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Econ-387 · economics · Economics

Repeated-observation interference on changing networks

LLM status Partially solved unverified

Statement

CI-61 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

For $t=1,\ldots,T$, observe all $N$ node outcomes and treatments and a predetermined symmetric matrix $B_t$ with $\|B_t\|_{\mathrm{op}}\leq1$. Given a recorded initial outcome vector, posit \[ Y_t=\alpha\mathbf1+(\rho I+\delta B_t)Y_{t-1} +(\beta I+\gamma B_t)W_t+U_t, \] where independent $U_t\sim N(0,\sigma^2 I)$ are independent of prior histories and current randomization. Treatment probabilities conditional on the recorded past are supplied and lie in $[\varepsilon,1-\varepsilon]$. Parameters lie in a fixed compact set, $\sigma\geq c>0$, and $\|\rho I+\delta B_t\|_{\mathrm{op}}\leq1-\eta$. Thus $\gamma$ describes contemporaneous treatment spillover and $\delta$ lagged outcome contagion. For supplied nonadaptive policy vectors $q_t^0,q_t^1$, potential outcomes follow the same recursion with independent policy assignments replacing $W_t$.

Formal question

Characterize identification and sharp squared-error risk for \[ \psi_{N,T}=\frac1{NT}\sum_{t=1}^T \mathbf1^\top\{\E[Y_t(q^1)]-\E[Y_t(q^0)]\} \] as $N,T$ grow. State design and network conditions ensuring sufficient information to distinguish $\rho,\delta,\beta,\gamma$, and obtain confidence intervals with uniform coverage and shrinking expected length on those classes. Determine how temporal persistence and network spectral structure change effective replication and precision.

Scope and status

Changing graphs are explicitly observed and predetermined in this benchmark. If graph formation reacts to previous outcomes or interventions, its invariant transition mechanism must be included when defining and identifying the policy counterfactual. Sequential randomization alone does not specify that mechanism. The lagged model makes the timing of outcome propagation explicit; extensions to nonlinear dynamics constitute additional research questions.

References for the source of the problem

Subhankar Bhadra and Michael Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808v1, 9 August 2025, Section 8.5; revised version arXiv:2508.06808v2, 9 August 2026, Section 8.3, Time-Indexed Data. https://arxiv.org/html/2508.06808v1#S8.SS5 https://arxiv.org/html/2508.06808v2#S8.SS3

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-387-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-387

Repeated-observation interference on changing networks: identification, spectral information and $1/(NT)$-sharp inference for the policy contrast

Abstract

In the linear dynamic benchmark, the outcome equation is, conditional on the recorded past and the current randomisation, a Gaussian linear regression with regressors $(\mathbf 1,Y_{t-1},B_tY_{t-1},W_t,B_tW_t)$. We prove four results. (i) Identification holds iff an explicit information matrix is nonsingular. Simple sufficient conditions are non-scalar graphs ($\mathrm{sv}(B_t)>0$, the spectral variance) for $\gamma$, and non-degenerate lagged outcomes for $\delta$. (ii) The information for $(\beta,\gamma)$ equals $\sum_t$ of $N\,\mathrm{tr}(D_t)$-weighted spectral moments. In particular, the information on $\gamma$ net of $\beta$ is at least $\epsilon(1-\epsilon)N\sum_t\mathrm{sv}(B_t)$. (iii) $\psi_{N,T}$ is an explicit smooth function of the parameters and the known graph sequence. Its gradient is bounded by $C/\eta^2$, so persistence enters through the factor $\eta^{-2}$. (iv) Delta-method intervals have uniform asymptotic coverage on classes with information bounded below by $\kappa NT$. Their expected length is $O(1/\sqrt{\kappa NT})$, and this rate matches the Cramér--Rao bound.

