{"id":"f91fb306-9f5b-420f-8b9d-b68d27a5c2fc","arxiv_id":"2607.06056","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":5,"one_line_summary":"Extending a vigilance cascade from direct neighbors to 2–4 circles of influence on multiplex networks substantially raises the critical temptation for defection, especially in sparse and layer-aligned topologies.","lead":"This paper shows that in a computer model of social cooperation, letting social pressure reach beyond your immediate neighbors to friends-of-friends substantially increases how much people cooperate. The result matters because online platforms now make distant contacts visible, and the model suggests this could reshape cooperative behavior in real populations.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The non-normalized vigilance kernel (Eq. 1) mechanically amplifies I_i as L grows, but the uncorrelated-multiplex control partially mitigates this concern: cooperation does not increase with L when layers are decorrelated, despite the same mechanical amplification being present.","rationale":"The reader identified the correct load-bearing concern: the non-normalized kernel in Eq. (1) means I_i mechanically increases with L, and by Eq. (2) this directly lowers temptation, making the direction of the headline result partly structural. This is a real issue that warrants the CONDITIONAL verdict.\n\nHowever, I partially disagree with the reader's framing because the paper already contains a control that mitigates the concern more than the reader acknowledges. The uncorrelated-multiplex results (Sec. III C, Fig. 4) show that when G_vig ≠ G_game, increasing L does NOT promote cooperation in ER z=4, even though the same mechanical amplification of I_i is present. This demonstrates that the amplification alone cannot account for the cooperation gain—the spatial coherence between vigilance and game layers is necessary. This is not a trivial control; it directly tests whether the mechanical increase in I_i is sufficient (it is not).\n\nThat said, the uncorrelated control does not fully resolve the concern. In the correlated case, mechanical amplification and dynamical feedback are entangled: increasing L both amplifies I_i and extends the reach of the vigilance cascade, and these effects cannot be separated without a normalized-kernel comparison. The reader's recommendation for such a comparison is well-placed.\n\nThe paper's other contributions—topology dependence, inter-layer correlation effects, the λ sensitivity threshold in dense BA networks, and the real-network validation—are more robust and do not depend on the normalization choice in the same way. The inter-layer correlation finding in particular is a genuine dynamical result, not a structural artifact.\n\nI recommend UNCHANGED because the CONDITIONAL verdict already reflects this concern, and the uncorrelated control provides partial mitigation that the reader did not weigh. A normalized-kernel test would strengthen the central claim but is not strictly necessary to accept the secondary findings.","tokens_in":13908,"tokens_out":4235,"duration_ms":262984,"concrete_test":"Re-run the main simulations for ER z=4 (correlated multiplex, θ=0.3, λ=0.5) with a normalized kernel: I_i = min(1, (Σ_{d=1}^{L} λ^{d-1} m_d/k_d) / (Σ_{d=1}^{L} λ^{d-1})). If the L=1→2 transition still shifts the critical temptation by a meaningful margin (e.g., from b≈1.3 to b≥1.5), the reach effect is genuine and not an artifact of amplification. If the critical temptation barely moves, the headline result is primarily driven by the non-normalized kernel design.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the core structural issue. Eq. (1) defines I_i = min(1, Σ_{d=1}^{L} λ^{d-1} m_d/k_d) with the kernel 'intentionally not normalized.' Each additional circle adds a non-negative term, so for any fixed vigilance configuration, I_i is non-decreasing in L. By Eq. (2), higher I_i directly lowers T_i, making defection less attractive. This means the direction of the headline result—cooperation increases with L—is partly guaranteed by construction rather than emerging purely from the dynamics. The finding that the L=1→2 transition accounts for 'most of the gain' is also consistent with the kernel weights (1, 0.5, 0.25, 0.125 at λ=0.5), where the largest mechanical jump occurs at L=1→2.\n\nHowever, the concern is less severe than it first appears because the paper includes a natural control: the uncorrelated multiplex (Sec. III C). In the uncorrelated case, I_i is still computed from G_vig and still mechanically increases with L, so T_i still decreases. Yet cooperation does NOT increase—the critical temptation in ER z=4 stays fixed near b≈1.3–1.5 for all L (Fig. 4). This demonstrates that mechanical amplification of I_i alone is insufficient to produce the cooperation gain; the spatial coherence between vigilance and game layers is essential. This partially addresses the normalization concern, though it does not fully isolate the 'reach' effect from the 'amplification' effect in the correlated case, where both are intertwined.