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Regression Analysis of Reciprocity in Directed Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper introduces the R²-Model, which regresses reciprocity on dyad covariates while removing node heterogeneity by conditioning on four-node subgraphs, and proves the resulting estimator is minimax rate-optimal.

desk verdict Solid, important extension of tetrad conditioning to directed networks with reciprocity; watch the bounded-heterogeneity assumption before you trust the advertised scope. read the letter →

arxiv 2507.21469 v1 pith:ORNZMUCH submitted 2025-07-29 stat.ME

Rui Feng , Chenlei Leng This is my paper · ORCID
classification stat.ME MSC 62F1262E2005C80
keywords directednetworksreciprocityconditionallikelihoodtetradsparsenodeheterogeneityasymptoticnormalityminimaxoptimality
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

The paper proposes a regression model for reciprocity in directed networks: the tendency of a pair of nodes to form mutual ties is written as $\rho_{ij} = \rho_n + V_{ij}^{\top}\gamma$, where $\rho_n$ is a baseline that may change with network size and $V_{ij}$ are observed dyad covariates. The model generalizes the classical $p_1$ model by keeping unrestricted sender- and receiver-specific effects for every node while allowing for network sparsity. The authors show that conditioning on four-node subgraphs (tetrads) that share the same degree sequence removes all node-specific nuisance parameters, leaving a likelihood that depends only on $(\rho_n, \gamma)$; the resulting estimator is consistent, asymptotically normal, and attains the minimax rate of convergence over the sparsity-indexed parameter class. If these claims hold, this is the first rigorous and rate-optimal way to regress reciprocity on covariates in sparse directed networks.

What carries the argument

The load-bearing object is the tetrad statistic $S_{ijkl}\in\{0,+1,-1\}$: three directed configurations on four nodes that are edge-rewirings of one another and therefore share the same in- and out-degree sequence. Conditioning on $S_{ijkl}\in\{0,\pm1\}$ cancels the density parameter and all $2n$ node heterogeneity terms, leaving a three-outcome likelihood with probabilities $1/(1+r)$, $f/(1+r)$, $g/(1+r)$ that depend only on $\rho_n$ and $\gamma$—this is the mechanism that isolates the reciprocity effects. The convergence rates are set by the sparsity indices $a$ and $b$: the three tetrad events have probabilities of order $n^{-4a}$ and $n^{-4a+2b}$, so the score and Hessian need different normalizations in different regimes, and asymptotic normality is proved by viewing the score as a $U$-statistic and showing its Hájek projection onto dyads dominates the variance.

What would settle it

Simulate the model with a fixed sparsity regime ($a,b$), node sizes growing from $n=50$ to $n=200$, and $\alpha_i,\beta_i$ drawn so that $\max_i|\alpha_i|$ grows with $n$ (e.g., Pareto tails), rather than bounded as Assumption 2 requires; compare the empirical frequencies of the three tetrad types against $n^{-4a}$ and $n^{-4a+2b}$, and check whether the Theorem 3 confidence intervals for $\rho_n$ and $\gamma$ keep their nominal 95% coverage. A widening coverage gap as $\max_i|\alpha_i|$ grows would show where the minimax claim breaks.

Watch

Extended reading notes

Core claim

The paper's central claim is that the reciprocity parameters can be estimated at the statistically optimal rate despite the presence of $2n$ nuisance parameters and sparsity. Explicitly, with sparsity indices $a,b$ defined by $\mu_n = -a\log n$ and $\rho_n = b\log n$, the tetrad-based conditional likelihood estimator satisfies $\sqrt{n^{2-2a+\min\{a,b\}}}(\hat{\rho}_n - \rho_{n0}) \to N(0,\sigma_\rho^2)$ and $\sqrt{n^{2-2a+b}}(\hat{\gamma}-\gamma_0) \to N(0,\Sigma_\gamma)$, and Theorem 4 proves that no estimator over the parameter class can achieve mean squared error smaller than $n^{-2+2a-\min\{a,b\}}$ for $\rho_n$ or $n^{-2+2a-b}$ for $\gamma$. The same construction gives a feasible inference procedure, Theorem 3, which does not require knowing $a$ or $b$.

Load-bearing premise

The theory assumes every node's sender and receiver effects are bounded by a constant independent of $n$; if a network has hubs or extremely inactive nodes, the tetrad event probabilities and all rates built on them no longer hold.

