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Econ-394 · economics · Economics

Identifiable effects of network topology and node attributes

LLM status Partially solved unverified

Statement

CI-59 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

A network intervention sets a specified graph $G$ and manipulable node attributes $A$, while retaining the remaining outcome mechanism. Fix a known symmetric weighting map $B(G,A)$ and feature matrix $Z(A)$ including an intercept. Require $\|B(G,A)\|_{\mathrm{op}}\leq1$ throughout the intervention class; do not require its rows to sum to one. Consider \[ Y(w;G,A)=(I-\delta B(G,A))^{-1} \{Z(A)\zeta+\beta w+\gamma B(G,A)w+U\}, \] with independent Gaussian innovations, $|\delta|\leq1-\eta$, and a parameter vector shared across intervention settings. Independent experimental populations supply observed $(G,A,W,Y)$; assignments of graphs, attributes, and treatments have supplied randomization laws. For treatment probability vector $q$, put \[ \mu(G,A,q)=N^{-1}\mathbf1^\top(I-\delta B(G,A))^{-1} \{Z(A)\zeta+\beta q+\gamma B(G,A)q\}. \]

Formal question

For two specified interventions, characterize design conditions identifying $\mu(G_1,A_1,q_1)-\mu(G_0,A_0,q_0)$, and establish estimators and sharp squared-error bounds as the number of independent experimental populations grows. Determine which topology contrasts distinguish stars, bridges, and communities after controlling the intervention on attributes. Characterize when extrapolation to an unassigned graph is justified by identification of the shared structural mechanism.

Scope and status

Structural invariance across graph interventions is an assumption requiring scientific justification. Randomizing individual treatment on one fixed graph does not, by itself, identify effects of changing that graph. Some row-normalized linear models make all-treated versus all-control effects topology invariant; this formulation avoids imposing that invariance by definition. Its explicit linear benchmark is not an assertion that every such subproblem is currently open.

References for the source of the problem

Subhankar Bhadra and Michael Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808v1, 9 August 2025, Section 8.3; revised version arXiv:2508.06808v2, 9 August 2026, Section 8.2, Effect of Network on Causal Effects. https://arxiv.org/html/2508.06808v1#S8.SS3 https://arxiv.org/html/2508.06808v2#S8.SS2

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-394-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-394

Identifiable effects of network topology and node attributes: a Möbius identification theorem and walk-moment characterisation of topology contrasts

Abstract

In the linear network model, the treatment reduced form on a fixed graph is the matrix function $\pi(B)$ of the Möbius map $\pi(x)=(\beta+\gamma x)/(1-\delta x)$. We prove the following. (i) Randomised treatment on one graph identifies $(\beta,\gamma,\delta)$ iff $\gamma\neq-\beta\delta$ and $B$ has at least three distinct eigenvalues whose eigenspaces are excited by the treatment design. (ii) All-or-none contrasts depend on topology only through the walk moments $m_k=N^{-1}\mathbf 1^\top B^k\mathbf 1$. Topologies with equal walk-moment sequences are indistinguishable by such contrasts, and stars, bridges and communities are separated exactly when their walk moments differ. (iii) With $J$ independent populations, the design-based contrast estimator has squared error $O(1/J)$. The structural plug-in has squared error $O(1/(JN))$ under the invariance assumption. A Cramér--Rao bound shows the latter rate is sharp. Extrapolation to an unassigned graph is justified exactly when the structural parameters are identified and invariant.

1. Reduced form

Write $(I-\delta B)Y=Z\zeta+(\beta I+\gamma B)W+U$, with $U\sim N(0,\sigma^2I)$, and $W$ independent of $U$ with $\mathrm{Var}(W)=D=\mathrm{diag}(q_i(1-q_i))$. Let $B$ be symmetric with spectral decomposition $B=\sum_\ell x_\ell P_\ell$, where the $x_\ell$ are distinct. Then \[ \mathbb E[Y\mid W]=c+\Pi W,\qquad \Pi=\pi(B)=\sum_\ell\pi(x_\ell)P_\ell,\quad c=(I-\delta B)^{-1}Z\zeta. \]