\n\nA normalized kernel (dividing by Σ λ^{d-1}) would redistribute rather than amplify total influence as L grows, cleanly separating whether it is the extended reach of the vigilance signal or the increased total pressure that drives cooperation. The paper does not provide this comparison.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper extends a two-layer multiplex model of social monitoring and cooperation (Pereda, 2016) from direct-neighbor vigilance to L circles of influence, with influence decaying geometrically as λ^{d-1}. The authors couple a weak Prisoner's Dilemma on a game layer to a Watts threshold cascade of vigilance on a vigilance layer, where a cooperator's temptation T_i is reduced in proportion to the vigilance influence I_i received from its L-hop neighborhood. Using the Fermi update rule (validated against replicator dynamics at L=1), the authors study correlated and uncorrelated multiplex networks on BA and ER topologies (z=4, 16), validate on the CKM physician network, and perform sensitivity analysis on λ. The central finding is that extending vigilance to L=2 already accounts for most of the cooperation gain, that the effect requires inter-layer correlation (except in BA networks), and that dense hub-dominated networks exhibit a sharp transition in cooperation as λ crosses a threshold.","tokens_in":14224,"tokens_out":1396,"duration_ms":330022,"significance":"The paper addresses a well-motivated question: whether social pressure from beyond direct neighbors promotes cooperation, grounded in empirical evidence for multi-hop social influence. Strengths include a thorough simulation protocol (100 replications, multiple topologies, adaptive stopping criterion), validation against the prior model at L=1 with two update rules, a real-network validation on the CKM physician data, a natural control via the uncorrelated multiplex, and publicly available code. The sensitivity analysis on λ revealing a topology-dependent sharp transition in dense BA networks is a notable non-trivial finding. The connection to empirical decay coefficients (λ≈0.65 from Miranda et al.) adds calibration credibility.","major_comments":[{"comment":"§II.B, Eq. (1): The vigilance kernel is 'intentionally not normalized,' so I_i = min(1, Σ_{d=1}^{L} λ^{d-1} m_d/k_d) is non-decreasing in L for any fixed vigilance configuration. By Eq. (2), T_i = 1 + (b-1)(1-I_i), so higher I_i directly lowers temptation. This means the direction of the headline result—cooperation increases with L—is partly guaranteed by construction in the correlated multiplex. The paper acknowledges this design choice but does not test a normalized alternative (e.g., dividing by Σ λ^{d-1}) to isolate the 'reach' effect from the 'amplification' effect. The uncorrelated-multiplex control (§III.C) partially mitigates this concern: cooperation does not increase with L when layers are decorrelated, despite the same mechanical amplification of I_i, demonstrating that amplification alone is insufficient. However, in the correlated case, both effects are intertwined and the '","section":null},{"comment":"§III.E, Fig. 6: The sharp jump in BA z=16 between λ=0.5 and λ=0.75 (⟨ρ⟩ from ~0.15 to ~0.87) is reported at a single parameter point (b=1.5, θ=0.5, L=4) and described as a transition between coexisting attractors. Given that the λ grid has only 5 values (0.1, 0.25, 0.5, 0.75, 0.9), the transition is resolved by a single step. This is a load-bearing claim for the conclusion that dense networks 'switch abruptly.' A finer λ sweep in the transition region, or at minimum an acknowledgment that the transition width is unresolved, would strengthen this finding.","section":null}],"minor_comments":[{"comment":"The author affiliations contain encoding artifacts (e.g., 'Mar´ ıa', 'Ingenier´ ıa', 'Barab´ asi'). These should be corrected.","section":null},{"comment":"§II.B: The justification for the geometric kernel over the linear kernel of Ref. [21] is reasonable, but the paper could note that the geometric kernel also has the property of being scale-free with respect to network diameter, which is the actual argument made—consider rephrasing for clarity.","section":null},{"comment":"Figures 2 and 4: The legend lists L=1 through L=4 with color/style assignments, but in Fig. 4 the correlated/uncorrelated distinction adds 8 curves per panel. Consider whether separating correlated and uncorrelated into sub-panels would improve readability.","section":null},{"comment":"§III.D: The choice of λ=0.65 for the CKM network validation is well-motivated, but the synthetic results use λ=0.5. A brief note on why the default λ=0.5 was chosen for the main results (beyond 'illustrative') would help the reader.","section":null},{"comment":"Table I and §II.G: The parameter space is large (11×11×4×2×2×2×2 = 7744 cells × 100 replications). A note on total computational cost or wall-clock time, perhaps referencing the GitHub repository's runtime notes, would contextualize the effort.","section":null},{"comment":"§III.A: The high variance in BA z=16 (σ≈0.41) is attributed to bistability. It would help to state explicitly whether the 100 replications use the same network realization with different initial conditions (which the text implies) or different network realizations, as this affects the interpretation of the variance.","section":null},{"comment":"The paper references Supplemental Material figures (S1–S9) but these are not included in the reviewed manuscript. Ensure they are available and properly cross-referenced.","section":null},{"comment":"§IV: The connection to higher-order interactions (hypergraphs, simplicial complexes) is mentioned as future work but feels somewhat tangential to the paper's actual contribution. Consider trimming or making the connection more concrete.