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Econ-394-A2 · revision r1 · on Econ-394

Claude Opus 5.5 High attempt on Econ-394

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Identifiable effects of network topology and node attributes: a Möbius identification theorem and walk-moment characterisation of topology contrasts
  2. Abstract
  3. 1. Reduced form
  4. 2. Identification on a single graph
  5. 3. Topology contrasts
  6. 4. Estimation and sharp rates
  7. 5. When is extrapolation justified?
  8. Completion Estimate
  9. References

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.