2. Identification on a single graph

Theorem: Assume $|\delta|\le1-\eta$ and $\|B\|_{\rm op}\le1$. Let $\mathcal L=\{\ell:P_\ell D\ne0\}$, the eigenspaces excited by randomisation. The covariance $\mathrm{Cov}(Y,W)=\Pi D$ identifies $(\beta,\gamma,\delta)$ iff $|\mathcal L|\ge3$ and $\gamma\ne-\beta\delta$. If $\gamma=-\beta\delta$, then $\pi\equiv\beta$, and only $\beta$ is identified from treatment variation.
Proof: $\Pi D=\sum_\ell\pi(x_\ell)P_\ell D$ reveals $\pi(x_\ell)$ for every $\ell\in\mathcal L$, because the $P_\ell$ are orthogonal projections. The map $\pi$ is a Möbius transformation with coefficient matrix $\bigl(\begin{smallmatrix}\gamma&\beta\\-\delta&1\end{smallmatrix}\bigr)$, whose determinant is $\gamma+\beta\delta$. If the determinant is nonzero, $\pi$ is a nondegenerate Möbius map, and such a map is determined by its values at three distinct points. With two points, a one-parameter family fits. If the determinant is zero, $\pi$ is constant. ∎
Corollary: (a) A disjoint union of equal-size cliques with row-normalised weights, or any graph whose $B$ has two eigenvalues (for example $B=J/N$), does not identify $\delta$ and $\gamma$ separately from treatment variation. (b) $\zeta$ and $\delta$ can alternatively be identified from $c=(I-\delta B)^{-1}Z\zeta$, which requires $Z$ to excite at least two eigenvalues. Combining both sources yields overidentification, which is testable.

3. Topology contrasts

Proposition: For $q=\mathbf 1$ versus $q=\mathbf 0$ with attributes fixed, \[ \mu(G,A,\mathbf 1)-\mu(G,A,\mathbf 0)=\sum_{k\ge0}\delta^k(\beta m_k+\gamma m_{k+1}),\qquad m_k=N^{-1}\mathbf 1^\top B^k\mathbf 1. \] Hence two graph interventions $G,G'$ (with $A$ fixed) have equal all-or-none contrasts for all parameter values iff $m_k(G)=m_k(G')$ for all $k\ge0$. For row-normalised $B$, $m_k=1$ and the contrast is $(\beta+\gamma)/(1-\delta)$ for every topology.
Proof: Expand $(I-\delta B)^{-1}=\sum_k\delta^kB^k$; this converges since $|\delta|\,\|B\|<1$. For the converse, the contrast is a power series in $\delta$ whose coefficients are $\beta m_k+\gamma m_{k+1}$. Equality for all $(\beta,\gamma,\delta)$ forces $m_k=m'_k$ for all $k$. ∎
Remark (Example): With spectral normalisation $B=\mathrm{Adj}/\lambda_{\max}$ on $N$ nodes: the star has $m_1=2\sqrt{N-1}/N$; the cycle has $m_1=1$; two cliques of size $N/2$ joined by one bridge edge have $m_1\approx1$, with corrections of order $1/N$ through the bridge. Hence the star versus cycle contrast is identified at first order, while communities versus a single clique differ only at order $1/N$. When treatment-induced variance per population is of order one, distinguishing them with the design-based estimator needs roughly $J\gtrsim N^2$ populations; the structural route avoids this.

