{"id":"30ca90a9-3579-4894-9bb7-257b96b28cd4","arxiv_id":"2510.12028","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Local neighbor comparisons can make a globally fair rule feel unfair, but the paper's main theorem does not actually prove the claimed linear amplification under demographic parity.","lead":"This paper introduces a model where people judge fairness by comparing their own outcomes with those of their network neighbors, not with society at large. It claims that even a rule satisfying demographic parity can look unfair when social ties are segregated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's proof derives a linear gap that vanishes under DP; the degree-exposure rescue is asserted without derivation, leaving the central advertised claim unproven.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: Theorem 3.1's proof derives a first-order term that vanishes under DP, and the degree-exposure rescue in Remark 3.2 is asserted without proof. This is an internal correctness issue, not a matter of outside consensus. The proof's use of unweighted group means in Eqs. (8)–(10) is a concrete error because Prop. 4.2 shows neighbor expectations are degree-weighted; the claimed Γ(h) includes a term from neighbor-exposure differences that the proof never derives. If this gap is not bridged, the paper's main advertised conclusion—that DP rules can show linearly growing perceived discrimination—remains unproven. The simulations do not rescue the theory because they generate outcomes that depend directly on group membership (α=0.7), so the global gap is not exactly zero and DP is not satisfied by construction. The verdict of REJECT remains appropriate; the preprint is not internally consistent enough to establish its central claim. I agree with the reader that the core idea is plausible and possibly salvageable, but the current formulation is unsupported.","tokens_in":10046,"tokens_out":7783,"duration_ms":83021,"concrete_test":"Re-derive the first-order expansion in Theorem 3.1 using the degree-weighted neighbor expectation, E[h(J)|J∈N(i),S_i=s]=E[d_J h(J)|S_J]/E[d_J|S_J], in the two-block SBM. For h satisfying DP (e.g., h(i)=0.5+β(d_i−E[d_i|S_i])), compute the coefficient of ρ in E[Δ1(h)] for πA≠πB. Numerically verify for n≥10^4, ρ∈[10^−3,10^−1]. If the coefficient is zero, the claimed linear mechanism is absent; if nonzero, the statement may hold but the proof is still incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that a DP-satisfying rule yields a perceived gap growing linearly in homophily — rests on Theorem 3.1. But the proof's expansion collapses under DP. In Step 3 the derived expression is ψσ(0)(μA−μB)[1−4ρπAπB]+o(ρ), which is o(ρ) when μA=μB (the DP condition). Equation (12) shows m_s is O(ρ) only if μA≠μB; under DP the leading term vanishes. Remark 3.1 admits this and points to degree-exposure effects (Remark 3.2), where E[Δ1] is claimed to be ρ·κ·Cov(h,d)+o(ρ), but no derivation is given. In fact, the proof's Eqs. (8)–(10) use unweighted group means μs for neighbors, implicitly assuming h is degree-independent within groups. The correct neighbor mean is degree-weighted (Prop. 4.2), and the difference between unweighted and degree-weighted means is exactly what could survive DP. Thus Theorem 3.1 as stated and proved does not establish the advertised result; the linear-in-ρ gap under DP is an asserted conjecture, not a theorem. Proposition 4.3 repeats the same flawed expansion. The central contribution—that network topology must be included in fairness audits—is unsupported without a valid derivation of the degree-exposure mechanism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a network-based model of 'perceived fairness,' in which individuals compare their own outcome to the average outcome of their network neighbors. The main advertised claim is that even a decision rule satisfying demographic parity (DP) can generate a nonzero and, under homophily, linearly growing perceived fairness gap. The paper proves that as neighborhoods grow, perceived fairness converges to objective fairness (Prop. 3.1/4.1); that degree–outcome correlation creates exposure bias (Prop. 4.2); and, as central results, that homophily linearly amplifies the perceived gap under DP (Thm. 3.1 and Prop. 4.3). Additional propositions claim that modularity bounds the gap and that clustering reduces dispersion.","tokens_in":10438,"tokens_out":11077,"duration_ms":88324,"significance":"If rigorously established, the central result would be valuable: it would show that standard demographic-parity audits are insufficient in networked environments and that network topology must be part of fairness assessment. The