{"id":"75da6163-2d78-4b8f-b594-bb2edbea106f","arxiv_id":"2411.12871","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In sparse directed networks, reciprocity is estimated with an effective sample size set by the slower of the two link-count scales, yielding a phase transition in the MLE's limiting distribution.","lead":"This paper derives how much statistical information a sparse directed network carries about reciprocity, the tendency of node pairs to form links in both directions. It shows that two parts of the model shrink at different rates, creating a phase transition, and it provides an inference method that works without knowing those rates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim hinges on exact power-law sparsity with exponents in (0,2); violations of this scaling — not just unknown a,b — invalidate the proposed 'sparsity-free' inference.","rationale":"I independently checked the core asymptotics. For the BR model, the per-dyad Fisher information is I_µµ ≈ 2e^{µ_n}, I_ττ ≈ e^{τ_n}, and the normalized cross term is O(n^{−(a+b)/2}) → 0, so Proposition 1's diagonal covariance and Corollary 1's rates are internally correct. Proposition 2's ρ-pivot simplifies to n(ρ̂ − ρ0)/√(4e^{−µ̂_n} + 2e^{−τ̂_n}), which is the right plug-in pivot in the exact power-law model. The mathematics under Assumption 1 is coherent, so I do not see grounds to reject. The genuine vulnerability is the exactness of the −a log n and −b log n scaling: the plug-in standardizers are not robust to slowly varying sparsity indices, and the paper concedes a and b are unidentifiable, leaving no data-driven check of the regime. This supports the reader's CONDITIONAL verdict: the results are valid as stated, but their practical reach is narrower than the abstract suggests. The additional inconsistencies (effective sample size differing by a factor n, Figure 1 exponent/OCR issues, and swapped covariate labels in Table 3) are presentation errors rather than defects in the main theorem.","tokens_in":11015,"tokens_out":29774,"duration_ms":305755,"concrete_test":"Simulate the BR model with µ_n = −a log n + c log log n and τ_n = −b log n + τ, with a = b = 0.5, c = ±1, and n ∈ {500, 1000, 5000}; construct 95% confidence intervals for µ_n, τ_n, and ρ_n using the Proposition 2 plug-in standardizers. Compare coverage against the exact power-law design (c = 0) at the same n. If the slow-varying design shows materially below-nominal coverage while the exact design attains it, the fixed-(a,b) scaling assumption is load-bearing for the claimed sparsity-free inference.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1 and Corollary 1, and their plug-in counterparts in Propositions 2–3, are valid only when the true sequences are exactly µ_n = −a log n + µ and τ_n = −b log n + τ with fixed a,b in (0,2). This is not merely an unidentified constant: the standardizers n√e^{µ̂_n}, n√e^{τ̂_n}, and the ρ-pivot in Proposition 2 approximate the Fisher information only under this exact power-law form. If the sparsity index varies slowly (e.g., µ_n = −a log n + c log log n), or if max(a,b) ≥ 2, the rates and pivots change or fail. The paper explicitly declares a and b unidentifiable, so a practitioner cannot verify the regime from data. Thus the headline claim that inference operates 'without prior knowledge of sparsity levels' is narrower than stated: it removes the need to know a,b only within the assumed power-law regime. The manuscript also gives inconsistent effective-sample-size statements (n^{1−a/2} versus n^{2−a}), which compounds the difficulty of extracting the answer to Question 1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the Bernoulli model with reciprocity (BR), in which each dyad (Aij, Aji) has probabilities proportional to 1, exp(mu_n), and exp(2 mu_n + rho_n) for the empty, single-arc, and mutual configurations, and asks what the effective sample size is for estimating the density parameter mu_n and the reciprocity parameter rho_n in sparse directed networks. The authors impose the scaling mu_n = -a log n + mu and tau_n = 2 mu_n + rho_n = -b log n + tau with fixed a, b in (0,2), so that the numbers of non-reciprocal and reciprocal dyads grow like n^{2-a} and n^{2-b}. Proposition 1 gives asymptotic normality of the MLE (mu_hat, tau_hat) with rates sqrt(n^{2-a}) and sqrt(n^{2-b}); Corollary 1 shows that rho_hat = tau_hat - 2 mu_hat converges at rate sqrt(n^{2-max{a,b}}) with three regime-dependent variances. Proposition 2 provides plug-in pivots based on n sqrt(exp(mu_hat_n)) and related quantities that do not require a or b to be known. Section 3 extends the analysis to a p1.5 model with node- and dyad-specific covariates: Theorem 1 derives rates sqrt(n^{2-a}) for mu_hat, sqrt(n^{2-min{a,b}}) for the covariate effects eta_hat, and sqrt(n^{2-b}) for (tau_hat, delta_hat), and Proposition 3 gives plug-in inference. The paper concludes with simulations and two data applications (Lazega's lawyer network and a trade network).","tokens_in":11194,"tokens_out":23401,"duration_ms":208820,"significance":"The contribution is substantial if the results hold. The paper gives the first systematic two-exponent treatment of effective sample size for reciprocity in sparse directed networks, including a phase transition in the joint limiting distribution of (mu_hat, rho_hat) across the regimes a < b, a = b, a > b, and it shows that reciprocity is identifiable as long as the number of mutual dyads diverges polynomially. The plug-in pivots of Propositions 2 and 3 are a genuine practical advance: the rho-pivot in Proposition 2 has the correct regime-dependent scaling in all three cases, and the rate statements of Proposition 1 and Corollary 1 are consistent with the expected dyad counts computed in Section 2. The p1.5 extension, with explicit rates for covariate effects in sparse directed networks, goes beyond the existing literature on the p1 model, which offers only implicit rates. The paper ships simulations and two real-data demonstrations, and the coverage results in Table 1 support the plug-in inference claim.","major_comments":[{"comment":"The answer to Question 1 is delivered in two incompatible conventions. The text after Proposition 1 states that n^{1-a/2} and n^{1-b/2} are the effective sample sizes for mu and tau, whereas the text after Corollary 1 states that the effective sample size for rho_hat is O(n^{2-max{a,b}}); these two conventions differ by a square, since Proposition 1's normalization sqrt(n^{2-a}) equals n^{1-a/2} on the standard-deviation scale, while n^{2-a} is the variance-normalization order and equals the expected number of non-reciprocal dyads computed in Section 2. The same shift appears in Section 3, where the effective sample size for delta is given as n^{2-b}. Because Question 1 is the paper's central claim, the authors must adopt a single convention (I would recommend the variance-normalization order n^{2-a}, n^{2-b}, n^{2-max{a,b}}, which matches the dyad counts) and state explicitly that the normalizers in Proposition 1, Proposition 2, and Corollary 1 are the square roots of those effective sample sizes.","section":"Section 2, after Proposition 1 and after Corollary 1"},{"comment":"The scope claim 'All our results hold under Assumption 1, meaning they apply to arbitrarily sparse networks' overstates what is proved. Assumption 1 restricts to the exact polynomial scaling mu_n = -a log n + mu, tau_n = -b log n + tau with fixed a, b in (0,2); the rates in Proposition 1 and Corollary 1 are tied to that form, so a slowly varying perturbation such as mu_n = -a log n + c log log n changes the rates, and a or b outside (0,2) changes the stated normalizations. The claim that consistency fails without a < 2 and b < 2 is also too strong: if b >= 2 but a < 2, mu remains estimable from the non-reciprocal dyads while only tau and rho fail, so the necessity statement should be phrased parameter by parameter. Because the paper itself notes that a, b, mu, and tau are not identifiable from a single network, the regime cannot be verified from the data, and the abstract's phrase 'operates without prior knowledge of sparsity levels' should be scoped accordingly: the procedure removes the need to know a and b only within the assumed