# TARGETING INTERVENTIONS IN NETWORKS

**Date**: 03/08/2026
**Domain**: social_sciences/economics
**Taxonomy**: academic/research_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Dependence on Quadratic Welfare Specification**

Theorem 1 and Corollary 1 rely on Property A, where welfare is proportional to the sum of squared actions. Online Appendix OA3.1 indicates that alternative welfare functions, such as those linear in actions, alter the optimality conditions for principal components. It would be helpful to qualify claims in the Abstract and Introduction to clarify that the clean spectral ordering is specific to this quadratic welfare structure rather than a general feature of network games.

**Sensitivity of Intervention Simplicity to Cost Curvature**

Proposition 2 shows optimal interventions become simple, aligned with a single principal component, for large budgets under quadratic costs. However, Online Appendix OA3.3 demonstrates that with linear costs, the optimal strategy targets a single individual rather than a network component. Readers might benefit from a discussion in Section 4.2 noting that the principal component targeting result depends critically on the marginal cost increasing with intervention magnitude.

**Sensitivity to Network Topology Measurement Error**

The optimal targeting strategy aligns interventions with specific eigenvectors of the adjacency matrix G, assuming perfect knowledge of the network topology. In networks with small spectral gaps, eigenvector stability theory suggests small measurement errors in G can cause large rotations in the estimated eigenvectors. It would be helpful to include a robustness analysis or discussion on how welfare degrades if the planner uses a perturbed adjacency matrix, particularly for empirical applications where links are estimated.

**Robustness to Strategic Parameter Estimation and Stability Boundaries**

Assumption 2 requires the spectral radius of beta G to be less than 1 for equilibrium stability, yet as beta approaches this boundary, the system becomes ill-conditioned. Additionally, Corollary 1 implies a discontinuous switch in targeting strategy based on the sign of beta. It would be useful to add remarks regarding the robustness of the optimal policy to small perturbations in beta when strategic effects are strong or near zero, as estimation errors could drastically alter the recommended intervention.

**Applicability of Centrality Interpretations to Directed Networks**

Assumption 1 restricts the adjacency matrix to be symmetric, ensuring real eigenvalues and orthogonal eigenvectors used to interpret eigenvector centrality. In many applications, networks are directed, where left and right singular vectors differ and the centrality interpretation is less direct. Readers might benefit from a clearer distinction in Section 4.1 between the undirected case and the directed case discussed in Online Appendix OA3.2, as the intuition regarding global structure relies on properties of symmetric matrices.

**Distinct Eigenvalues Assumption Violated in Network Illustrations**

Assumption 2 requires that all eigenvalues of G are distinct, stating this 'holds generically.' However, the network structures used in the key figures violate this assumption. Figure 1 depicts a circle network with 14 nodes and equal link weights—a highly symmetric structure known to produce repeated eigenvalues due to rotational symmetry. Similarly, Figure 2's two-hub network with symmetric structure likely has repeated eigenvalues. Since Theorem 1, Corollary 1, and Propositions 1-2 all depend on Assumption 2, the numerical illustrations demonstrating these results technically operate outside the formal assumptions. Readers might note that while the distinct eigenvalue condition holds for randomly drawn matrices, the pedagogically useful symmetric networks used in figures do not satisfy it. It would be helpful to either (a) verify that the results extend to networks with repeated eigenvalues (which they likely do, given the 'generic' qualification), or (b) use asymmetric network structures in illustrations that satisfy Assumption 2, or (c) explicitly note in figure captions that these illustrations relax the distinct eigenvalue requirement.

**Symmetric Network Assumption Limits Application to Directed Networks**

Assumption 1 requires the adjacency matrix G to be symmetric, meaning all network relationships are undirected and reciprocal. This assumption underpins the eigenvector decomposition used throughout Sections 3-4 and the main results (Theorem 1, Corollary 1, Propositions 1-2). However, many real-world networks where intervention targeting is relevant are inherently directed: citation networks, Twitter follower relationships, supply chain dependencies, and information flow networks all feature asymmetric relationships. While Online Appendix OA3.2 extends the analysis to non-symmetric matrices using singular value decomposition (SVD), the main theorems and their clean interpretations (e.g., 'target eigenvector centrality for complements') do not directly apply. The consequences are that practitioners working with directed networks cannot directly use the main results without implementing the SVD extension, which changes the interpretation from eigenvectors to singular vectors. It would be helpful to either (a) present the SVD-based results in the main text alongside the symmetric case, or (b) provide clearer guidance on when the symmetric approximation is reasonable for approximately reciprocal networks, or (c) include empirical validation showing how much the optimal intervention changes when symmetry is relaxed.