paper's setting is well motivated, and the definitional framework (local comparison operators, exposure bias) is a useful contribution. However, the central theorem's proof is invalid for the DP case, and the advertised linear amplification under DP is not actually derived; it is asserted in a remark. The paper also relies on unproven or externally deferred results for other propositions. Because the main claim is the paper's raison d'être, the significance of the current manuscript is limited until a correct proof is supplied.","major_comments":[{"comment":"The proof's Step 3 derives E[Δ1] = ψσ(0)(μA−μB)[1−4ρπAπB] + o(ρ). Under DP, μA=μB, so this expression is o(ρ) — there is no linear-in-ρ term. Remark 3.1 concedes this and points to a degree-exposure mechanism in Remark 3.2, but that mechanism is asserted without derivation. The 'transcribing' step that converts the coefficient into c(πA,πB)ρΓ(h) is algebraically inconsistent: with μA=μB the displayed expression has no linear term. Moreover, Eqs. (8)–(10) use unweighted group means, implicitly assuming h is degree-independent within groups, even though Γ(h) includes degree-weighted neighbor exposures. The theorem's central claim is therefore not established.","section":"§3.6, Theorem 3.1"},{"comment":"This proposition repeats the same linear-response expansion as Theorem 3.1. It again uses unweighted group means and does not handle degree-dependent h, despite the statement of Γ(h) involving degree-weighted exposures. No new argument is provided to overcome the DP cancellation identified in Remark 3.1. The claimed result that homophily linearly amplifies the perceived gap under DP is thus unsupported.","section":"§4.2, Proposition 4.3"},{"comment":"These results are used in the paper's summary of 'takeaways' but are not proven. Proposition 3.2 is given only as a proof idea ('expand the group difference using B...'). Proposition 3.3's proof is deferred to Charpentier and Ratz (2025), a same-author arXiv preprint, without reproducing the argument. A theoretical result whose proof is entirely in a different, not-yet-published source is not an established result in this manuscript.","section":"§3.7–3.8, Propositions 3.2 and 3.3"},{"comment":"The simulation's data-generating process violates DP: H_i = 0.7 H_group + 0.3 H_degree + ε, with H_group ~ Beta(4,2) for group A and Beta(2,4) for group B. These Beta distributions have means 2/3 and 1/3, so E[H|S] is substantially different across groups. The figure legend states the global fairness gap 'remains nearly constant,' not that it is zero. The simulation therefore cannot confirm the DP-specific claim of Theorem 3.1.","section":"§5, Simulation section"}],"minor_comments":[{"comment":"The perception indicator uses a strict/non-strict inequality; the smooth-approximation argument later assumes a continuous density at 0. The exposition would benefit from a precise convention on the tie case, though this does not affect the main criticism.","section":"§3.4, Eq. (3)"},{"comment":"No error bars, confidence intervals, or number of simulation replications are reported. Since the claim is about a linear trend, some measure of variability is needed.","section":"§5, Figure 1"},{"comment":"The terms 'objective fairness,' 'global fairness,' and 'demographic parity' are sometimes used interchangeably; the paper should consistently distinguish the decision rule property (DP) from the population-level perception limit in Prop. 3.1.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central theorem (Thm. 3.1) is not proven: the expansion derived in the proof cancels under DP, and the degree-exposure mechanism that would rescue the claim is only asserted. The same flaw propagates to Prop. 4.3. The paper also relies on a same-author unpublished work for Prop. 3.3 and only a proof idea for Prop. 3.2. Given that the advertised contribution rests entirely on the unproven linear-amplification claim, I cannot recommend publication in its current form. If the authors can provide a rigorous derivation of the degree-exposure term and prove all three main propositions self-containedly, a resubmission might be considered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me cut to the chase. This paper brings a genuinely nice idea: define perceived fairness as a local, network-dependent comparison and show that global parity can coexist with local disparity. The formal setup—the perception operator, neighborhood expectations, the distinction between node and edge averages—is clean and well motivated. The convergence result (Prop 3.1/4.1) is straightforward and correct. Prop 4.2, the degree-weighted exposure bias identity, is a clean and useful statement. The paper cites the right literature on friendship paradox and homophily.