power-law regime.","section":"Section 2, after Assumption 1; Abstract"},{"comment":"Theorem 1 is the main result of Section 3, but its content, the covariance matrix 2H^{-1}(theta) with its regime-dependent block structure, is deferred entirely to the Supplementary Materials, which are not part of the posted manuscript. Because the displayed vector mixes three different rates (sqrt(n^{2-a}) for mu_hat, sqrt(n^{2-min{a,b}}) for eta_hat, sqrt(n^{2-b}) for phi_hat), the key information is which blocks of H are non-degenerate in each of the cases a < b, a = b, a > b, and the follow-up text in Section 3 makes assertions about correlations and rates that can only be checked against that matrix. The main text should display at least the block form of H(theta) for the three regimes, or the authors should ensure the supplement is available with the submission so that the theorem can be verified.","section":"Section 3, Theorem 1"}],"minor_comments":[{"comment":"The typeset rho-pivot in Proposition 2 is ambiguous about which factors sit under the square roots; the intended formula appears to be n sqrt( exp(mu_hat_n) exp(2 mu_hat_n + rho_hat_n) / (2 exp(mu_hat_n) + 4 exp(2 mu_hat_n + rho_hat_n)) ) (rho_hat_n - rho_n0) -> N(0,1), which does have the correct scaling in all three regimes, but it should be typeset with explicit braces.","section":"Section 2, Proposition 2"},{"comment":"The notation exp^{-1}(2 mu0 + rho0) and exp^{-1}(mu0) is ambiguous, since exp^{-1} could be read as the inverse function log rather than the reciprocal; the variances should be written as e^{-(2 mu0 + rho0)} and e^{-mu0}.","section":"Corollary 1 and Figure 1"},{"comment":"The sentence claiming that H_n(theta_hat_n) 'does not converge to a deterministic matrix' is misleading; after scaling by n^{min{a,b}} the Hessian does converge to a deterministic matrix depending on mu0, tau0, and the covariate distribution, so the convergence rate is explicit up to that scaling.","section":"Section 3, discussion after Proposition 3"},{"comment":"The claim that no formal inference procedures are currently available for the p1 model is in tension with the literature review, which credits Yan et al. (2016, 2019) with asymptotic properties of the p1 maximum likelihood estimator; the sentence should be clarified to distinguish estimators from formal inference procedures.","section":"Section 1.1 and the literature review"},{"comment":"The statement that the rates 'are explicit for arbitrarily sparse networks as long as a, b > 0' is inaccurate on two counts: the rates are explicit only under exact power-law scaling with a, b in (0,2), and a, b > 0 alone does not suffice.","section":"Section 3, remark after Theorem 1"},{"comment":"The QQ plots in Figure 2 are provided only for the case a = b = 0.5; adding a setting with a != b would directly illustrate the regime-dependent rates in Theorem 1, and Table 1 should report the number of replications or standard errors for the coverage estimates.","section":"Section 4, simulations"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of a top statistics methodology journal and is a natural continuation of Chen et al. (2021) and Krivitsky and Kolaczyk (2015); I see no circularity concern in the self-citations, which are to background work in the same research program. My recommendation of major revision rests on the two main-text issues: the inconsistent effective-sample-size convention and the gap between the 'no prior knowledge of sparsity' claim and the actual power-law regime; both are fixable with local revision. The supplementary materials containing the proofs and the explicit H(theta) matrix need to be available at review time, since Theorem 1 is not fully checkable from the main text as posted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has a genuinely new and useful result—the effective sample size analysis for directed networks when the sparsity of reciprocal and non-reciprocal links can differ—and the main asymptotics look correct to me. But the manuscript is in a state where several presentation mistakes need to be fixed before the formulas can be used as stated.