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REVIEW 3 major objections 6 minor 20 references

Modelling Directed Networks with Reciprocity

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In sparse directed networks, reciprocity can be estimated at a rate set by the sparser of the two link configurations.

desk verdict 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. read the letter →

arxiv 2411.12871 v1 pith:FPBUBG5B submitted 2024-11-19 stat.ME

Rui Feng , Chenlei Leng This is my paper · ORCID
classification stat.ME MSC 62F1262E2005C80
keywords directednetworksreciprocityeffectivesamplesizesparsemaximumlikelihoodestimatorBernoullimodelwithphasetransitionp1.5
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper 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.

What carries the argument

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 $b$.

What would settle it

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.

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Extended reading notes

Core claim

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 $b$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $\sqrt{n^{2-b}}$.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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).

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 (3)
  1. [Section 2, after Proposition 1 and after Corollary 1] 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.
  2. [Section 2, after Assumption 1; Abstract] 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.
  3. [Section 3, Theorem 1] 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.
minor comments (6)
  1. [Section 2, Proposition 2] 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.
  2. [Corollary 1 and Figure 1] 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}.
  3. [Section 3, discussion after Proposition 3] 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.
  4. [Section 1.1 and the literature review] 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.
  5. [Section 3, remark after Theorem 1] 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.
  6. [Section 4, simulations] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the MLE asymptotics are derived from the stated BR/p1.5 likelihood under an explicit sparsity scaling, with self-citations used only for background and interpretation.

full rationale

The paper's central results (Proposition 1, Corollary 1, Theorem 1, and the plug-in pivots in Propositions 2 and 3) are derived from the negative log-likelihoods of the stated Bernoulli-with-reciprocity and p1.5 models under the explicit scaling mu_n = -a log n + mu and tau_n = -b log n + tau, with a,b in (0,2). The asymptotic variances and rates are obtained by direct Taylor expansion of the likelihood, not by fitting parameters and then reusing them as predictions. The plug-in results in Proposition 2 are algebraically equivalent to Proposition 1 under the same scaling, because n sqrt(exp(mu_n0)) = n^{1-a/2} sqrt(exp(mu0)) and similarly for tau; this is a legitimate reparametrization, not a circular definition. Self-citations (e.g., Chen et al. 2021 and Krivitsky & Kolaczyk 2015) are invoked for background, for the scaling convention, and for interpreting effective sample sizes after the rates are already derived; none of these citations supplies the load-bearing theorem. The claim that inference works 'without prior knowledge of sparsity levels' is a consequence of the plug-in pivots, and concern about whether the assumed power-law scaling holds in practice is a robustness/correctness issue, not circularity. No step reduces by construction to its own inputs, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The p1.5 model is a new statistical model, but it introduces no new physical or mathematical entities beyond parameters and covariates. No invented particles, forces, or conserved quantities appear. The asymptotic theory uses no fitted free parameters; the sparsity indices a and b are part of the assumed scaling regime, not numbers fit to data.

assumptions (5)
  • domain assumption The Bernoulli model with reciprocity is correctly specified and dyads are independent across disjoint node pairs.
    This is the data-generating model stated in Section 2. If the true network has dyadic dependence beyond reciprocity, the MLEs are still defined but the stated covariance matrix is wrong.
  • ad hoc to paper Sparsity scaling mu_n = -a log n + mu and tau_n = -b log n + tau with fixed a,b in (0,2) and true values in compact interiors.
    This asymptotic regime is chosen to match the Erdos-Renyi scalability and makes the expected link counts polynomial. It is not derived from data.
  • domain assumption Assumption 2: true covariate coefficients lie in a compact set and covariate dimensions are fixed.
    Standard compactness assumption for maximum likelihood asymptotics in the p1.5 model.
  • domain assumption Assumption 3: covariates are centered, i.i.d., uniformly bounded, independent across disjoint dyads, and the covariance of (Z_ij + Z_ji, V_ij) is positive definite.
    Needed for the law of large numbers and the nonsingular information matrix for the covariate parameters.
  • standard math Standard maximum likelihood regularity: the log-likelihood is smooth and the parameter space is compact, so the argmin exists and consistency follows.
    The paper does not state a full set of regularity conditions but relies on them implicitly for Propositions 1 through 3.

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Cite this review

Pith. "Pith review of Modelling Directed Networks with Reciprocity." pith.science (2026). https://pith.science/paper/FPBUBG5B

@misc{pith2026241112871,
  author       = {Pith},
  title        = {Pith review of: Modelling Directed Networks with Reciprocity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FPBUBG5B}},
  note         = {Machine review of arXiv:2411.12871}
}
read the original abstract

Asymmetric relational data is increasingly prevalent across diverse fields, underscoring the need for directed network models to address the complex challenges posed by their unique structures. Unlike undirected models, directed models can capture reciprocity, the tendency of nodes to form mutual links. In this work, we address a fundamental question: what is the effective sample size for modeling reciprocity? We examine this by analyzing the Bernoulli model with reciprocity, allowing for varying sparsity levels between non-reciprocal and reciprocal effects. We then extend this framework to a model that incorporates node-specific heterogeneity and link-specific reciprocity using covariates. Our findings reveal intriguing interplays between non-reciprocal and reciprocal effects in sparse networks. We propose a straightforward inference procedure based on maximum likelihood estimation that operates without prior knowledge of sparsity levels, whether covariates are included or not.

Figures

Figures reproduced from arXiv: 2411.12871 by the authors.

Figure 1
Figure 1. Asymptotic distributions illustrating the phase transition phenomenon driven by the sparsity indices a and b. The dashed red line is the 45 degree line where a = b. While Proposition 1 and Corollary 1 provide valuable theoretical insights, it is important to note that, in practice, the sparsity level parameters a and b are typically unknown. This lack of information complicates the solution of equation (3) and the e… view at source ↗
Figure 2
Figure 2. The QQ plots of the standardized estimators for a = b = 0.5 are shown for n = 200 (first row) and n = 1000 (second row). The x-axis represents the theoretical quantiles, while the y-axis displays the sample quantiles. Next, we examine the normal approximation for our estimator using Proposition 3. In the following table, we keep θ0 and the generating scheme for {Xi}i≥1, {Yj}j≥1, and {Vij}i̸=j fixed while varying the… view at source ↗

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Reference graph

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Reviewed August 12, 2026 · model on record in the stance chip above.