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The core structural concern about the non-normalized kernel is real but is partially addressed by the uncorrelated-multiplex control. I judge that the paper's contribution—the topology-dependent phenomenology, the real-network validation, and the λ-sensitivity result—stands on its own merits even acknowledging the kernel design. A normalized-kernel comparison would strengthen the paper but is not strictly necessary for publication if the authors add a clear discussion of the amplification-vs-reach issue. The paper fits the journal's scope well. The single-author-plus-student citation pattern is unremarkable for this type of computational study."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive reading of our manuscript. Both major comments identify genuine gaps that we will address in the revised version. Below we respond point by point.","responses":[{"response":"The referee is correct that the non-normalized kernel makes I_i non-decreasing in L by construction, and that this mechanically lowers T_i as L grows. We acknowledge this design choice in the manuscript but agree that we have not adequately separated the 'reach' effect (accessing vigilant agents at greater distances who were previously invisible) from the 'amplification' effect (adding more weight on top of existing influence). The uncorrelated control demonstrates that amplification alone is insufficient—cooperation does not increase with L when layers are decorrelated despite the same mechanical amplification—but the referee is right that in the correlated case the two effects remain confounded. We will address this in revision by running a normalized-kernel variant, I_i^norm = min(1, Σ_{d=1}^{L} λ^{d-1} m_d/k_d / Σ_{d=1}^{L} λ^{d-1}), which holds the total kernel weight at unity for all L and isolates the reach effect. We will report these results alongside the existing ones for the correlated multiplex on all four topologies. If the cooperation gain persists under normalization (which we expect it will, at least partially, because the reach effect is what enables the Watts threshold to be met by previously invisible vigilant neighbors), this will clarify that the result is not purely an artifact of amplification. We will also add an explicit discussion in §II.B distinguishing the two effects and noting that the non-normalized kernel was chosen to model the empirically motivated scenario in which wider awareness does increase total social pressure, not merely redistribute it.","revision_made":"yes","referee_comment":"§II.B, Eq. (1): The vigilance kernel is intentionally not normalized, so I_i is non-decreasing in L for any fixed vigilance configuration. By Eq. (2), higher I_i directly lowers temptation. This means the direction of the headline result—cooperation increases with L—is partly guaranteed by construction in the correlated multiplex. The paper does not test a normalized alternative to isolate the 'reach' effect from the 'amplification' effect. The uncorrelated-multiplex control partially mitigates this concern but in the correlated case both effects are intertwined."},{"response":"The referee is correct that the current λ grid (5 values) resolves the transition in BA z=16 by a single step, which is insufficient to characterize the transition width or confirm that it is genuinely sharp rather than steep but smooth. We will address this in two ways. First, we will run a finer λ sweep in the transition region (λ = 0.50, 0.55, 0.60, 0.65, 0.70, 0.75) at the representative parameter point (b=1.5, θ=0.5, L=4) for BA z=16, and add the resulting curve to Figure 6. This will allow us to determine whether the transition is genuinely discontinuous (consistent with the bistability interpretation we propose) or merely steep. Second, regardless of the finer sweep's outcome, we will revise the language in §III.E and §IV to acknowledge that the transition width is not fully resolved by the original grid and to temper the claim of abruptness accordingly. If the finer sweep reveals a smooth but steep transition, we will describe it as such rather than as a sharp switch between attractors.","revision_made":"yes","referee_comment":"§III.E, Fig. 6: The sharp jump in BA z=16 between λ=0.5 and λ=0.75 is reported at a single parameter point and resolved by a single step of the λ grid. This is a load-bearing claim for the conclusion that dense networks 'switch abruptly.' A finer λ sweep in the transition region, or at minimum an acknowledgment that the transition width is unresolved, would strengthen this finding."