\n\nThe real contribution is allowing a and b to differ in the Bernoulli-with-reciprocity model. Krivitsky and Kolaczyk only handled equal effective sample sizes; here the two sparsity indices are decoupled, and Corollary 1's phase transition for rho-hat is a nice piece of theory. The plug-in pivots in Proposition 2 are also genuinely useful: n*sqrt(e^{mu-hat_n}) and its analogues give valid inference without knowing a or b, and the p1.5 covariate model is a reasonable extension that should find applications.\n\nThe soft spots are mostly presentational, but they are real. First, the effective sample size is defined as n^{1-a/2} in the discussion after Proposition 1 and later as n^{2-max{a,b}} after Corollary 1. Those differ by a factor of n. The n^{2-a} quantity is the expected number of non-reciprocal links and is the right effective sample size; n^{1-a/2} is the square root of that and should be called a standardizer, not the effective sample size. Second, Figure 1 has errors: in the a > b case the relation should be rho-hat - rho_0 = -2(mu-hat - mu_0) + o_p, and the a = b covariance matrix does not match Corollary 1. Third, Theorem 1's H(theta) is deferred entirely to the Supplementary Materials; for a paper whose selling point is explicit rates, that leaves a gap in the main text. Fourth, the literature review claims no cited directed-network paper considered reciprocity, but Yan et al. (2016) studies the p1 model, which includes a reciprocity parameter. The novelty should be framed as allowing unequal sparsity regimes, not as being first to incorporate reciprocity.\n\nThe stress-test note worries that the plug-in inference only works under the exact power-law scaling. I think that is overstated: the pivots are self-normalizing and should hold for any sequence with p_n -> 0 and diverging counts, not just exact power laws. The paper only proves the narrower statement, but the concern is not load-bearing.\n\nWho is this for? Statisticians doing asymptotics for sparse directed networks, and applied researchers who want a reciprocity model with covariates and straightforward confidence intervals. It deserves a serious referee; I would send it out. It needs a careful revision before acceptance, but the core is sound.","headline":"Worth a serious referee: the decoupled-sparsity reciprocity result is new and the asymptotics look right, but the paper needs careful revision to fix effective-sample-size definitions, figure errors, and an overbroad novelty claim.","tokens_in":11764,"tokens_out":11321,"would_cite":true,"duration_ms":99770,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F12","62E20","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"In sparse directed networks, reciprocity can be estimated at a rate set by the sparser of the two link configurations.","keywords":["directed networks","reciprocity","effective sample size","sparse networks","maximum likelihood estimator","Bernoulli model with reciprocity","phase transition","p1.5 model"],"falsifier":"Simulate the BR model with known sparsity indexes $a$ and $b$, compute the MLE, and check the predicted correlations and rates: when $a<b$, the studentized $\\hat{\\mu}$ and $\\hat{\\rho}$ should be asymptotically independent, while when $a>b$ they should satisfy $\\hat{\\rho}-\\rho_0 = -2(\\hat{\\mu}-\\mu_0) + o_p(1)$; a decisive numerical check is whether the empirical variance of $\\hat{\\rho}$ scales as $n^{-(2-\\max\\{a,b\\})}$ exactly across the three regimes.","tokens_in":10723,"feed_emoji":"🔁","tokens_out":8864,"duration_ms":72729,"temperature":0.7,"pith_summary":"This paper addresses a fundamental question about directed network models that include reciprocity: when the network is sparse, how many dyads actually inform estimation of the baseline tendency to form directed links and the extra tendency to form mutual links? Working in the Bernoulli model with reciprocity, the authors allow the two tendencies to have different sparsity levels, measured by constants $a$ and $b$ in a $\\log(n)$ scaling, and show that the maximum likelihood estimators converge at rates $\\sqrt{n^{2-a}}$ and $\\sqrt{n^{2-b}}$, so the effective sample sizes are $n^{2-a}$ and $n^{2-b}$. The reciprocity parameter itself is estimated at the slower of these two rates, $\\sqrt{n^{2-\\max\\{a,b\\}}}$, with a phase transition in its asymptotic distribution when $a=b$. The same framework extends to a covariate model, called the p1.5 model, and inference can be run without knowing the sparsity indexes. If correct, reciprocity is identifiable in any sparse directed network whose expected number of mutual links diverges.","feed_headline":"Reciprocity in sparse networks has explicit effective sample size","feed_subtitle":"Two sparsity indexes set the MLE rates; reciprocity is identifiable as long as mutual links diverge.","key_machinery":"The engine of the analysis is the Bernoulli model with reciprocity, reparametrized by $(\\mu_n, \\tau_n)$ where $\\tau_n = 2\\mu_n + \\rho_n$. The scaling $\\mu_n = -a\\log n + \\mu$, $\\tau_n = -b\\log n + \\tau$ makes the expected numbers of non-reciprocal and reciprocal dyads grow as $n^{2-a}$ and $n^{2-b}$, so $a$ and $b$ act as sparsity indexes for the two kinds of links. Proposition 1 establishes asymptotic normality of the MLE with Fisher information $\\mathrm{diag}(e^{\\mu_0}, e^{\\tau_0}/2)$, and Corollary 1 reads off the rate and phase-transition behavior of $\\hat{\\rho} = \\hat{\\tau} - 2\\hat{\\mu}$. The practical inference procedures in Proposition 2 and Proposition 3 use the estimated Hessian to avoid requiring knowledge of $a$ and $b$.","core_discovery":"The central claim is that, under the scaling $\\mu_n = -a\\log n + \\mu$ and $\\tau_n = -b\\log n + \\tau$ with fixed $a,b \\in (0,2)$, the Bernoulli model with reciprocity is identifiable in sparse directed networks: the MLE of $(\\mu,\\tau)$ is consistent and asymptotically normal with rates $\\sqrt{n^{2-a}}$ and $\\sqrt{n^{2-b}}$, and the limiting covariance matrix is diagonal with entries $e^{\\mu_0}$ and $e^{\\tau_0}/2$. Consequently, the reciprocity parameter $\\rho = \\tau - 2\\mu$ is estimated at rate $\\sqrt{n^{2-\\max\\{a,b\\}}}$, and its limiting distribution undergoes a phase transition: when $a<b$ it is independent of $\\hat{\\mu}$, when $a>b$ it is perfectly negatively correlated with $\\hat{\\mu}$, and when $a=b$ an additional variance term appears. The paper also establishes these rates for a covariate extension, the p1.5 model, and proves that studentized inference is valid without knowing $a$ or $b$.","pith_inferences":["A practical consequence the paper does not spell out: when $a>b$, confidence intervals for reciprocity should be interpreted as intervals for the density difference, but the data cannot separate $\\rho$ from $\\mu$; applied work on mutual links should either report the pair or impose a model constraint.","The phase transition suggests a diagnostic: estimate the sparsity indexes from observed counts of non-mutual versus mutual dyads, then check which regime the network occupies; the paper's rates translate directly into such a diagnostic without requiring a new fitting procedure.","The p1.5 framework assumes covariates fully account for node heterogeneity, but the rates imply that if node-level fixed effects were added, their effective sample size would be $n^{2-\\min\\{a,b\\}}$, potentially making inference very difficult for sparse networks; the covariate parametrization is thereby a deliberate trade-off.","For multiple networks observed over time, the explicit rates allow pooling: if each network contributes an effective sample size of order $n^{2-\\max\\{a,b\\}}$, a pooled estimator of reciprocity would improve at that same rate, a testable prediction for longitudinal network data."],"forward_implications":["The effective sample size for the link-density parameter $\\mu$ is $n^{2-a}$; for the mutual-link parameter $\\tau$ it is $n^{2-b}$; for reciprocity $\\rho$ it is the smaller of the two, $n^{2-\\max\\{a,b\\}}$.","When $a<b$, $\\hat{\\mu}$ and $\\hat{\\rho}$ are asymptotically independent; when $a>b$, $\\hat{\\rho}$ is asymptotically a deterministic linear transform of $\\hat{\\mu}$, so the data do not separately identify reciprocity beyond