comment":"The phrase 'for simplicity of representation α → αβ' is confusing; please clarify that α has been rescaled by absorbing β, and state the resulting dimensions of α.","section":"Main text, after Eq. (2)"},{"comment":"Reference [33] (Granovetter 1978) is a duplicate of Reference [17]; please remove or distinguish them.","section":"References"},{"comment":"Footnote [44] calls Eq. (4) a mean-field approximation, but Eq. (4) is the individual-based model; the terminology is confusing and should be clarified.","section":"Supplemental Material, Section IV.A and footnote [44]"},{"comment":"The inset shows the degree distribution but the caption does not state the mean and variance of the degree distribution; adding these values would help the reader judge the validity of the 'narrow and symmetric' assumption.","section":"Fig. 1(b) inset"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the uncontrolled mean-field reduction, which is load-bearing for the paper's main analytical claim. The numerical simulations on the modular network do appear to show the described explosive transitions, so the phenomenon is not in doubt; what is not established is the analytical mechanism. The paper is within the scope of physics.soc-ph, and the self-citations for spectral perturbation results are appropriate. I would be willing to review a revised version that either makes the approximation controlled (e.g., by deriving bounds on the dropped terms and verifying them numerically on the networks used) or clearly reframes the contribution as a numerical study with a heuristic mean-field explanation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper gives a clean mechanism story — nonlinear incidence in SIS opinion dynamics flips a supercritical bifurcation to subcritical, producing explosive polarization and hysteresis. The simulations on a modular network and in the SM on BA and WS networks show the phenomenon robustly. That much is worth taking seriously.\n\nWhat is genuinely new: applying nonlinear incidence, standard in epidemiology, to opinion spreading with the sign of the quadratic term in the normal form linked to the perception parameter α. The α>1 subcritical condition for d=2 is a nice, readable result. The paper also connects polarization patterns to modular eigenvectors via spectral perturbation, and the SM is honest about the gap between the individual-based and mean-field critical points.\n\nThe soft spot is the mean-field reduction from Eq. (2) to Eq. (3). The dropped correlation terms aren't obviously small. Under the paper's own xi~ki ansatz, δxi ≈ (<x>/<k>)δki, so Σ_j δx_j δk_j is first order in <x> and can dominate near the critical point where the linear term vanishes. The SM itself concedes that the critical-point difference depends on that sum. The other dropped term, Σ Aij δxiδxj, could renormalize the coefficient of <x>^2 — exactly the coefficient that decides subcritical vs supercritical. So the analytical derivation of the α>1 condition is not controlled. That doesn't refute the phenomenon; the numerics stand on their own. But the paper's claim to have established the minimal mechanism via mean-field is stronger than the math supports.\n\nMinor issue: the abstract and conclusion talk about 'minimal conditions' and practical implications without calibrating to any empirical data. That is overreach but not a fatal flaw for a physics letter.\n\nOverall: worth a serious referee. A good referee should ask for a more careful treatment of the correlation terms, or a clear statement that Eq. (3) is a heuristic normal-form model rather than a controlled reduction. If the authors tighten that, the paper would be a solid contribution to opinion dynamics. I'd treat the main qualitative claims as conditional, not refuted.","headline":"A plausible and interesting mechanism for explosive opinion transitions, but the mean-field derivation that carries the α>1 subcritical claim drops correlation terms that can dominate near the critical point.","tokens_in":15982,"tokens_out":2809,"would_cite":false,"duration_ms":29324,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Asymmetric perception of others' opinions turns smooth opinion change into an abrupt, hysteresis-locked polarization transition.","keywords":["asymmetric perception","explosive polarization","hysteresis","nonlinear incidence","mean-field","opinion dynamics","modular networks","SIS model"],"falsifier":"Run the individual-based model (Eq. 2) on a modular network with a broad, asymmetric degree distribution, with $d=2$ and $\\alpha>1$, sweeping the reversion rate $\\gamma$ upward and downward; if the equilibrium $\\langle x\\rangle$ shows no discontinuous jump and no bistable window, then the mean-field subcritical bifurcation is not the actual mechanism.","tokens_in":14887,"feed_emoji":"💥","tokens_out":9717,"duration_ms":92633,"temperature":0.7,"pith_summary":"The paper asks whether a minimal change in how people perceive the opinions of others can turn gradual opinion change into abrupt, explosive polarization. It builds an epidemic-style susceptible-infected-susceptible model in which the adoption rate of a new opinion depends nonlinearly on the number of neighbors already holding it (a term proportional to $1+\\alpha x_j^{d-1}$), and it claims that when perception underestimates the true prevalence ($\\alpha>1$), the model undergoes a first-order phase transition: the new opinion takes over discontinuously, and once established it stays until the control