Editorial extensions

If this is right

  • Practitioners can test whether reciprocity varies with covariates using confidence intervals computed from Theorem 3 without knowing the sparsity parameters $a$ and $b$.
  • In the classical $p_1$ model (no covariates), the estimator reduces to a closed-form log-ratio of tetrad counts, providing a simple method-of-moments-style estimator of baseline reciprocity.
  • When reciprocity is strong ($b > a$), the covariate slopes are estimated at a strictly faster rate than the baseline reciprocity, because comparisons between the two $\pm1$ tetrad configurations carry information about $\gamma$ only.
  • The minimax lower bounds apply to all estimators, not just conditional likelihood ones, so no alternative estimator can beat the proposed rates in the moderate-heterogeneity regime.

Reading between the lines

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

  • The same tetrad-conditioning design should transfer to models with edgewise homophily covariates, because the sufficiency lemma only requires configurations sharing the same degree sequence; a direct implementation would give inference on the homophily slope without node-effect estimation.
  • A simulation with heavy-tailed $\alpha_i,\beta_i$ (for example, Pareto-distributed) would likely show tetrad frequencies and coverage degrading, marking the empirical boundary of Assumption 2; this is a concrete testable extension the paper does not run.
  • Although the theory is stated for a single network snapshot, the log-odds structure suggests the estimator could be adapted to panel or multi-layer directed networks by treating each layer's degree sequences as the conditioning statistic—an extension the authors do not pursue.
  • The closed-form estimator of Proposition 1 (a log ratio of tetrad counts) hints at an interpretation of the R²-Model as comparing two oriented 4-cycles; this connection suggests the methodology could be exported to any dyadic outcome model where a sufficient statistic can be frozen by degree-preserving rewirings.
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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 / 4 minor

Summary. The paper introduces the R²-Model, a generalized p1 model for directed networks that incorporates reciprocity through a baseline parameter ρ_n and a covariate vector γ, while allowing node-specific sender and receiver heterogeneity. To avoid estimating 2n+2+q parameters, the authors propose a conditional likelihood estimator based on tetrad configurations that conditions out the degree sequence and isolates (ρ_n, γ). They prove consistency, asymptotic normality, and minimax rate optimality under sparsity assumptions, and support the results with simulations and two empirical applications. The proof strategy relies on U-statistic projections and a case-by-case analysis of tetrad dependence, with complete proofs in the supplementary material.

Significance. If the results hold, this is a useful and nontrivial extension of Graham's tetrad-based conditional likelihood approach to directed networks with reciprocity and covariates. The paper provides machine-checkable auxiliary material in the form of detailed proofs, and it gives a constructive proof of minimax optimality via Le Cam's two-point method, which is a strong selling point. The closed-form estimator for the p1 submodel and the real-data applications increase the paper's practical appeal. However, the theoretical guarantees are established only under a bounded-heterogeneity assumption, which is narrower than the paper's advertised 'unrestricted' heterogeneity, and the numerical studies lack a description of how the computationally heavy tetrad enumeration was performed.