1. Regression form

Let $\vartheta=(\alpha,\rho,\delta,\beta,\gamma)$ and $X_t=[\mathbf 1,\ Y_{t-1},\ B_tY_{t-1},\ W_t,\ B_tW_t]\in\mathbb R^{N\times5}$. Then $Y_t=X_t\vartheta+U_t$, and $U_t\sim N(0,\sigma^2I)$ is independent of $X_t$: $X_t$ depends only on the recorded past, the predetermined $B_t$ and the current randomisation. The conditional likelihood is Gaussian. The MLE is OLS on the stacked data, and the conditional Fisher information is $\mathcal I_{NT}=\sigma^{-2}\sum_tX_t^\top X_t$.
Theorem (Identification): $\vartheta$ is identified iff $\Pr(\sum_tX_t^\top X_t\text{ nonsingular})>0$ under the design. Sufficient conditions: some $B_t$ with $t\ge2$ is not a multiple of $I$; treatment probabilities lie in $[\varepsilon,1-\varepsilon]$; $\sigma>0$; and, conditionally on $B_t$, the innovation $U_{t-1}$ keeps a nondegenerate Gaussian law. The last condition holds, for example, when $B_t$ is fixed before $U_{t-1}$ is realised.
Proof: If $X^\top X$ is a.s. singular, then some fixed nonzero $c$ has $Xc=0$ a.s., and $\vartheta$ and $\vartheta+c$ give the same law. Conversely, with nonsingular information, OLS recovers $\vartheta$. For sufficiency, suppose $Xc=0$ a.s. Since $W_t$ has independent nondegenerate coordinates, the coefficients on $W_t$ and $B_tW_t$ must cancel coordinatewise: $c_4I+c_5B_t=0$, so $c_4=c_5=0$ when $B_t\notin\mathbb RI$. Then $U_{t-1}$ enters $Y_{t-1}$ with nondegenerate Gaussian law conditionally on $B_t$. Since $(c_2I+c_3B_t)Y_{t-1}=-c_1\mathbf 1$ a.s., the matrix $c_2I+c_3B_t$ must annihilate a nondegenerate Gaussian vector, so $c_2I+c_3B_t=0$, and $c_2=c_3=0$. Finally $c_1\mathbf 1=0$ gives $c_1=0$. ∎

2. Spectral information

Proposition: Condition on the graphs, and let $D_t=\mathrm{diag}(q_{it}(1-q_{it}))\succeq\varepsilon(1-\varepsilon)I$. The $(\beta,\gamma)$ block of $\sigma^2\mathbb E\,\mathcal I$, after partialling out the treatment means, is \[ \sum_t\begin{pmatrix}\mathrm{tr}D_t&\mathrm{tr}(B_tD_t)\\ \mathrm{tr}(B_tD_t)&\mathrm{tr}(B_tD_tB_t)\end{pmatrix}. \] With $D_t=dI$, the Schur complement for $\gamma$ is $d\sum_t[\mathrm{tr}(B_t^2)-(\mathrm{tr}B_t)^2/N]=dN\sum_t\mathrm{sv}(B_t)$, where $\mathrm{sv}$ is the variance of the eigenvalues of $B_t$. In general it is at least $\varepsilon(1-\varepsilon)N\sum_t\mathrm{sv}(B_t)$ when the $D_t$ are scalar, and at least the same bound times $\min q(1-q)/\max q(1-q)$ otherwise.
Proof: $\mathbb E[(W_t-q_t)^\top M(W_t-q_t)]=\mathrm{tr}(MD_t)$, applied with $M\in\{I,B_t,B_t^2\}$. The Schur complement of a $2\times2$ block is computed directly. The eigenvalue variance identity is $\mathrm{tr}(B^2)/N-(\mathrm{tr}B/N)^2$. ∎
So the effective replication for the spillover is $N\sum_t\mathrm{sv}(B_t)$. Graphs close to regular with $B_t\approx\kappa I$ carry little information about $\gamma$, whatever $N$ and $T$ are. For $(\rho,\delta)$ the analogous block involves $\mathbb E[Y_{t-1}^\top\{I,B_t,B_t^2\}Y_{t-1}]$. Its stochastic part is $\sigma^2\,\mathrm{tr}(\Sigma_{t-1}\{\cdot\})$, where $\Sigma_{t-1}=\sum_{s<t}\Phi(t-1,s)\Phi(t-1,s)^\top+\dots$ grows with persistence, up to the order $\eta^{-1}$.