parameter is lowered past a second, smaller critical point, producing hysteresis. The authors derive a one-dimensional mean-field normal form that changes from transcritical/supercritical to subcritical exactly when $\\alpha>1$, and they show that continuous transitions localize the new opinion inside one community while discontinuous transitions erase that pattern. If the claim is right, the model supplies a single mechanism for both explosive polarization and abrupt, hard-to-reverse depolarization.","feed_headline":"Opinion perception can make polarization explosive and irreversible","feed_subtitle":"A small change in how people perceive consensus turns gradual opinion change into abrupt polarization with hysteresis.","key_machinery":"The load-bearing object is the one-dimensional mean-field equation (Eq. 3), $\\langle \\dot{x}\\rangle = (\\tilde{\\beta}-\\gamma)\\langle x\\rangle - \\tilde{\\beta}\\langle x\\rangle^2 + \\alpha\\tilde{\\beta}(1-\\langle x\\rangle)\\langle x\\rangle^d$, with $\\tilde{\\beta}=\\beta\\langle k\\rangle$. It is derived from the full network equations by assuming $x_i\\sim k_i$ and a narrow, symmetric degree distribution, so that deviations $\\delta x_i$ and $\\delta k_i$ average out. For $d=2$ this becomes $\\langle \\dot{x}\\rangle = (\\tilde{\\beta}-\\gamma)\\langle x\\rangle + \\tilde{\\beta}(\\alpha-1)\\langle x\\rangle^2 - \\alpha\\tilde{\\beta}\\langle x\\rangle^3$, and the sign of the quadratic term at small $\\langle x\\rangle$ decides whether the bifurcation is transcritical ($\\alpha=0$), supercritical ($\\alpha<1$ and $\\alpha=1$), or subcritical ($\\alpha>1$). The second piece of machinery is spectral perturbation theory for modular networks: the eigenvectors belonging to the $M$ largest eigenvalues are localized on individual modules, which is what makes the opinion pattern polarize along community lines; the Perron-Frobenius theorem guarantees the principal eigenvector is the only positive one and therefore the relevant critical mode.","core_discovery":"The central claim is that asymmetric perception, encoded as nonlinear incidence in a metanode SIS model, is sufficient to produce explosive polarization and explosive depolarization. Each node contains $N$ opinion units that flip between the old opinion $S$ and the new opinion $I$; the standard linear contagion term $\\beta(1-x_i)\\sum_j A_{ij}x_j$ is modified to $\\beta(1-x_i)\\sum_j A_{ij}x_j(1+\\alpha x_j^{d-1})$, where $d>1$ means underestimated and $0<d<1$ overestimated perception. Linearization shows the initial instability is set by $\\beta\\lambda_{\\max}^A>\\gamma$, independent of $\\alpha$. For $d=2$, the mean-field equilibrium equation reduces to a cubic normal form whose quadratic coefficient is $\\tilde{\\beta}(\\alpha-1)$; when $\\alpha>1$ the bifurcation becomes subcritical, producing a bistable region, a discontinuous jump in the average opinion as $\\gamma$ is lowered, hysteresis on the return path, and an abrupt switch back to the old opinion at a second critical point. For $0<d<1$, the new opinion is always embraced and abrupt depolarization is mitigated. Continuous transitions leave the spatial pattern close to the critical eigenvector, so the new opinion concentrates in one network community; discontinuous transitions deviate from that pattern and can produce a uniform opinion shift.","pith_inferences":["Beyond the paper: because nonlinear incidence is a generic feature of binary-state dynamics, the same $\\alpha>1$ condition may produce first-order cascades in rumor spreading, behavioral adoption, and financial herding, where a threshold-curve shape similar to the opinion model appears.","Beyond the paper: the mean-field proxy is derived under a narrow symmetric degree distribution and by cancelling fluctuation products; on empirical networks with hubs, quantitative discrepancies are expected, and measuring the actual equilibrium curve would separate perception-driven bistability from purely structural effects.","Beyond the paper: a testable intervention follows directly: if platforms make perceived consensus more proportional to true prevalence (pushing $\\alpha$ toward $1$ or below), the model predicts the explosive jump and hysteresis should disappear; this could be probed with online experiments that modulate exposure salience.","Beyond the paper: fitting the effective exponent $d$ to adoption time series would place a community in the underestimation ($d>1$) or overestimation ($0<d<1$) regime, and the two regimes make opposite predictions about depolarization that observational data could discriminate."],"forward_implications":["For $\\alpha>1$ (underestimated perception), adoption of a new opinion is discontinuous: once the reversion rate $\\gamma$ passes the critical value, the average opinion jumps to a higher level, with the jump size growing with $\\alpha$.","The backward path is hysteretic: lowering $\\gamma$ back to its original value does not undo the shift; the new opinion persists until a second, smaller critical point, so depolarization happens suddenly and the new opinion is resistant to reversal.","For $0<d<1$ (overestimated perception), the new opinion is always embraced and the abrupt depolarization is suppressed, so perception bias determines not just whether but how abruptly opinions change.","In the continuous regime ($\\alpha\\le 1$), the final opinion pattern near criticality mirrors the critical eigenvector and thus localizes in a single community; in the discontinuous regime the pattern