major comments (3)
  1. [Section 1, p.2; Section 2.2, Assumption 2] The paper repeatedly claims to allow 'unrestricted node-level heterogeneity' (Section 1) and the abstract states the model 'accommodates both network sparsity and node heterogeneity,' but Assumption 2 requires max_i{|α_i|,|β_i|} ≤ C for a constant independent of n. This uniform boundedness is load-bearing: Lemma 1's event probabilities, Lemma 2's derivative bounds, Proposition 2's covariance orders, and the rates in Theorems 2 and 3 all use it. If max_i|α_i| were to grow with n, say as C log n, then the probabilities P(S_ijkl=0) ∼ n^{-4a} and P(S_ijkl=±1) ∼ n^{-4a+2b} would fail, and the U-statistic variance decomposition would collapse. The authors should either weaken the stated claims to 'moderate degree heterogeneity' or extend the theory to slowly growing heterogeneity with explicit rate conditions.
  2. [Section 3.1, simulations] The numerical study reports 1000 replications for n = 50, 100, 150, 200 without explaining how the estimator L_n, which sums over all C(n,4) tetrads, was computed. For n=200 there are approximately 6.5×10^7 tetrads per replication, making the reported 1000 replications (about 6.5×10^10 tetrad evaluations per setting) computationally infeasible with the described full enumeration. If a random subsample of tetrads or another approximation was used, that estimator is not covered by Theorems 1-3, and the finite-sample claims and confidence-interval coverages need to be justified for the implemented procedure. The paper should state the exact algorithm and, if subsampling is used, provide a corresponding theoretical statement.
  3. [Supplementary, proof of Theorem 2, Case C] In the proof of Theorem 2, Case C, the text states 'ˆρ − ρ0 = ... = OP(1)' after having shown n^{4a}Ψ_{n,ρ}(ϑ0) = OP(n^{(-2+a)/2}) and that the scaled Hessian is Θ(1). This implies the difference should be OP(n^{-1+a/2}), not OP(1). This is a typographical error, but it obscures the convergence rate and should be corrected in the supplementary material.
minor comments (4)
  1. [Section 2.2, Theorem 3] Theorem 3 uses the normalization 'n · (ˆρ_n − ρ_n0) / sqrt(Σ̂_1,1)', which is correct because the variance estimator Σ̂ automatically scales as n^{2a-b} (or n^a in the b>a case), but the statement would be clearer with explicit parentheses, e.g., 'n(ˆρ_n − ρ_{n0})'. The stray 'q' artifacts in the displayed formulas should also be cleaned.
  2. [Supplementary, proof of Proposition 1] The proof of Proposition 1 derives ˆρ_n = (1/2) log( Σ I(S=1) / Σ I(S=0) ), while the proposition states the numerator as Σ I(S=±1). The numerator should be I(S=±1) to match the first-order condition; the current text is inconsistent.
  3. [Section 1, model definition] The covariate V_ij is defined for i<j in Assumption 1, but the model formulas use V_il and V_jk for the S=-1 configuration. Since these expressions are only well-defined if V_ij = V_ji, the paper should state explicitly that dyadic covariates are symmetric.
  4. [Supplementary, Proposition 2 statement] Proposition 2 states Var(τ_n ⊙ τ_n ⊙ Ψ_n(ϑ)) = o(1), but the proof analyzes Var(τ_n ⊙ Ψ_n(ϑ)). The extra 'τ_n ⊙' appears to be a typo and should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the estimator and minimax rates are derived from the model, with no fitted parameter renamed as a prediction.

full rationale

The derivation chain is self-contained. Lemma 1 computes the tetrad conditional probabilities directly from the model in (2), eliminating the nuisance parameters and leaving a likelihood in (rho, gamma). The score bounds in Lemma 2 and the variance decomposition in Proposition 2 are proved from the model's event probabilities P(S_ijkl = 0) ~ n^{-4a} and P(S_ijkl = +/-1) ~ n^{-4a+2b}, so the rates in Theorems 1-3 are consequences of the model rather than assumptions. The minimax lower bound in Theorem 4 uses Le Cam's two-point method over the parameter class Theta_n, with explicit perturbations chosen so that the KL divergence is bounded; the matching upper bounds come from Theorem 2, so the optimality claim does not reduce to a fitted constant or to a self-citation. The self-citations to Feng and Leng (2025), including the phrase 'effective sample size', are contextual: the proofs cite no theorem from that paper, and the scaling is derived in Lemmas 2 and Proposition 2. The bounded-heterogeneity restriction in Assumption 2 is a stated scope limitation rather than a circular step; it narrows the regime but does not define the conclusions in terms of the inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central theory rests on model-based independence, bounded node heterogeneity, a moderately sparse regime, and standard M-estimation limits. No new physical entities are introduced. The sparsity exponents a and b are assumed regime parameters, not estimated from data.

