Linear regression with probabilistic networks for network-linked data

• Binary adjacency weights treat observed neighbors equally, while conventional response models take the observed network as fixed despite uncertainty in edge formation.The paper identifies both as limitations of social interaction models and related approaches.Source: §1 · p. 3
• The related subspace projection approach captures low-dimensional network structure but omits explicit peer-response dependence, and its flexible node effects may absorb other heterogeneity.Source: §1 · p. 4
• The proposed NLDA framework uses degree-normalized connection probabilities for peer responses and latent positions for network cohesion while modeling network formation probabilistically.It estimates the network structure first, then uses fitted probabilities and latent positions in a Gaussian quasi-likelihood for responses.Source: §1 · p. 4
• Under suitable restrictions, the paper establishes network-recovery rates and consistent, jointly asymptotically normal regression estimators, with the classical spatial autoregressive convergence rate attainable despite estimated network weights.The introduction states that the results clarify the requirements for accurate regression inference beyond those needed for network recovery.Source: §1 · p. 5

2 Probabilistic network-linked data model

• The network model treats pairwise edges as independent Bernoulli variables with probabilities determined by node-specific effects and latent positions.The latent dimension is fixed and small relative to the number of nodes.Source: §2 · p. 6
• Degree-normalized connection probabilities replace binary adjacency weights, allowing peer influence to vary across node pairs while keeping each row of weights summing to one.Latent positions represent network cohesion, while individual node effects represent variation in connection propensity.Source: §2 · p. 7
• Peer-response dependence and latent similarity are distinct response components; the latent-position term can capture similar responses even when the peer-effect coefficient is zero.The passage notes that omitting either component may lead to model misspecification and obscure their distinction.Source: §2 · pp. 7–8

2.1 Computing algorithm

• The algorithm estimates network effects and fitted edge probabilities from the adjacency matrix, then estimates response parameters using the fitted network quantities and observed covariates and responses.For response fitting, it profiles the quasi-likelihood over the peer-effect coefficient on a deterministic grid and obtains the other parameters at the selected value.Source: §2.1 · pp. 8–10

3.1 Network recovery

• Under bounded heterogeneity, the network recovery bounds yield uniform parameter consistency when ρ_n ≥ c_d n^(-1+ε_d), matching the sparsity regime in Li et al. [2025].The theory also permits node-effect heterogeneity to grow with n, subject to Assumption 2.Source: §3.1 · p. 13
• The recovery guarantees concern local likelihood estimators; extending convergence guarantees to Algorithm 1 under both sparsity and increasing node-effect heterogeneity needs additional conditions and separate optimization analysis.The paper leaves that investigation for future work.Source: §3.1 · pp. 13–14

3.2 Response inference

• Theorem 2 establishes consistency and joint asymptotic normality for the aligned response estimator under its stated assumptions.Its rate combines response variation of order n^-1/2 with network-estimation error of order K_n^2/s_n.Source: §3.2 · p. 16
• The aligned regression estimator retains the classical root-n rate despite estimating both the weight matrix and latent positions, when network-estimation error is negligible at that scale.The sufficient conditions for this rate are stronger on network sparsity and node-effect heterogeneity than those needed for consistency.Source: §3.2 · pp. 16–17

4.1 Estimation performance

• The estimation experiments compare NLDA with SIM, RNC, and SP across three synthetic examples and report average RelMSE over 50 replications.The examples vary network density and latent-structure strength, and include responses generated from SIM.Source: §4 · p. 17, §4.1 · pp. 17–18
• NLDA has the smallest average RelMSE in every setting of Examples 1 and 2; in Example 3, SIM usually leads, while NLDA generally beats SP and RNC.NLDA's RelMSE decreases with sample size across all three examples; denser networks and stronger latent structure also reduce its estimation error in the specified settings.Source: §4.1 · pp. 18–19, Table 1 · p. 19, Table 2 · p. 19

4.2 Inference performance

• Inference performance is assessed using 95% confidence intervals for response parameters, with latent coordinates aligned to the truth when evaluating η.The reported coverage, bias, and standard deviation average across 13 coordinates and 500 replications.Source: §4.2 · p. 20, Table 4 · p. 21
• 92.4%–94.8% empirical coverage falls below the nominal 95% level, but generally approaches it as sample size or network density increases.Absolute bias and standard deviation also decrease with sample size; the authors relate improving coverage to more accurate network estimation and a more accurate normal approximation.Source: §4.2 · pp. 20–21, Table 4 · p. 21