3. The target as a function of parameters

Let $\Phi(t,s)=\prod_{r=s+1}^{t}(\rho I+\delta B_r)$, with the empty product equal to $I$. The initial condition cancels in the contrast. With nonadaptive policies $q^1,q^0$, \[ \psi_{N,T}(\vartheta)=\frac{1}{NT}\sum_{t=1}^T\sum_{s=1}^{t}\mathbf 1^\top\Phi(t,s)(\beta I+\gamma B_s)(q_s^1-q_s^0). \]
Lemma: With $\|\rho I+\delta B_r\|_{\rm op}\le1-\eta$, $\|q^1_s-q^0_s\|_\infty\le1$ and parameters in a compact set, $|\psi|\le C/\eta$ and $\|\nabla_\vartheta\psi\|\le C/\eta^2$, with $C$ independent of $N$ and $T$.
Proof: $\|\Phi(t,s)\|\le(1-\eta)^{t-s}$ and $|\mathbf 1^\top\Phi v|\le\sqrt N\|\Phi v\|_2\le N(1-\eta)^{t-s}\|v\|_\infty$; the geometric sum is at most $1/\eta$. Differentiating $\Phi$ with respect to $(\rho,\delta)$ produces sums of $t-s$ terms of the same form. Since $\sum_kk(1-\eta)^{k-1}=\eta^{-2}$, the gradient bound follows. ∎

4. Inference

Theorem: Let $\mathcal P_{N,T}(\kappa)$ be the class of parameters and graph sequences with $\lambda_{\min}(\sum_t\mathbb E X_t^\top X_t)\ge\kappa NT$, a uniform $\eta$, and $\sigma\in[c,C]$. With the OLS estimate $\widehat\vartheta$, $\widehat V=\widehat\sigma^2(\sum_tX_t^\top X_t)^{-1}$, and $C_{NT}=\psi(\widehat\vartheta)\pm z_{\alpha/2}\sqrt{\nabla\psi^\top\widehat V\nabla\psi}$, we have \[ \liminf_{NT\to\infty}\inf_{\mathcal P_{N,T}(\kappa)}\Pr(\psi_{N,T}\in C_{NT})\ge1-\alpha, \] and $\mathbb E|C_{NT}|=O(\eta^{-2}(\kappa NT)^{-1/2})$. Conversely, the Cramér--Rao bound gives $\mathrm{Var}(\widetilde\psi)\ge\nabla\psi^\top\mathcal I_{NT}^{-1}\nabla\psi$ for every unbiased $\widetilde\psi$, so the rate is sharp whenever $\|\nabla\psi\|$ is bounded below.
Proof[Proof sketch]: The scores $X_t^\top U_t$ form a martingale difference sequence. Under the information lower bound and bounded fourth moments, which hold for Gaussian innovations and a stable recursion, the martingale CLT with the Cramér--Wold device gives $\mathcal I_{NT}^{1/2}(\widehat\vartheta-\vartheta)\Rightarrow N(0,I)$. This holds uniformly on compacts, because all constants are uniform. The ratio $(\sum X_t^\top X_t)/\mathbb E(\cdot)\to I$ by a martingale LLN, using geometric ergodicity from $\eta$. Lemma Lemma and the delta method conclude. ∎

5. Interpretation and open parts

Temporal persistence (small $\eta$) increases the information on $(\rho,\delta)$ but inflates the sensitivity of $\psi$ by $\eta^{-2}$; the net effect on interval length is design-specific. Spectral spread of the $B_t$ is what identifies $\gamma$; network size alone does not. Open: (i) graph processes that react to outcomes, which require modelling the graph kernel, as the statement notes; (ii) adaptive designs that choose $q_t$ to maximise $\mathrm{sv}$-weighted information, which is an optimal-design problem in $(q_t)$ we have not solved; (iii) misspecification-robust versions without Gaussianity (QMLE sandwich).

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The dynamic policy formula and spectral treatment-information calculations describe how network variation can separate direct and spillover coefficients. Identification and uniform interval claims assume joint treatment independence and design concentration beyond the catalogue conditions, leaving sharp risk on general admissible designs unresolved.

References

  • [bs] S. Bhadra, M. Schweinberger, Causal Inference Under Network Interference, arXiv:2508.06808 (v1 2025, Section 8.5; v2 2026, Section 8.3).
  • [hh] P. Hall, C. C. Heyde, Martingale Limit Theory and Its Application, Academic Press (1980).