free parameters (2)
  • sparsity exponent a = not fitted; assumed (simulations use a=0.2 or 0.3)
    Defined by mu_n = -a log n + mu. The rates and identifiability conditions in Assumptions 3 and 5 depend on a, but it is not estimated from data. The paper claims inference in Theorem 3 does not require knowing a.
  • reciprocity exponent b = not fitted; assumed (simulations use b=0.1, -0.1, or 0.5)
    Defined by rho_n = b log n + rho. The convergence rates and the distinction between regimes rely on b, but it is not estimated. It is part of the theoretical parameter class rather than a fitted constant.
assumptions (5)
  • domain assumption Dyads (A_ij, A_ji) are mutually independent conditional on covariates and node-specific parameters alpha and beta.
    This is the core model assumption stated in Section 1. It defines the likelihood in equation (2) and is necessary for the tetrad probability calculations.
  • domain assumption Assumption 1: covariates V_ij are centered, i.i.d., and uniformly bounded; true parameter lies in the interior of a compact set.
    Used to control the gradient and Hessian bounds in Lemma 2 and to ensure identifiability in the asymptotic analysis. The i.i.d. assumption excludes spatially or network-dependent covariates.
  • domain assumption Assumption 2: max_i {|alpha_i|, |beta_i|} <= C for a constant C independent of n.
    Uniform boundedness of node heterogeneity is required for the tetrad event probability bounds P(S=0) ~ n^{-4a} and P(S=+/-1) ~ n^{-4a+2b}. This is the weakest assumption in the paper and limits the theory to moderate heterogeneity.
  • domain assumption Assumption 3: 0 < a < 1 and 0 < 2a - b < 2; Assumption 5: a < 2/3 for asymptotic normality.
    These sparsity conditions ensure that informative tetrads appear frequently enough for identification and that the Hajek projection dominates the variance. They exclude very sparse networks.
  • standard math Assumptions 4 and 6: the scaled score and Hessian converge to non-random limits, and the limiting information matrix is positive definite.
    These are standard M-estimation regularity conditions, analogous to assuming a well-behaved limiting Fisher information matrix; they are invoked in the proofs of Theorems 1 and 2.

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Pith. "Pith review of Regression Analysis of Reciprocity in Directed Networks." pith.science (2026). https://pith.science/paper/ORNZMUCH

@misc{pith2026250721469,
  author       = {Pith},
  title        = {Pith review of: Regression Analysis of Reciprocity in Directed Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORNZMUCH}},
  note         = {Machine review of arXiv:2507.21469}
}
abstract

Reciprocity--the tendency of individuals to form mutual ties--is a fundamental structural feature of many directed networks. Despite its ubiquity, reciprocity remains insufficiently integrated into statistical network models, particularly in relation to covariate information. In this paper, we introduce the $R^{2}$-Model, a novel and flexible framework that explicitly models reciprocity while incorporating covariate effects. Built upon a generalized $p_1$ model, our framework accommodates both network sparsity and node heterogeneity, offering the most comprehensive parametrization of reciprocity to date--capturing not only its baseline level but also how it systematically varies with observed covariates. To address the challenges posed by high dimensionality and nuisance parameters, we develop a conditional likelihood estimator that isolates and consistently estimates the reciprocity effects. We establish its theoretical guarantees, including consistency, asymptotic normality, and minimax optimality under broad sparsity regimes. Extensive simulations and real-world applications demonstrate the $R^{2}$-Model's flexibility, interpretability, and strong finite-sample performance, highlighting its practical utility for uncovering covariate-driven patterns of reciprocity in directed networks.

Figures

Figures reproduced from arXiv: 2507.21469 by the authors.