5 Diplomatic network-linked data

• The analysis uses a connected 2005 diplomatic network of 187 countries and 4,507 edges, with log life expectancy as the response and three country characteristics as covariates.The covariates are log GDP per capita, log population and log urban population percentage.Source: §5 · p. 21
• Observed edge frequencies and mean fitted edge probabilities show similar patterns across five continental groups, suggesting the fitted model captures the graph’s main regional connectivity structure.The groups are Africa, the Americas, Asia, Europe and Oceania.Source: §5 · p. 22, Figure 1 · p. 22
• NLDA achieved the best average prediction on both measures across 50 country splits, with paired comparisons against each competitor yielding one-sided Wilcoxon p-values below 0.001.The measures were rMSE and out-of-sample R2.Source: §5 · p. 23, Table 6 · p. 23
• The estimated diplomatic-tie coefficient is significantly positive, the three observed covariate coefficients are positive, and GDP per capita has the largest coefficient.The joint Wald test indicates a significant contribution of latent positions (p-value 0.000475).Source: §5 · p. 23, Table 7 · p. 24

6 Summary

• NLDA weights peer responses with degree-normalized connection probabilities and includes latent positions as a network-cohesion component.This replaces binary adjacency weights in the peer-response term.Source: §6 · p. 24
• The theory establishes recovery rates for a local network likelihood maximizer under increasing node-effect heterogeneity and consistency of the plug-in regression estimator.Source: §6 · p. 24
• Under sub-Gaussian errors and nondegeneracy conditions, √nK2_n/s_n → 0 ensures joint asymptotic normality of the aligned regression estimator with consistently estimable covariance.With bounded node-effect variation and average density bounded away from one, this condition reduces to d_n/√n → ∞.Source: §6 · p. 24
• The diplomatic-network analysis illustrates an application of the framework to network-linked data.Source: §6 · p. 24

Supplementary Materials

• The supplement first introduces notation and definitions used in its proofs.Source: Supplementary Materials for “Linear… · p. 29
• It presents proofs of Theorems 1 and 2.Source: Supplementary Materials for “Linear… · p. 29
• It also contains auxiliary lemmas and their proofs.Source: Supplementary Materials for “Linear… · p. 29

Section I. Notation

• Constants c and C may vary across inequalities and depend only on fixed quantities in the assumptions and the fixed integer k.Source: Section I. Notation · p. 29
• The network parameter pair is denoted ν = (α, Z), with fitted pair bν = (bα, bZ).Source: Section I. Notation · p. 29
• For i ≠ j, θ_ij and p_ij denote the logit and edge probability, while w_ij = p_ij(1 − p_ij).Source: Section I. Notation · pp. 29–30

Section II. Proofs of the main theorems

• The aligned local maximizer has maximum latent-row error O_p(δ_{n,k}), and fitted pairwise predictor errors are also O_p(δ_{n,k}).The proof uses bounded true and fitted latent rows to control the predictor errors.Source: Section II. Proofs of the main theorems · p. 33
• The fixed margins and vanishing maximum latent-row error keep fitted node effects within b_n and fitted latent rows within R_Z with probability tending to one.Source: Section II. Proofs of the main theorems · p. 33
• With probability tending to one, the fitted latent matrix has smallest singular value at least c√n.This follows from the latent-matrix error bound and Weyl’s inequality.Source: Section II. Proofs of the main theorems · p. 34
• The estimated normalized-weight matrix satisfies ∥Ŵ − W*∥F = O_p(K_n/√s_n), completing (14).Source: Section II. Proofs of the main theorems · p. 35
• The cited result gives n U_n + o_p(1) converging in distribution to N(0, I_{p+r+2}), and Slutsky’s theorem then proves (22).Source: Section II. Proofs of the main theorems · pp. 39–40

Lemma A1

• Across the stated parameter set, edge probabilities and Bernoulli variances lie within envelopes, while expected degrees have a lower bound and the adjacency error is controlled with high probability.The degree lower bound is at least s_n/2 for n ≥ 2; the spectral error is at most C√t_n with probability at least 1 − n^-3 for sufficiently large n.Source: Section III. Auxiliary lemmas and proofs · pp. 40–42