**Property A Requirement Restricts Welfare Function Applicability**

The main characterization results (Theorem 1, Corollary 1, Propositions 1-2) all require Property A: that aggregate equilibrium utility is proportional to the sum of squares of equilibrium actions (W = w·(a*)ᵀa*). While Examples 1-2 and the beauty contest game satisfy this property, the paper acknowledges in Online Appendix OA3.1 that important applications violate it. Example OA1 (peer effects in smoking/alcohol consumption) explicitly shows a welfare function with both quadratic and linear terms in actions that does not satisfy Property A. The extension in Theorem OA1 is more complex and yields different comparative statics (e.g., the first principal component receives different treatment than other components). The consequence is that the clean, interpretable main results cannot be applied to settings with general non-strategic externalities without substantial modification. Readers might note that many policy-relevant applications involve externalities that violate Property A. It would be helpful to (a) provide a checklist of common economic environments that do vs. do not satisfy Property A, (b) quantify how much the optimal intervention differs when Property A is violated, or (c) present the OA3.1 extension more prominently given its practical importance for applications with heterogeneous externalities.

**Status**: [Pending]

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## Detailed Comments (19)

### 1. Overstatement of Intervention Simplicity in Abstract

**Status**: [Pending]

**Quote**:
> For large budgets, optimal interventions are simple — they involve a single principal component.

**Feedback**:
The Appendix proof indicates that as $C\to\infty$, the similarity $x_{\ell}^{*}$ converges to a non-zero value for $\ell\neq 1$. This suggests the intervention retains weight on multiple components, which differs from the claim that it involves a 'single' component. It would be helpful to rewrite the quote as 'For large budgets, optimal interventions are approximately simple — they are dominated by a single principal component' to align with the technical derivation.

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### 2. Generalization of Local Structure for Bottom Components

**Status**: [Pending]

**Quote**:
> In games of strategic complements (substitutes), interventions place more weight on the top (bottom) principal components, which reflect more global (local) network structure.

**Feedback**:
In bipartite networks, which are explicitly analyzed in Section 4.1, the bottom eigenvector is a global sign-flip of the top eigenvector, reflecting global structure rather than local. Readers might note that adding a qualifier such as 'in non-bipartite networks' would acknowledge the exception demonstrated in the paper's own figures.

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### 3. Similarity Claim Lacks Budget Condition Qualification

**Status**: [Pending]

**Quote**:
> In similarity terms, this means that the optimal intervention has a cosine similarity of nearly 1 to the first or last principal component (depending on the case), and a similarity of nearly 0 to all other principal components.

**Feedback**:
Footnote 3 states the optimal intervention has cosine similarity 'nearly 1' to the first/last principal component. However, Proposition 1 shows this extreme similarity only holds in limiting budget cases. It would be helpful to clarify that 'nearly 1' and 'nearly 0' characterize limiting cases, not the general optimal intervention.

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### 4. Principal Component Ordering Mismatches Eigenvalue Sorting

**Status**: [Pending]

**Quote**:
> The first principal component of $\bm{G}$ is defined as the $n$-dimensional vector that minimizes the sum of squares of the distances to the columns of $\bm{G}$.

**Feedback**:
The text defines principal components via variance maximization, which orders components by the magnitude of eigenvalues. However, the text subsequently indexes PCs by $\ell$ corresponding to eigenvalues sorted by signed value. It would be helpful to clarify that 'principal components' here refers to the eigenvectors ordered by $\lambda_\ell$, noting that this matches the statistical PC ordering only when $|\lambda_\ell|$ is decreasing.