Figure 1
Figure 1. Diagrams illustrating the values of Sijkl: Sijkl = 0 (left), Sijkl = 1 (middle), and Sijkl = −1 (right). Dashed lines indicate no restrictions on the presence of edges. non-reciprocal parameters and identify only ρn and γ. We present this argument rigorously in Lemma 1. Let Vijkl = {Vij , Vjk, Vkl, Vli, Vik, Vjl}, define fijkl(ρn, γ) = exp 2ρn + (V ⊤ ij + V ⊤ kl )γ  , gijkl(ρn, γ) = exp 2ρn + (V ⊤ il + V ⊤ jk)γ  ,… view at source ↗
Figure 2
Figure 2. Illustration of configurations corresponding to S˜ijkl = 0 (left) and S˜ijkl = 1 (right). Solid arrows represent required edge directions, while dashed lines indicate unrestricted edges. Following similar arguments as in Theorems 1 and 2, we obtain the asymptotic normality: p n2−2a+min{a,b}(ϑˇ − ϑ0) d −→ N(0, Σ1), for some positive definite matrix Σ1. Crucially, this rate reflects a key difference in the performance… view at source ↗
Figure 3
Figure 3. MAE of our estimator and the MLE for the true parameters ρn and γk (k = 1, 2) across different network sizes. The decreasing trend illustrates improved estimation accuracy as the number of nodes n increases. Beyond point estimation accuracy, we assess the validity of the asymptotic normality result stated in Theorem 2 via quantile-quantile (QQ) plots. Due to space constraints, we present the QQ-plots for n = 200 in … view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Quantile-quantile (QQ) plots of the standardized estimators for ρn and γk (k = 1, 2) when n = 200. The alignment of empirical quantiles with the theoretical normal quantiles supports the asymptotic normality of the estimators. Moreover, we evaluate the practical utilit…
Figure 5
Figure 5. Figure 5: MAE of our estimator and MLE in the p1.5 model across different network sizes. With unobserved degree heterogeneity, fitting the data using the p1.5 model will be biased. Overall, these simulation results demonstrate not only the consistency and efficiency of our estim…
Figure 6
Figure 6. Figure 6: Boxplot of the log distance covariate grouped by dyad configurations. Dyads with mutual trade links exhibit lower average log distances, highlighting the role of geographic proximity in reciprocal trade relationships. Finally, we fit the full model incorporating the lo…
Figure 7
Figure 7. Figure 7: Illustration of (Case 2.1): when Sijk1l1 = Sijk2l2 = 0 (left), Sijk1l1 = Sijk2l2 = 1 (middle), and Sijk1l1 = Sijk2l2 = −1 (right). Next, we compute the joint probabilities: P(Sijk1l1 = 1, Sijk2l2 = 1) = P(AijAji)P(Al1k1Ak1l1 )P(Al2k2Ak2l2 ) · P((1 − Ajk1 )(1 − Ak1j ))P…
Figure 8
Figure 8. Figure 8: Illustration of (Case 2.3): when Sijk1l1 = Si,k2,l2,j = 0 (left), Sijk1l1 = Si,k2,l2,j = 1 (middle), and Sijk1l1 = Si,k2,l2,j = −1 (right). Therefore, E(∇ϑℓijk1l1 (ϑ)∇⊤ ϑ ℓik2l2j (ϑ)) = E(∇ϑℓijk1l1 (ϑ)∇⊤ ϑ ℓik2l2j (ϑ) | Sijk1l1 = 1, Sik2l2j = −1) P(Sijk1l1 = 1, Sik2l2j…
Figure 9
Figure 9. Figure 9: Illustration of (Case 2.4): when Sijk1l1 = Si,k2,j,l2 = 0 (left), Sijk1l1 = Si,k2,j,l2 = 1 (middle), and Sijk1l1 = Si,k2,j,l2 = −1 (right). Then E [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Illustration of (Case 2.7): when Sijk1l1 = Sj,i,k2,l2 = 0 (left), Sijk1l1 = Sj,i,k2,l2 = 1 (middle), and Sijk1l1 = Sj,i,k2,l2 = −1 (right). Then E [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Illustration of (Case 2.9): when Sijk1l1 = Sj,k2,l2,i = 0 (left), Sijk1l1 = Sj,k2,l2,i = 1 (middle), and Sijk1l1 = Sj,k2,l2,i = −1 (right). Similarly to (Case 2.1), when b > 0, Cov ∇ρℓijk1l1 (ϑ), ∇ρℓj,k2,l2,i(ϑ)  = O(n −6a−min{a,b} ), Cov ∇γℓijk1l1 (ϑ), ∇γℓj,k2,l2,i(…
Figure 12
Figure 12. Figure 12: Illustration of (Case 2.10): when Sijk1l1 = Sj,k2,i,l2 = 0 (left), Sijk1l1 = Sj,k2,i,l2 = 1 (middle), and Sijk1l1 = Sj,k2,i,l2 = −1 (right). when Sijk1l1 = −1, P(Sijk1l1 = −1, Sk2,i,j,l2 = 0 or − 1) = 0, P(Sijk1l1 = −1, Sk2,i,j,l2 = 1) = O(n −8a+4b ). Therefore, in th…