4. Estimation and sharp rates

Let populations $j=1,\dots,J$ be independent, with graph and attribute interventions $(G_j,A_j)$ drawn from a known design and treatments randomised.
Proposition: (a) Design-based. If $(G_1,A_1)$ and $(G_0,A_0)$ are each assigned to $J/2$ populations with treatment probabilities $q_1,q_0$, the difference in mean outcomes is unbiased for the $\mu$-contrast. Its variance is $O(1/J)$, with constant $\max_g N^{-2}\mathbf 1^\top\mathrm{Var}(Y_g)\mathbf 1\le C(\sigma^2/N+v_W)$, where $v_W$ collects treatment-induced variance. (b) Structural. Suppose the parameters are invariant and identified (Theorem Theorem holds in some assigned graph). Then the Gaussian MLE $\widehat\vartheta$ satisfies $\sqrt{JN}(\widehat\vartheta-\vartheta)\Rightarrow N(0,\mathcal I^{-1})$, where $\mathcal I$ is the average per-node Fisher information. The plug-in $\mu(\cdot;\widehat\vartheta)$ has squared error $\nabla\mu^\top\mathcal I^{-1}\nabla\mu/(JN)+o(1/(JN))$ for any target graph, including unassigned ones. (c) Sharpness. For every unbiased estimator, the Cramér--Rao bound gives $\mathrm{Var}\ge\nabla\mu^\top(JN\mathcal I)^{-1}\nabla\mu$. So (b) is rate- and constant-efficient.
Proof: (a) Independence across populations. (b) Standard maximum-likelihood theory for the Gaussian SAR model with exogenous regressors $(Z,W,BW)$ and Jacobian term $\log\det(I-\delta B)$. Under $|\delta|\le1-\eta$, $\|B\|\le1$, and the identification of Theorem Theorem, the information matrix is positive definite and scales linearly in total nodes. (c) Cramér--Rao. ∎

5. When is extrapolation justified?

Extrapolating $\mu$ to an unassigned $(G^\dagger,A^\dagger)$ is valid under exactly two conditions. First, the mechanism $(\zeta,\beta,\gamma,\delta,\sigma)$ must be invariant across graph interventions; this is the scientific assumption the statement flags, and it cannot be tested from data on one graph. Second, it must be identified from the assigned designs. Invariance is testable across multiple assigned graphs: estimate the parameters separately in each design class and compare them, a Hausman-type test with $O(1/\sqrt{JN})$ power. Proposition Proposition shows which unassigned graphs are "close": those whose walk-moment sequences are close in a $\delta$-weighted $\ell_1$ norm. This gives a bound $|\mu^\dagger-\mu|\le\sum_k|\delta|^k(|\beta||m_k^\dagger-m_k|+|\gamma||m_{k+1}^\dagger-m_{k+1}|)$ for all-or-none contrasts, even without exact invariance of everything else. Open: nonlinear outcome maps, and graph interventions that change $N$.

Completion Estimate

44%

This is a post hoc, heuristic estimate for the full catalogue target. Treatment reduced-form identification and weighted-walk formulas clarify which topology contrasts can be distinguished, with a design-based mean comparison. Uniform structural risk scaling lacks quantitative information conditions, and covariance-based identification and target-specific extrapolation are not fully characterized.

References

  • [bs] S. Bhadra, M. Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808 (v1 2025, Section 8.3; v2 2026, Section 8.2).
  • [lee] L.-F. Lee, Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models, Econometrica 72 (2004), 1899--1925.

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Econ-394-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-394

Identifiable effects of network topology and node attributes: a research attempt

Target and outcome

Status: partial. In a finite-dimensional homoskedastic Gaussian specialization I derive structural identification from covariance, a necessary-and-sufficient linear target criterion once contagion is identified, and a sharp $n^{-1}$ squared-error rate for a fixed-size regular experimental design. I also derive explicit topology contrasts and an extrapolation counterexample. General innovation covariance, weak designs, and growing graph size are not covered by the rate theorem.

The inspected primary network-interference review discusses effects of network structure and generalization to other graphs [bs]. The algebra below concerns the catalogue's specified symmetric bounded weighting map, with one additional explicit assumption: $U\sim N(0,\sigma^2I)$ with a common unknown $\sigma>0$ across experimental populations. Independence of Gaussian coordinates with unrestricted unequal variances would not alone imply this assumption.

Covariance identifies contagion under an informative graph

Write $R_B=(I-\delta B)^{-1}$, $|\delta|\leq1-\eta$, and $\|B\|\leq1$. All eigenvalues of $I-\delta B$ are positive and at least $\eta$. Conditional on $(B,Z,W=w)$, \[ m_B(w)=R_B(Z\zeta+\beta w+\gamma Bw), \qquad \Sigma_B=\sigma^2R_B^2. \] The unique symmetric positive-definite inverse square root satisfies \[ \boxed{\Sigma_B^{-1/2}=\sigma^{-1}I-(\delta/\sigma)B.}\tag{1} \] This identity uses positive definiteness; choosing arbitrary eigenvalue signs would create spurious observational equivalences.