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Econ-387-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-387

Repeated-observation interference on changing networks: a research attempt

Target and outcome

Status: partial. For exogenous recorded graph paths, I derive the policy functional, exact linear identification conditions, a sharp $1/(NT)$ squared-error rate on a quantitatively excited design class, and uniformly valid shrinking intervals. The excitation condition is made explicit rather than inferred from marginal treatment overlap. The larger problem of deriving optimal rates for every weakly excited, adaptive graph sequence remains unresolved.

The source's time-indexed discussion motivates separating treatment propagation from outcome propagation [bs]. The Gaussian autoregression and the particular theorems below are calculations for the catalogue's explicit benchmark. Throughout the main results $B_1,\ldots,B_T$ are fixed exogenous matrices, or assigned independently before treatment with a known policy-invariant law. Merely being measurable from an observed past does not justify holding an outcome-responsive graph path fixed under another policy.

The finite-horizon functional

Put $A_t=\rho I+\delta B_t$, $D_t=\beta I+\gamma B_t$, and $d_t=q_t^1-q_t^0$. Because the two interventions share the same recorded initial outcome, their mean difference satisfies \[ h_0=0,\qquad h_t=A_th_{t-1}+D_td_t. \] Thus, with the empty matrix product equal to $I$, \[ h_t=\sum_{s=1}^t A_tA_{t-1}\cdots A_{s+1}D_sd_s, \qquad \psi=\frac1{NT}\sum_{t=1}^T\mathbf1^\top h_t. \] The order of the factors matters: changing symmetric networks need not commute. The intercept and initial level cancel from the contrast, but the transition parameters do not. For deterministic $B_t$, policy assignments affect expected outcomes only through their supplied means; no independence across nodes is needed for this mean recursion.

Let $H$ bound $|\beta|+|\gamma|$, and assume the uniform stability margin $\|A_t\|\leq1-\eta$. Since $\|d_t\|\leq\sqrt N$, $\|h_t\|\leq H\sqrt N/\eta$. For two admissible parameter vectors, recursive subtraction gives \[ \frac{\|h_t-\widetilde h_t\|}{\sqrt N} \leq \frac{|\Delta\beta|+|\Delta\gamma|}{\eta} +\frac{H(|\Delta\rho|+|\Delta\delta|)}{\eta^2}. \] Consequently $\psi$ is uniformly Lipschitz in the four relevant coefficients, with a constant $L$ depending on $H,\eta$ but not $N,T$. This bound also states precisely why near-unit persistence is excluded from the rate result.

Identification is a rank question with predictable regressors

Write $\theta=(\alpha,\rho,\delta,\beta,\gamma)^\top$ and \[ X_t=[\mathbf1,\;Y_{t-1},\;B_tY_{t-1},\;W_t,\;B_tW_t],\quad V=\sum_tX_t^\top X_t,\quad M=NT. \] Conditional on the past and current assignment, $Y_t$ has mean $X_t\theta$ and covariance $\sigma^2I$. Two coefficient vectors with the same variance generate the same sequential law exactly when their mean differences vanish almost surely on every history having positive probability. Equivalently, their difference $v$ satisfies ${\mathbb{E}}[v^\top Vv]=0$. Positive definiteness of ${\mathbb{E}} V$ is sufficient for full coefficient identification. If it fails, a policy effect can still be identified when $\psi$ is constant over the compatible coefficient class.

For example, if all $B_t=bI$, only $\rho+b\delta$ and $\beta+b\gamma$ enter the law. The four coefficients are not separately identified, although a policy contrast on the same fixed matrix may be. If all $B_t=0$, spillovers are absent from the observed law. On a nontrivial fixed symmetric matrix, treatment assignment varying along at least two eigenspaces helps distinguish own treatment from neighboring treatment. Marginal probabilities in $[\varepsilon,1-\varepsilon]$ alone are insufficient: assigning every node the same Bernoulli variable can leave $W_t$ and $B_tW_t$ collinear on a regular network.

A quantitative class with a sharp rate

Fix a compact coefficient set $\Theta$, $\sigma\in[\underline\sigma,\overline\sigma]$, stability margin $\eta$, and constants $K,K_0,\kappa>0$. Impose, uniformly over the class, \[ {\mathbb{E}}\operatorname{tr}(V)\leq KM,\qquad {\mathbb{P}}\{\lambda_{\min}(V)<\kappa M\}\leq K_0/M.\tag{D} \] These are declared information conditions, not automatic consequences of treatment randomization. The first follows from stability and an initial second-moment bound ${\mathbb{E}}\|Y_0\|^2=O(N)$ for many bounded-treatment designs. The second requires persistent excitation and concentration of the graph-specific regressors. Its failure captures weak network modes directly.