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### 5. Missing Square in Assumption 3 Budget Condition

**Status**: [Pending]

**Quote**:
> problem:
> 
> ###### Assumption 3.
> 
> Either $w<0$ and $C<\|\hat{\bm{b}}\|$, or $w>0$.
> 
> Let $\bm{b}

**Feedback**:
There appears to be an inconsistency in the budget constraint condition. Earlier in the section, the text states the first-best is achievable when '$C\geq\|\hat{\bm{b}}\|^{2}$'. Since the budget constraint is $\|\bm{y}\|^2 \leq C$, the cost of setting $\bm{b}=\mathbf{0}$ is $\|\hat{\bm{b}}\|^2$. For dimensional consistency, Assumption 3 should read '$C<\|\hat{\bm{b}}\|^2$'.

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### 6. Incorrect Alpha Definition in Similarity Ratio Discussion

**Status**: [Pending]

**Quote**:
> The second factor, $\frac{w\alpha_{\ell}}{\mu-w\alpha_{\ell}}$, is determined by two quantities: the eigenvalue corresponding to $\bm{u}^{\ell}(\bm{G})$ (via $\alpha_{\ell}=\frac{1}{1-\beta\lambda_{\ell}}$), and the budget $C$ (via the shadow price $\mu$).

**Feedback**:
The definition of $\alpha_\ell$ here appears to be missing the square. Earlier in the section, $\alpha_\ell$ is correctly defined as '$\alpha_{\ell}=\frac{1}{(1-\beta\lambda_{\ell})^{2}}$'. It would be helpful to rewrite as '(via $\alpha_{\ell}=\frac{1}{(1-\beta\lambda_{\ell})^{2}}$)' to match the correct definition.

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### 7. Figure Reference Inconsistency for Spectral Gap Illustrations

**Status**: [Pending]

**Quote**:
> We illustrate Proposition 2 with a comparison of the two graphs in Figure 4(C).

**Feedback**:
The figure captions in this section show Figure 3 contains parts (A) Large/small spectral gap, (B) Cosine similarity and strategic complements, (C) Large/small bottom gap, and (D) Cosine similarity and strategic substitutes. It would be helpful to rewrite references to Figure 4(C) as Figure 3(C) to match the actual figure captions.

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### 8. Ambiguous Language About Last Eigenvalue Magnitude

**Status**: [Pending]

**Quote**:
> In other words, the last eigenvalue is large when nodes can be partitioned into two sets and most of the connections are across sets, and thus $|\lambda_{n}|$ is maximized in a bipartite graph.

**Feedback**:
The text switches between discussing $|\lambda_n|$ (the magnitude) and $\lambda_n$ (the eigenvalue itself) without clarity. Since $\lambda_n$ is the minimum eigenvalue and is typically negative, saying 'the last eigenvalue is large' is ambiguous. It would be helpful to rewrite as 'the magnitude $|\lambda_n|$ is large' to avoid confusion about sign.

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### 9. Proposition 4 Lacks Explicit Budget Constraint Specification

**Status**: [Pending]

**Quote**:
> eigenvalues and therefore the same trace.###### Proposition 4 (Variance control).
> 
> Consider problem (IT-G) with the cost of intervention satisfying Assumption 5. Suppose Assumptions 1 and 2 hold and the network game satisfies Property A. Let the optimal intervention be $\mathcal{B}

**Feedback**:
Proposition 4 claims a specific ordering of variances across principal components at the optimum, but the problem formulation does not explicitly state what constraint limits the planner's choice of $\mathcal{B}$. It would be helpful to rewrite Proposition 4 to explicitly state the budget constraint: 'Consider problem (IT-G) with cost function satisfying Assumption 5 and budget constraint $K(\mathcal{B}) \leq C$'.

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### 10. Incorrect PCA Variance Terminology in Conclusion

**Status**: [Pending]

**Quote**:
> corresponding to a given principal component quantifies the residual variation explained by that vector.

**Feedback**:
The sentence fragment appears to define the metric associated with a principal component. In standard Principal Component Analysis (PCA) and spectral decomposition, the eigenvalue quantifies the variation explained by that component, not the residual variation. It would be helpful to rewrite 'quantifies the residual variation explained by that vector' as 'quantifies the variation explained by that vector'.