the baseline density.","Studentized maximum likelihood inference works without prior knowledge of $a$ or $b$, so practitioners do not need to estimate the sparsity level first.","In the p1.5 covariate model, node-specific outgoingness and incomingness parameters are estimated at rate $\\sqrt{n^{2-\\min\\{a,b\\}}}$, and link-specific reciprocity covariate effects at rate $\\sqrt{n^{2-b}}$."],"supporting_citations":[{"why":"Introduced the Bernoulli model with reciprocity and posed the effective sample size question, giving the special case $a=b=1$ that this paper generalizes.","marker":"Krivitsky & Kolaczyk (2015)"},{"why":"Supplies the sparse-network scaling and effective sample size methodology that the authors extend to two sparsity indexes.","marker":"Chen et al. (2021)"},{"why":"Defined the p1 model with reciprocity, which the p1.5 covariate model builds upon.","marker":"Holland & Leinhardt (1981)"},{"why":"Represents the directed network models without reciprocity whose implicit convergence rates are compared against the explicit rates obtained here.","marker":"Yan et al. (2019)"},{"why":"Established asymptotics for directed exponential random graph models without reciprocity, providing the comparison case for the p1.5 results.","marker":"Yan et al. (2016)"}],"fun_headline_variants":["Explicit effective sample size for sparse directed reciprocity","Reciprocity identifiable in sparse networks: explicit rates","Two sparsity exponents control MLE rates for reciprocity","Sparse reciprocity: effective sample size is explicit","Reciprocity has an explicit effective size in sparse nets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theory assumes the sparsity indexes $a$ and $b$ are fixed constants in $(0,2)$, so both the expected number of non-reciprocal links and the expected number of mutual links diverge polynomially; if $a$ or $b$ falls outside this range, the stated rates and the inference procedure do not apply.","fun_headline_variants_meta":{"raw":{"variants":["Explicit effective sample size for sparse directed reciprocity","Reciprocity identifiable in sparse networks: explicit rates","Two sparsity exponents control MLE rates for reciprocity","Sparse reciprocity: effective sample size is explicit","Reciprocity has an explicit effective size in sparse nets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000868,"raw_usage":{"total_tokens":3739,"prompt_tokens":904,"completion_tokens":2835,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":2758}},"tokens_in":520,"tokens_out":2835,"duration_ms":19551,"temperature":1.0,"reasoning_tokens":2758,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:07:29.148969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the BR model with known sparsity indexes $a$ and $b$, compute the MLE, and check the predicted correlations and rates: when $a<b$, the studentized $\\hat{\\mu}$ and $\\hat{\\rho}$ should be asymptotically independent, while when $a>b$ they should satisfy $\\hat{\\rho}-\\rho_0 = -2(\\hat{\\mu}-\\mu_0) + o_p(1)$; a decisive numerical check is whether the empirical variance of $\\hat{\\rho}$ scales as $n^{-(2-\\max\\{a,b\\})}$ exactly across the three regimes.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the Bernoulli model with reciprocity and posed the effective sample size question, giving the special case $a=b=1$ that this paper generalizes."},{"cited_title":"& Leng, C","cited_arxiv_id":null,"evidence_quote":"Supplies the sparse-network scaling and effective sample size methodology that the authors extend to two sparsity indexes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defined the p1 model with reciprocity, which the p1.5 covariate model builds upon."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Represents the directed network models without reciprocity whose implicit convergence rates are compared against the explicit rates obtained here."},{"cited_title":"& Zhu, J","cited_arxiv_id":null,"evidence_quote":"Established asymptotics for directed exponential random graph models without reciprocity, providing the comparison case for the p1.5 results."}],"review_version":1}