If $B$ is not a scalar multiple of $I$, the two matrices $I,B$ are linearly independent. Regressing the entries of the known covariance transformation on these two matrices identifies $a=1/\sigma$ and $b=-\delta/\sigma$, and hence $\delta=-b/a$. Equivalently, two distinct eigenvalues $\lambda_1,\lambda_2$ give two equations $s_j^{-1}=a+b\lambda_j$, where $s_j^2$ is the outcome innovation variance in that eigenmode.

With several assigned graph settings, stack (1). It identifies $(a,b)$ whenever the stack of $I$ and the stack of $B_j$ are linearly independent. Even scalar graphs $B_j=c_jI$ can satisfy this condition if the scalars differ across settings and the innovation variance is shared. If every graph is the same $cI$, the covariance identifies only $(1-\delta c)/\sigma$; the scale and contagion remain confounded unless mean restrictions or known variance provide additional information.

A target can be identified without every coefficient

Once $\delta$ is identified, transform every observed conditional mean: \[ (I-\delta B_j)m_j(w) =[Z_j,\ w,\ B_jw]\,\xi, \qquad \xi=(\zeta^\top,\beta,\gamma)^\top. \] Stack the rows over graph settings and treatment vectors with positive assignment probability to obtain $D\xi=r$. Full column rank identifies all mean coefficients. Independent Bernoulli treatments with probabilities strictly between zero and one give all treatment vectors positive probability; with a nonscalar $B$, varying those vectors identifies $\beta I+\gamma B$. Identification of $\zeta$ additionally requires sufficient rank of the observed feature matrices.

For the requested two interventions, let \[ \ell_\delta^\top=\frac1N\mathbf1^\top \left\{R_{B_1}[Z_1,q_1,B_1q_1] -R_{B_0}[Z_0,q_0,B_0q_0]\right\}. \] The contrast equals $\ell_\delta^\top\xi$. In an unconstrained linear coefficient class, or locally around an interior point of a compact class, it is identified exactly when \[ \ell_\delta\in\operatorname{row}(D).\tag{2} \] If (2) holds, write $\ell_\delta^\top=v^\top D$ and the target is $v^\top r$. If it fails, the fundamental subspace theorem supplies $h\in\ker D$ with $\ell_\delta^\top h\ne0$, so $\xi$ and $\xi+th$ have the same observed means but distinct targets for small feasible $t$. At a boundary with restrictive inequality constraints, the exact criterion is constancy on the feasible fiber rather than the full nullspace; this qualification matters.

When covariance does not identify $\delta$, a computable finite-design identified-set procedure is to retain all $(\delta,\sigma)$ satisfying the covariance equations, solve the corresponding linear mean equations with the declared coefficient constraints, and optimize $\ell_\delta^\top\xi$ over those feasible solutions. For fixed $\delta$ and polyhedral coefficient constraints, this last step is linear programming. This reduces the nonlinear ambiguity to a scalar contagion search, but does not promise a closed-form range for every design.

Estimation and a sharp fixed-size rate

Fix $N$, finitely many assigned graph settings, and finite treatment support, each relevant cell having probability at least $p_*>0$. Put parameters in a compact set with $\sigma$ bounded above and away from zero, stability margin $\eta>0$, a positive lower singular-value bound for the covariance design in (1), and a positive lower singular-value bound for the required mean design. All constants are held fixed as the number $n$ of independent populations grows.

Estimate conditional means and a common conditional covariance within each graph from the observed cells. Conditional Gaussian fourth moments are uniformly bounded. Cell sample means and covariances have mean squared errors $O(n^{-1})$; binomial cell-count concentration makes the event of fewer than a fixed fraction of expected observations exponentially small. On that event use a bounded fallback. Clip covariance eigenvalues to an a priori compact positive interval containing their possible population values before taking inverse square roots.