}],"tokens_in":13649,"tokens_out":1232,"duration_ms":83622,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this is a well-executed simulation paper that extends Pereda's 2016 two-layer monitoring model to multi-hop vigilance with geometric decay. The main soft spot is the non-normalized kernel, which the reader and stress-test correctly flag, but the paper's own uncorrelated-multiplex control does meaningful work against that concern. It deserves a serious referee who should push for one additional experiment. What's genuinely new: the extension from L=1 to L circles of influence with decay, the Fermi update rule adaptation (validated against replicator at L=1), the inter-layer correlation analysis showing cooperation gains require aligned game and vigilance layers, and the CKM physician network validation. The λ sensitivity analysis revealing a sharp transition in dense BA networks between λ=0.5 and 0.75 is a nice find. Code is public. 100 replications across a thorough parameter sweep. The paper earns credit for transparency about its design choices and for testing on real network data rather than stopping at synthetic topologies. The soft spot: Eq. (1) is intentionally not normalized, so I_i mechanically increases with L, and Eq. (2) guarantees higher I_i lowers temptation. The direction of the headline result is partly built into the equations. The paper says this is intentional—agents in larger vigilant neighborhoods should feel more pressure—but doesn't test a normalized alternative. That said, the concern is less severe than it looks. The uncorrelated-multiplex control (Sec. III C) is a natural counterfactual: I_i still mechanically increases with L there, yet cooperation does not improve. This shows that mechanical amplification alone is insufficient—the spatial coherence between layers matters. So the kernel isn't doing all the work. Still, a normalized-kernel comparison would cleanly separate whether it's the extended reach or the increased total pressure driving the result. That's the one experiment I'd ask for. The detailed findings—topology dependence, the L=1→2 transition accounting for most gain, the λ threshold effect in dense networks, the inter-layer correlation effect—are more robust than the headline and constitute the real contribution. This paper is for researchers in evolutionary game theory on networks who care about mechanism design. It's a simulation study with a stylized binary model, not theory, and should be read as such. Recommend: accept into peer review. The normalized-kernel comparison is the key revision to request.","headline":"Solid simulation study extending vigilance-based cooperation to multi-hop neighborhoods. The non-normalized kernel is a real but partially mitigated concern.","tokens_in":14808,"tokens_out":577,"would_cite":false,"duration_ms":109449,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Second-circle vigilance boosts cooperation by 30%","keywords":[],"falsifier":"Replace the non-normalized vigilance kernel with a normalized one (e.g., dividing by Σ λ^{d-1}) and check whether cooperation still increases with L. If the gain disappears, the result is an artifact of the kernel's construction rather than a property of long-range social pressure.","tokens_in":13958,"feed_emoji":"","tokens_out":1209,"duration_ms":61842,"temperature":0.7,"pith_summary":"This paper claims that allowing social pressure — the awareness of being watched — to reach beyond one's immediate neighbors to the second and third circles of social contacts substantially raises the level of cooperation that can be sustained in a population, even when the temptation to defect is high. The authors couple a Prisoner's Dilemma on one layer of a two-layer network to a vigilance cascade on the other, where influence from each circle of contacts decays geometrically with distance. The central mechanism is that accumulated vigilance from a wider neighborhood lowers the effective temptation to defect against any given cooperator, making defection less attractive. The paper shows that the single step from first-circle to second-circle vigilance already captures most of the achievable gain, consistent with empirical measurements of how social influence decays with network distance. The effect is strongest in sparse networks where local monitoring alone fails, requires that the people watching you are the same people you interact with, and reproduces on a real physician social network. In dense, hub-dominated networks, cooperation can switch abruptly between low and high regimes depending on how fast influence decays with distance.","feed_headline":"","feed_subtitle":"","key_machinery":"The vigilance kernel I_i = min(1, Σ_{d=1}^{L} λ^{d-1} · m_d/k_d) aggregates the fraction of vigilant agents across L circles of influence with geometric decay. This feeds into the temptation T_i = 1 + (b-1)(1-I_i), so higher