Let $\widehat\theta$ minimize $\sum_t\|Y_t-X_t\vartheta\|^2$ over $\vartheta\in\Theta$; compactness ensures attainment. Put $S=\sum_tX_t^\top U_t$. Because current innovations are independent of predictable history and current randomization, cross-time martingale terms have zero expectation and \[ {\mathbb{E}}\|S\|^2=\sigma^2{\mathbb{E}}\operatorname{tr}(V) \leq\overline\sigma^2KM. \] One must not condition on the complete future design and then assert an ordinary fixed-design regression distribution: future $X_t$ contain past innovations.

Proposition (Uniform risk under (D)): For a measurable constrained least-squares minimizer, \[ \sup{\mathbb{E}}\|\widehat\theta-\theta\|^2 \leq\frac{4\overline\sigma^2K}{\kappa^2M} +\frac{\operatorname{diam}(\Theta)^2K_0}{M}. \] The plug-in estimator has squared-error risk at most $L^2$ times this bound.
Proof: The least-squares basic inequality gives $\Delta^\top V\Delta\leq2\Delta^\top S$. On the good-design event, $\|\Delta\|\leq2\|S\|/(\kappa M)$. Bound its squared expectation using the martingale identity. On the complement use the compact diameter and (D). The Lipschitz bound proves the target claim. ∎

For sharpness additionally require the class to contain an interior submodel varying $\beta$ alone, with $|\partial\psi/\partial\beta|\geq g_0>0$. Two alternatives $\beta_0\pm h/\sqrt M$ have sequential Gaussian Kullback--Leibler divergence \[ \frac{(2h/\sqrt M)^2}{2\sigma^2} {\mathbb{E}}\sum_t\|W_t\|^2\leq2h^2/\underline\sigma^2. \] For small fixed $h$, their total variation stays below a constant less than one, whereas their targets are separated by at least $2hg_0/\sqrt M$. The two-point testing inequality yields risk at least $c/M$. Thus $M^{-1}$ is a sharp squared-error rate on these nondegenerate classes. If both policy paths coincide, the target is identically zero and this lower bound does not apply.

Uniform confidence without a false conditional-normality claim

For confidence error $a\in(0,1)$, form \[ \mathcal C=\left\{\vartheta\in\Theta: \left\|\sum_tX_t^\top(Y_t-X_t\vartheta)\right\| \leq r_M\right\},\qquad r_M=\sqrt{\overline\sigma^2KM/a}. \] Markov's inequality applied to $\|S\|^2$ gives finite-sample coverage at least $1-a$ uniformly. Report the interval from the infimum to supremum of $\psi(\vartheta)$ over $\mathcal C$; if empty, report an empty set. On the good-design event its length is at most $2Lr_M/(\kappa M)$. On the complement it is bounded by $L\operatorname{diam}(\Theta)$. Its expected length is therefore $O(M^{-1/2})$. Constants are conservative but explicit; the result supplies both coverage and shrinking length.

Unresolved design and transport questions

Condition (D) is directly stronger than qualitative identification. Deriving its best constants from spectral variation, joint assignment probabilities, and graph transitions is still required for a particular application. If $\eta$ shrinks, the Lipschitz constants and regressor moments change, so the displayed uniform result does not imply a universal $NT$ effective sample size. Outcome-responsive graph formation also requires its own intervention-invariant transition law. These are substantial remaining parts of the original growing-network question.

Completion Estimate

72%

This is a post hoc, heuristic estimate for the full catalogue target. The policy recursion, exact information-rank criterion, sharp excited-design risk rate and finite-sample shrinking confidence construction are proved without conditioning on future regressors. Deriving spectral excitation guarantees and precision for weak or adaptive network designs remains incomplete.

References

  • [bs] S. Bhadra and M. Schweinberger (2025), Causal Inference Under Network Interference, arXiv:2508.06808v1, Section 8.5. Primary HTML time-indexed discussion inspected on 9 October 2026; claims here do not rely on the later revision. https://arxiv.org/html/2508.06808v1#S8.SS5.

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