\n\nThe problem is the central advertised result. Theorem 3.1 claims that for any DP-satisfying rule, the perceived gap E[Δ1] grows linearly with homophily ρ. But the proof's own Step 3 derives E[Δ1] = ψσ(0)(μA−μB)[1−4ρπAπB] + o(ρ). Under demographic parity, μA = μB, so that first-order term vanishes. The author acknowledges this in Remark 3.1 and then points to a degree-exposure mechanism in Remark 3.2, where E[Δ1] is asserted to equal ρ·κ·Cov(h,d) + o(ρ). No derivation is given. That is the load-bearing step, and it is missing. Proposition 4.3 repeats the same expansion and has the same gap. So the paper does not establish the linear-in-ρ gap under DP. The claim is a conjecture dressed as a theorem.\n\nThe simulations do not rescue it. The outcome model has group differences in the Beta base distributions, so global fairness does not hold by construction. The figure shows a rising perceived gap, but it is not testing the DP scenario. Error bars are absent, and the code link is a placeholder. A few supporting propositions are proof ideas or deferred to a same-author companion paper (Charpentier and Ratz 2025).\n\nWhat is salvageable: the operator formalism and the qualitative intuition that degree-outcome correlation plus homophily distort local perceptions are plausible and likely correct. But the proof needs a real derivation of the degree-exposure term, or a different theorem. As it stands, the central claim is unproven.\n\nThe paper deserves a serious referee—the question is timely and the framework is useful—but an editor should send it out expecting that major revision will be required. I would recommend reject-and-resubmit or a strong 'major revisions.'","headline":"Useful formal framework for perceived fairness in networks, but the central theorem that DP rules yield a linear perception gap is not proven—the paper's own expansion cancels under DP and the rescue mechanism is asserted, not derived.","tokens_in":10873,"tokens_out":3479,"would_cite":false,"duration_ms":30998,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91D30","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A decision rule can satisfy demographic parity population-wide and still appear systematically discriminatory when individuals judge fairness by comparing their own outcome with the average outcome of their network neighbors.","keywords":["perceived fairness","demographic parity","homophily","stochastic block model","friendship paradox","network topology","algorithmic fairness","assortative mixing"],"falsifier":"Simulate a two-block network with equal group averages and outcomes independent of degree, then measure the perceived fairness gap as the homophily index increases; if the gap does not grow linearly, the asserted degree-exposure mechanism fails. Alternatively, compute the exact first-order coefficient in rho under demographic parity — the paper leaves it as an unspecified kappa(pi_A, pi_B) — and check whether it is nonzero.","tokens_in":9934,"feed_emoji":"⚖️","tokens_out":6251,"duration_ms":51598,"temperature":0.7,"pith_summary":"The paper argues that fairness as experienced by people is a local, network-dependent quantity rather than a global statistic. It defines perceived fairness as the fraction of individuals in a group whose own outcome is at least the average outcome of their neighbors, and studies how this quantity behaves under homophily. In a two-group network with assortative mixing, the paper claims that even a rule satisfying demographic parity produces a perceived fairness gap between groups that grows linearly with the homophily index. It also proves that perceived fairness converges to objective fairness as comparison neighborhoods expand, that degree-outcome correlation creates exposure bias, and that clustering dampens the dispersion of perceived fairness. If the main theorem is right, standard fairness audits that check only population-level parity can certify a rule while a perception of discrimination persists in segregated networks.","feed_headline":"Segregation makes globally fair rules feel unfair","feed_subtitle":"When people judge by neighbors' outcomes, segregation creates a discrimination gap that global parity misses.","key_machinery":"The central object is the fairness perception indicator F^(d)(i;h) = 1{E_i^(d)[h] <= h(i)}, which records whether individual i's own acceptance probability is at least the average acceptance probability in its d-neighborhood. The argument is carried by the contrast between two exposure averages: the node average and the edge-weighted average, whose difference equals