Lemma A2

• The lemmas provide tangent-space likelihood-information and orthogonal-projector bounds under Assumption 1.The proof also identifies the latent-coordinate projection components as translations and rotations.Source: Section III. Auxiliary lemmas and proofs · pp. 43–44

Lemma A3

• For nearby parameter endpoints, the likelihood-information matrix has uniform curvature and controlled off-diagonal blocks and inverse.The bounds apply over the specified neighborhood and are uniform over its endpoints.Source: Section III. Auxiliary lemmas and proofs · pp. 46, 49

Lemma A4

• The observed and expected information matrices differ by an endpoint-independent projected network residual, whose spectral norm is bounded by the adjacency error; uniform curvature and inverse bounds also hold.The curvature bounds hold for sufficiently large n, and the conclusions are uniform over the endpoints.Source: Section III. Auxiliary lemmas and proofs · pp. 51–53

Lemma A5

• With probability tending to one, the network likelihood has a local maximizer in the relative interior of the centered parameter space, with projected score zero and aligned error satisfying the stated bound.The passage does not include the bound’s formula.Source: Section III. Auxiliary lemmas and proofs · p. 53

Proofs of auxiliary lemmas

• A fixed-point construction yields a maximizer over the constrained set and, after orthogonal alignment, a local maximizer over the centered parameter space.The construction also establishes the stated error bounds, including n^-1∥δ∥2^2 = Op(s_n^-1) and ∥δ − δ(1)∥bl,∞ = Op(δn,k).Source: Section III. Auxiliary lemmas and proofs · pp. 55, 59
• The response score has mean zero under the stated assumptions, and the auxiliary bounds control its coefficient matrix and quadratic-form terms.The proof derives the mean-zero result coordinatewise and uses bounds on the network-related matrices and error moments for the subsequent covariance calculations.Source: Section III. Auxiliary lemmas and proofs · pp. 61–62
• With probability tending to one, interpolating between the true and estimated network matrices preserves row-stochasticity and the stated entrywise bound, yielding uniform matrix bounds.The proof also gives ∥y∥2 + ∥W* y∥2 = Op(√n).Source: Section III. Auxiliary lemmas and proofs · pp. 64–65
• Under Assumptions 1–4, the response information matrix is uniformly bounded above and away from zero; if condition (19) also holds, Vn has the same uniform eigenvalue bounds.The passage states cI ⪯ Jy,n ⪯ CI and, under (19), cI ⪯ Vn ⪯ CI for sufficiently large n.Source: Section III. Auxiliary lemmas and proofs · p. 66

Lemma A11

• In particular, the fitted residual sum of squares (RSS) is uniformly bounded below by c n; this is the stated lower-bound result for the fitted RSS.Source: Section III. Auxiliary lemmas and proofs · p. 73
• The response-model estimates satisfy error rates O_p(K_n/√s_n), while the fitted gamma score coordinate is O_p(n^-1).The bounds cover the regression coefficients, gamma, and variance parameter; K_n/√s_n=o(1).Source: Section III. Auxiliary lemmas and proofs · pp. 76–77

Lemma A12

• Lemma A12 assumes Assumptions 1–4 and concerns any random eζ satisfying ∥eζ − ζ∗∥₂ = op(1); the supplied passage does not state its conclusion.Source: Section III. Auxiliary lemmas and proofs · p. 77

Lemma A13

• Lemma A13 establishes network-likelihood derivative bounds under Assumptions 1–2, with additional bounds under Assumptions 3–4 and for every score coordinate.Its proof uses degree normalization and Taylor expansion of fitted dyad probabilities.Source: Section III. Auxiliary lemmas and proofs · pp. 80–81

Section III. Auxiliary lemmas and proofs

• The network perturbation remainder has Frobenius norm Op(Kn/sn), while directional derivatives are controlled at scale Kn/√n, supporting the response expansion and uniform score-coordinate bound.The response expansion uses a Taylor remainder controlled by the squared perturbation size.Source: Section III. Auxiliary lemmas and proofs · pp. 83, 85, 86
• Under Assumptions 1–4, the auxiliary argument defines a unique direction in the tangent space through the positive-definite information operator.The direction is characterized by the projected score and information operator.Source: Section III. Auxiliary lemmas and proofs · pp. 86–87
• Residual variance, third and fourth moments, and the aligned covariance estimator are consistent; under condition (19), the estimated covariance is positive definite with probability tending to one.The consistency errors are op(1).Source: Section III. Auxiliary lemmas and proofs · p. 90