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### 11. Lagrangian Missing Squared Term on Baseline

**Status**: [Pending]

**Quote**:
>  flipping the sign of $x_{\ell}^{*}$.
> 
> We now complete the proof. Observe that the Lagrangian corresponding to the maximization problem is
> 
> $\mathcal{L}=w\sum_{\ell}\alpha_{\ell}(1+x_{\ell})^{2}\hat{\underline{b}}_{\ell}+\mu\left[C-\sum_{\ell}\hat{\underline{b}}_{\ell}^{2}x_{\ell}^{2}\right].$
> 
> Taking our observation above that the constraint is binding at $\bm{x}=\bm{x}^{*}$, t

**Feedback**:
The Lagrangian objective term appears to have $\hat{\underline{b}}_{\ell}$ not $\hat{\underline{b}}_{\ell}^{2}$. From the transformed problem stated earlier, the objective is $\max _ {\boldsymbol {x}} \quad w \sum_ {\ell} \alpha_ {\ell} (1 + x _ {\ell}) ^ {2} \hat {\underline {{b}}} _ {\ell} ^ {2}$. It would be helpful to rewrite the Lagrangian to include the squared term.

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### 12. Contradictory Variance Equality in Proposition 4 Proof

**Status**: [Pending]

**Quote**:
> and so $\operatorname{Var}(\underline{b}_k^{**}) = \operatorname{Var}(\underline{b}_k^*)$ for all $k \notin \{\ell, \ell'\}$ and $\operatorname{Var}(\underline{b}_{\ell}^{*}) = \operatorname{Var}(\underline{b}_{\ell'}^{*}) > \operatorname{Var}(\underline{b}_{\ell'}^{*}) = \operatorname{Var}(\underline{b}_{\ell}^{*})$.

**Feedback**:
This equation contains a logical contradiction. The statement claims a quantity equals itself while simultaneously being strictly greater than itself. The correct statement should describe how the NEW intervention's variances relate to the OLD ones after the permutation. It would be helpful to correct the inequality to reflect the variance swap.

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### 13. Inconsistent Covariance Matrix Notation in Proposition 4

**Status**: [Pending]

**Quote**:
>  and $\pm m{P}$ are both orthogonal. Hence, by Assumption 5, $K(\mathcal{B}^{*}) = K(\mathcal{B}^{**})$. Furthermore, the matrix
> 
> $$
> \boldsymbol {\Sigma} _ {\mathcal {B} ^ {* *}} = \boldsymbol {P} \boldsymbol {\Sigma} _ {\mathcal {B} ^ {*}} \boldsymbol {P} ^ {\top}
> $$
> 
> and so $\operatorname{Var}(\underline{b}_k^{**}) = \operatorname{Var}(\underline{b}_k^*)$ for all $k \notin \{\ell, \ell'\}$ and $\operatorname{Var}(\underline{

**Feedback**:
There is a notation inconsistency between the covariance matrix equation and the variance statements that follow. The permutation matrix P operates in the principal component basis, but the covariance equation uses $\Sigma_B$ (original basis). It would be helpful to rewrite the covariance matrix equation using underlined notation to maintain consistency.

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### 14. First-Order Condition Equation Contains Typo

**Status**: [Pending]

**Quote**:
> tility if and only if $a_j < a_i$. It can be verified that the first-order condition for individual $i$ is given by
> 
> $$
> a_i = \frac{\tilde{b}_i}{1 + \gamma} + \frac{\tilde{b}_i + \gamma}{1 + \gamma} \sum g_{ij} a_j.
> $$
> 
> By defining $\beta = \frac{\tilde{\beta} + \gamma}{1 

**Feedback**:
The coefficient of the summation term in the first-order condition appears to be inconsistent with the derivation from the payoff function. Deriving the FOC yields $a_i = \frac{\tilde{b}_i}{1 + \gamma} + \frac{\tilde{\beta} + \gamma}{1 + \gamma} \sum g_{ij} a_j$. It would be helpful to rewrite the equation to replace $\tilde{b}_i$ with $\tilde{\beta}$ in the numerator of the second term.