The inverse-square-root map is Lipschitz on that interval. Least-squares recovery of $(a,b)$ and then $\xi$ is Lipschitz under the singular-value conditions, with ratios protected by $a=1/\sigma$ bounded away from zero. The resolvent identity \[ R_B(\delta')-R_B(\delta) =(\delta'-\delta)R_B(\delta')BR_B(\delta) \] gives a target Lipschitz constant bounded in terms of $\eta$ and the compact primitives. The plug-in target therefore has uniform squared-error risk at most $C/n$.

For a matching lower bound, require an interior one-dimensional parameter submodel along which the target derivative is bounded away from zero. Two parameters separated by $h/\sqrt n$ induce Gaussian conditional laws whose product Kullback--Leibler divergence remains bounded by a constant times $h^2$. Their target separation is order $n^{-1/2}$. Choosing fixed small $h$ and applying the two-point testing bound yields risk at least $c/n$. Thus the squared-error rate is sharp on this regular nonconstant-target class. This argument gives rates, not an exact minimax constant. Constants can deteriorate rapidly with $N$, weak treatment-cell probabilities, or nearly indistinguishable graph spectra; no growing-$N$ claim is hidden in the result.

Which topology differences matter?

Hold attributes fixed, set $q=\mathbf1$, and consider the treatment component of the mean, \[ \phi(B)=\frac1N\mathbf1^\top R_B(\beta I+\gamma B)\mathbf1. \] Writing $m_k(B)=N^{-1}\mathbf1^\top B^k\mathbf1$, the convergent Neumann series gives \[ \phi(B)=\beta+(\beta\delta+\gamma) \sum_{k\geq1}\delta^{k-1}m_k(B). \] Thus equal edge counts need not imply equal effects: weighted walks of length two and longer can differ. For four nodes use one common normalization $B=A/3$, where $A$ is an unweighted adjacency matrix. A star and a four-node path both have three edges and $m_1=1/2$, but their degree-square sums are $12$ and $10$, giving $m_2=1/3$ and $5/18$. With $\beta=1,\gamma=1/5,\delta=1/2$, direct inversion gives respectively $83/55$ and $43/29$; the star's treatment mean is higher by $42/1595$. The normalization is held fixed across graphs, so this is not an artifact of changing the units of an edge.

For a symmetric perturbation $\Delta B$, differentiation simplifies to \[ d\phi(B)[\Delta B]=\frac{\beta\delta+\gamma}{N} \mathbf1^\top R_B\Delta B R_B\mathbf1. \] A bridge edge joining nodes $i,j$ with common small weight increment has derivative proportional to $2(R_B\mathbf1)_i(R_B\mathbf1)_j$. Its effect depends on the surrounding network and on the signs of the structural coefficients. Community and star labels alone do not determine a universal ranking. If $B\mathbf1=\mathbf1$ for every graph, the expression collapses to $(\beta+\gamma)/(1-\delta)$; this is the familiar topology invariance that the catalogue avoids imposing.

Extrapolation and the remaining gap

If every assigned graph has $B=0$, observed outcomes are independent of $\delta$ and $\gamma$. Two models agreeing on $(\zeta,\beta,\sigma)$ but differing in these coefficients have identical experimental laws and can predict different effects on an unassigned connected graph. Individual randomization on the empty graph cannot repair that failure. Conversely, (1)--(2), stability, and the assumed shared mechanism can justify extrapolation to a new graph even if that graph is not directly assigned. Scientific invariance across graph and attribute interventions remains an assumption; it cannot be proved by the observed covariance identity.

The remaining original tasks concern broader Gaussian innovation structures, optimal weak-design bounds, and scientifically justified structural transport. The finite homoskedastic results solve substantial benchmark subproblems while leaving those extensions open.

Completion Estimate

75%

This is a post hoc, heuristic estimate for the full catalogue target. Homoskedastic covariance identifies contagion, target row-space conditions characterize remaining ambiguity, and regular fixed-size experiments achieve a sharp risk rate with explicit topology contrasts. Unequal innovation variances, weak designs and scientifically justified structural transport remain beyond the established benchmark.

References

  • [bs] S. Bhadra and M. Schweinberger (2025), Causal Inference Under Network Interference, arXiv:2508.06808v1, Section 8.3. Primary HTML inspected on 9 October 2026; the later revision is not needed for the claims made here. https://arxiv.org/html/2508.06808v1#S8.SS3.

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