decouples from that eigenvector and can become uniform, meaning explosive transitions also change who ends up holding the new opinion.","The condition for the new opinion to start spreading, $\\beta\\lambda_{\\max}^A>\\gamma$, is independent of $\\alpha$, so network structure alone sets the initial threshold while perception asymmetry controls the order of the transition."],"supporting_citations":[{"why":"Supplies the network-theoretic notions (degree-based mean-field, modular structure, eigenvalue spectrum) that the model and its reductions rely on.","marker":"[7]"},{"why":"Provides the pattern-formation result that near a continuous instability the nonlinear pattern follows the critical eigenvector, which the paper uses to link polarization to communities.","marker":"[25]"},{"why":"Gives the normal forms for transcritical and pitchfork bifurcations used to classify the transitions and identify the subcritical case.","marker":"[30]"},{"why":"Establishes the SIS-on-networks framework and its critical condition, which the opinion model extends with nonlinear incidence.","marker":"[31]"},{"why":"Introduces the degree-based mean-field assumption $x_i\\sim k_i$ used to reduce the system to one dimension.","marker":"[35]"},{"why":"Provides the Rayleigh-quotient and Perron-Frobenius results used to bound $\\lambda_{\\max}^A$ and to justify the principal-eigenvector pattern argument.","marker":"[39]"},{"why":"Demonstrates that modular networks have eigenvectors localized on single communities, the mechanism behind opinion polarization in the model.","marker":"[41]"},{"why":"Further establishes spectral signatures of modularity used to interpret how polarization patterns emerge and shift.","marker":"[42]"}],"fun_headline_variants":["Asymmetric perception triggers explosive opinion polarization and hysteresis","Why small perception shifts cause abrupt opinion polarization with hysteresis","Perception asymmetry makes opinion polarization explosive with hysteresis","Explosive polarization from asymmetric perception in opinion dynamics","Nonlinear perception leads to explosive opinion switching with hysteresis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analytic story rests on a simplification that assumes each person's opinion is proportional to how connected they are and that random fluctuations cancel out, so the one-dimensional equation, not the full network, is what predicts the explosive transition.","fun_headline_variants_meta":{"raw":{"variants":["Asymmetric perception triggers explosive opinion polarization and hysteresis","Why small perception shifts cause abrupt opinion polarization with hysteresis","Perception asymmetry makes opinion polarization explosive with hysteresis","Explosive polarization from asymmetric perception in opinion dynamics","Nonlinear perception leads to explosive opinion switching with hysteresis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00054,"raw_usage":{"total_tokens":2588,"prompt_tokens":939,"completion_tokens":1649,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":1575}},"tokens_in":555,"tokens_out":1649,"duration_ms":10826,"temperature":1.0,"reasoning_tokens":1575,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:46:53.245930+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the individual-based model (Eq. 2) on a modular network with a broad, asymmetric degree distribution, with $d=2$ and $\\alpha>1$, sweeping the reversion rate $\\gamma$ upward and downward; if the equilibrium $\\langle x\\rangle$ shows no discontinuous jump and no bistable window, then the mean-field subcritical bifurcation is not the actual mechanism.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the network-theoretic notions (degree-based mean-field, modular structure, eigenvalue spectrum) that the model and its reductions rely on."},{"cited_title":"de Kemmeter, A","cited_arxiv_id":null,"evidence_quote":"Provides the pattern-formation result that near a continuous instability the nonlinear pattern follows the critical eigenvector, which the paper uses to link polarization to communities."},{"cited_title":"Kuramoto, in International Symposium on Mathemat- ical Problems in Theoretical Physics: January 23–29, 1975, Kyoto University, Kyoto/Japan (Springer, 1975) pp","cited_arxiv_id":null,"evidence_quote":"Gives the normal forms for transcritical and pitchfork bifurcations used to classify the transitions and identify the subcritical case."},{"cited_title":"Strogatz, Nonlinear dynamics and chaos: with appli- cations to physics, biology, chemistry, and engineering , repr","cited_arxiv_id":null,"evidence_quote":"Establishes the SIS-on-networks framework and its critical condition, which the opinion model extends with nonlinear incidence."},{"cited_title":"Horn and R","cited_arxiv_id":null,"evidence_quote":"Introduces the degree-based mean-field assumption $x_i\\sim k_i$ used to reduce the system to one dimension."},{"cited_title":"Barth´ elemy, A","cited_arxiv_id":null,"evidence_quote":"Provides the Rayleigh-quotient and Perron-Frobenius results used to bound $\\lambda_{\\max}^A$ and to justify the principal-eigenvector pattern argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates that modular networks have eigenvectors localized on single communities, the mechanism behind opinion polarization in the model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Further establishes spectral signatures of modularity used to interpret how polarization patterns emerge and shift."}],"review_version":1}