Cov(d,h) / E[d], the friendship-paradox identity. Homophily enters through a two-block stochastic block model parameterized by rho = (p_in - p_out)/(p_in + p_out), and a first-order expansion of the group-level perception probability in rho is what converts topology into a perceived fairness gap. The degree-exposure mechanism des","core_discovery":"The paper's central claim is Theorem 3.1: in a homophilous two-block stochastic block model, the expected depth-1 perceived fairness gap E[Delta_1(h)] between groups equals c(pi_A, pi_B) rho Gamma(h) + o(rho), so it is generically non-zero and grows linearly with homophily even when the decision rule h satisfies demographic parity. The paper interprets this as evidence that perceived discrimination is not a sign of a biased decision rule but an emergent property of local observation: people benchmark themselves against neighbors, and homophily makes those benchmarks differ systematically by group. A corollary (Proposition 3.2) bounds the gap by the network's modularity, and Proposition 4.4 s","pith_inferences":["This suggests a direct empirical test: in lending, insurance, or workplace data, self-reported perceived discrimination should track neighborhood composition and degree, not just global outcome gaps.","A policy consequence the paper leaves implicit is that changing what people can compare — for instance, by publishing global statistics or adding cross-group ties — could reduce perceived unfairness without altering the allocation rule.","The linear-response technique should extend to directed, weighted, or temporally evolving networks; the degree-covariance term would become a measure of how strongly outcome correlates with neighbor-sampling probability.","The convergence theorem implies a measurement caveat: in large sparse networks the diameter is large, so at practical observation depths the perceived gap can remain substantial even though it eventually vanishes."],"forward_implications":["Fairness audits that ignore network structure can approve a rule that every group in a segregated network perceives as systematically unfair.","The perceived fairness gap grows with the homophily index, so more segregated interaction networks deepen perceived discrimination without any change in the decision rule.","Deepening the comparison neighborhood reverses the effect: on a connected graph, perceived fairness converges to demographic parity as the neighborhood radius grows.","If outcomes are positively correlated with degree, the average person sees neighbors with better outcomes than the population average, a downward bias in perceived fairness.","Degree-equalizing or clustering-reducing rewires weaken the dispersion of perceived fairness, so network design can shape fairness experience without changing allocations."],"fun_headline_variants":["Even fair algorithms look biased in homophilous networks","Local peers, not global stats, drive perceived discrimination","Homophily inflates fairness gaps despite parity","Perceived unfairness emerges from network structure alone","Fair rules, unfair neighborhoods: the perception gap"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central result relies on an unproved degree-exposure lemma: with equal average outcomes between groups, neighbor sampling biased by degree must still create a perceived gap that grows with segregation — if that mechanism fails, parity-fair rules show no such gap.","fun_headline_variants_meta":{"raw":{"variants":["Even fair algorithms look biased in homophilous networks","Local peers, not global stats, drive perceived discrimination","Homophily inflates fairness gaps despite parity","Perceived unfairness emerges from network structure alone","Fair rules, unfair neighborhoods: the perception gap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000422,"raw_usage":{"total_tokens":1961,"prompt_tokens":656,"completion_tokens":1305,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":400,"completion_tokens_details":{"reasoning_tokens":1246}},"tokens_in":400,"tokens_out":1305,"duration_ms":8486,"temperature":1.0,"reasoning_tokens":1246,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:00:14.236904+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a two-block network with equal group averages and outcomes independent of degree, then measure the perceived fairness gap as the homophily index increases; if the gap does not grow linearly, the asserted degree-exposure mechanism fails. Alternatively, compute the exact first-order coefficient in rho under demographic parity — the paper leaves it as an unspecified kappa(pi_A, pi_B) — and check whether it is nonzero.","supporting_citations":[],"review_version":1}