Figure 13
Figure 13. Figure 13: Illustration of (Case 2.13): when Sijk1l1 = Sk2,i,j,l2 = 0 (left), Sijk1l1 = Sk2,i,j,l2 = 1 (middle), and Sijk1l1 = Sk2,i,j,l2 = −1 (right). (Case 2.14): π ′ (i, j, k2, l2) = (k2, i, l2, j). From [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: Illustration of (Case 2.14): when Sijk1l1 = Sk2,i,l2,j = 0 (left), Sijk1l1 = Sk2,i,l2,j = 1 (middle), and Sijk1l1 = Sk2,i,l2,j = −1 (right). When Sijk1l1 = −1, we have P(Sijk1l1 = −1, Sk2,j,i,l2 = 0 or − 1) = 0, P(Sijk1l1 = −1, Sk2,j,i,l2 = 1) = O(n −8a+4b ). Therefor…
Figure 15
Figure 15. Figure 15: Illustration of (Case 2.15): when Sijk1l1 = Sk2,j,i,l2 = 0 (left), Sijk1l1 = Sk2,j,i,l2 = 1 (middle), and Sijk1l1 = Sk2,j,i,l2 = −1 (right). (Case 2.16): π ′ (i, j, k2, l2) = (k2, j, l2, i). From [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: Illustration of (Case 2.16): when Sijk1l1 = Sk2,j,l2,i = 0 (left), Sijk1l1 = Sk2,j,l2,i = 1 (middle), and Sijk1l1 = Sk2,j,l2,i = −1 (right). When Sijk1l1 = −1, P(Sijk1l1 = −1, Sk2,l2,i,j = 0 or 1) = 0, P(Sijk1l1 = −1, Sk2,l2,i,j = −1) = O(n −8a+4b ). Therefore, in thi…
Figure 17
Figure 17. Figure 17: Illustration of (Case 2.17): when Sijk1l1 = Sk2,l2,i,j = 0 (left), Sijk1l1 = Sk2,l2,i,j = 1 (middle), and Sijk1l1 = Sk2,l2,i,j = −1 (right). (Case 2.18): π ′ (i, j, k2, l2) = (k2, l2, j, i). From [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]
Figure 18
Figure 18. Figure 18: Illustration of (Case 2.18): when Sijk1l1 = Sk2,l2,j,i = 0 (left), 1 (middle), and −1 (right). same as (Case 2.14). (Case 2.21): π ′ (i, j, k2, l2) = (l2, j, i, k2). Due to the symmetry between j2 and k2, this case is the same as (Case 2.15). (Case 2.22): π ′ (i, j, k…
Figure 19
Figure 19. Figure 19: Illustration of (Case 3.1): when Sijkl1 = Sijkl2 = 0 (left), Sijkl1 = Sijkl2 = 1 (middle), and Sijkl1 = Sijkl2 = −1 (right). By Assumptions 1, 2, and 3, we have P(Sijkl1 = 0, Sijkl2 = 0) = O(n −6a ), and P(Sijkl1 = −1, Sijkl2 = −1) = O(n −6a+3b ), P(Sijkl1 = 1, Sijkl2…
Figure 20
Figure 20. Figure 20: Illustration of (Case 4.1): when Sijkl = 0 (left), Sijkl = 1 (middle), and Sijkl = −1 (right). By Assumptions 1, 2, and 3, we have P(Sijkl = 1) = O(n −4a+2b ), P(Sijkl = −1) = O(n −4a+2b ), P(Sijkl = 0) = O(n −4a ). 31 [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: shows the absolute estimation errors across different network sizes for the setting a = 0.2 and b = −0.1, while [PITH_FULL_IMAGE:figures/full_fig_p049_21.png]
Figure 22
Figure 22. Figure 22: QQ plots of standardized estimators for n = 200, a = 0.2, and b = −0.1. 50 100 150 200 Network size 0.0 0.2 0.4 0.6 0.8 1.0 Absolute error Conditional likelihood in proposed model Maximum likelihood in p1.5 model (a) Absolute error for ρˆn 50 100 150 200 Network size …
Figure 23
Figure 23. Figure 23: Comparison with p1.5 model across different network sizes when a = 0.2, and b = −0.1. 50 100 150 200 Network size 0.0 0.2 0.4 0.6 0.8 1.0 Absolute error Conditional likelihood Maximum likelihood (a) Absolute error for ρˆn 50 100 150 200 Network size 0.0 0.1 0.2 0.3 0.…
Figure 24
Figure 24. Figure 24: Absolute estimation error across various network sizes for a = 0.3 and b = 0.5. n = 50 n = 100 n = 150 n = 200 Coverage Width Coverage Width Coverage Width Coverage Width ρˆn 98.0% 0.837 96.2% 0.373 96.3% 0.241 96.0% 0.179 γˆ1 97.5% 0.727 95.8% 0.317 96.6% 0.200 96.1%…
Figure 25
Figure 25. Figure 25: QQ plots of standardized estimators for n = 200, a = 0.3, and b = 0.5. 50 [PITH_FULL_IMAGE:figures/full_fig_p050_25.png]
Figure 26
Figure 26. Figure 26: Comparison with p1.5 model across different network sizes when a = 0.3, and b = 0.5. 51 [PITH_FULL_IMAGE:figures/full_fig_p051_26.png]

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Reviewed August 6, 2026 · model on record in the stance chip above.