vigilance lowers the payoff for defecting. Vigilance itself spreads via a Watts threshold cascade: cooperators become vigilant when their influence index exceeds a threshold θ, creating a feedback loop between monitoring and cooperation.","core_discovery":"The paper establishes that the critical temptation to defect — the payoff threshold above which cooperation collapses — shifts upward by roughly 30% when vigilance extends from the first to the second circle of influence in sparse networks, and by over 50% at four circles. The L=1 to L=2 transition alone accounts for most of the gain, matching the empirically measured decay coefficient of social influence (λ ≈ 0.65). The effect depends on the vigilance layer and the game layer being structurally aligned: when the people monitoring your behavior are not the same people you play against, the benefit of extending vigilance largely disappears. In dense, hub-dominated networks, the outcome is not","pith_inferences":["If the non-normalized kernel is replaced by a normalized version that redistributes rather than amplifies total influence, the qualitative result that cooperation increases with L may weaken or vanish, since the mechanism partly relies on I_i growing with L by construction rather than emerging from the dynamics.","The model's binary vigilance and strategy states may understate the effect: graded vigilance levels could produce smoother transitions and potentially larger cooperative regions, since agents near the threshold would be partially protected rather than all-or-nothing.","The absence of monitoring cost means the model sidesteps the second-order free-rider problem — if vigilance were costly, the wider reach might paradoxically dilute individual incentives to remain vigilant, since the marginal contribution of any single monitor shrinks as L grows."],"forward_implications":["Online platforms that make distant contacts' behavior visible could meaningfully shift cooperative norms in populations where local monitoring is insufficient, even if the added visibility only reaches one additional circle of contacts.","Interventions aimed at sustaining cooperation in sparse communities — small towns, professional networks, online forums — may benefit more from widening the monitoring radius than from increasing the intensity of local monitoring.","The finding that layer alignment is necessary suggests that privacy-preserving designs which decouple who-watches from who-interacts may inadvertently weaken the cooperative benefits of social monitoring.","The sharp regime switch in hub-dominated networks implies that small changes in how influence decays — perhaps driven by platform design choices about information visibility — could tip entire populations between cooperative and defective equilibria."],"fun_headline_variants":["Extending social vigilance beyond direct neighbors boosts cooperation in networks","Second-circle observation strongly promotes cooperation in sparse multiplex networks","Long-range social pressure raises the threshold for defection in multiplex networks","Extending social vigilance to second-degree contacts shifts defection thresholds","Second-circle awareness accounts for most gains in networked cooperative behavior"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The vigilance kernel is deliberately not normalized: each additional circle of influence adds non-negative terms to the total influence score, so the score mechanically increases as L grows. This means the headline result — that cooperation increases with vigilance range — is partly built into the model's construction rather than emerging purely from the dynamics. A normalized kernel that redistributes the same total influence across circles could yield a different conclusion","fun_headline_variants_meta":{"raw":{"variants":["Extending social vigilance beyond direct neighbors boosts cooperation in networks","Second-circle observation strongly promotes cooperation in sparse multiplex networks","Long-range social pressure raises the threshold for defection in multiplex networks","Extending social vigilance to second-degree contacts shifts defection thresholds","Second-circle awareness accounts for most gains in networked cooperative behavior","Long-range observation promotes cooperation when vigilance and game layers align","Extended social pressure reshapes cooperative behavior in sparse network topologies"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1210,"prompt_tokens":542,"completion_tokens":668,"prompt_tokens_details":null},"tokens_in":542,"tokens_out":668,"duration_ms":45914,"temperature":1.0,"reasoning_tokens":656,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T17:43:19.611599+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Replace the non-normalized vigilance kernel with a normalized one (e.g., dividing by Σ λ^{d-1}) and check whether cooperation still increases with L. If the gain disappears, the result is an artifact of the kernel's construction rather than a property of long-range social pressure.","supporting_citations":[],"review_version":1}