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### 15. Incorrect Condition for Positive Utility Effect

**Status**: [Pending]

**Quote**:
> This is a game of strategic complements; moreover, an increase in $j$'s action has a positive effect on individual $i$'s utility if and only if $a_j < a_i$.

**Feedback**:
The condition for $\frac{\partial U_i}{\partial a_j} > 0$ is derived as $a_j < \frac{\tilde{\beta} + \gamma}{\gamma} a_i$. Since $\tilde{\beta} > 0$, the threshold factor is strictly greater than 1. The claim 'if and only if $a_j < a_i$' appears to be inaccurate. It would be helpful to rewrite the claim to reflect the correct threshold.

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### 16. Welfare Coefficient w1 Missing Factor of 1/2

**Status**: [Pending]

**Quote**:
> where $w_{1}=1+m_{2}+m_{5}+(n-1)m_{4}$, $w_{2}=nm_{5}(n-2)$, and $w_{3}=\sqrt{n}[m_{1}+(n-1)m_{3}]$.

**Feedback**:
The coefficient w1 appears to miss a factor of 1/2 from the strategic utility component. At equilibrium, $\sum_i \hat{U}_i = \frac{1}{2}(\bm{a}^*)^T\bm{a}^*$. For Example OA1 where all m parameters equal 0, the stated formula gives w1=1, but Example OA1 equation (OA-1) shows the welfare has coefficient 1/2. It would be helpful to rewrite as '$w_{1}=\frac{1}{2}+m_{2}+m_{5}+(n-1)m_{4}$'.

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### 17. Lemma OA2 Domain Bound Typo in Upper Limit

**Status**: [Pending]

**Quote**:
> expression in part 2, and the fact that $\mu>w_{1}\alpha_{\ell}$. This concludes the proof of Lemma OA2. ∎
> 
> Lemma OA2 implies that $\mu$ as a function of $x_{1}\in\left[-C/\hat{b}_{1},C\hat{b}_{1}\right]$ is U-shaped; the slope is $-\infty$ at $x_{1}=-C/\hat{b}_{1}$ and $+\infty$ at $x_{1}=C/\hat{b}_{1}$; and it reaches a minimum at $x_{1}=0$.
> 
> For $w_

**Feedback**:
The constraint is $\sum_{\ell}\hat{\underline{b}}_{\ell}^{2}x_{\ell}^{2}\leq C$. For x1 alone, this gives $\hat{b}_1^2 x_1^2 \leq C$, so $|x_1| \leq \sqrt{C}/\hat{b}_1$. The upper bound 'C\hat{b}_{1}' should be '$\sqrt{C}/\hat{b}_{1}$' as correctly stated in Lemma OA2 part 1. It would be helpful to rewrite the interval for consistency with the budget constraint.

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### 18. Duplicate Sentence in Theorem OA1 Part 2 Proof

**Status**: [Pending]

**Quote**:
> Suppose, toward a contradiction, that$x_{1}^{*}<0$. Suppose, toward a contradiction, that $x_{1}^{*}<0$. By inspection of the maximization problem

**Feedback**:
The sentence 'Suppose, toward a contradiction, that $x_{1}^{*}<0$' appears twice consecutively in the proof of Theorem OA1 Part 2. This appears to be a duplication error from editing. It would be helpful to remove the first occurrence to improve readability.

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### 19. Optimization Variable Mismatch in Rescaled Problem

**Status**: [Pending]

**Quote**:
> y}$. We will now define a re-scaled version of the problem, $\hat{\text{IT}}(C)$.
> 
> $\max_{\bm{b}}\ C^{-1}\Delta(C^{1/2}\check{\bm{y}})$ ($\hat{\text{IT}}(C)$)
> $\text{s.t.}\ C^{-1}\kappa(C^{1/2}\check{\bm{y}})\leq 1.$
> 
> This is clearly equivalent to the original problem. Let $\check{\bm{y}}^{*}(C)$ be the (possibly 

**Feedback**:
The optimization problem is written with $\max_{\bm{b}}$ but the objective and constraint are expressed entirely in terms of $\check{\bm{y}}$. Since $\bm{y} = \bm{b} - \hat{\bm{b}}$ and $\check{\bm{y}} = C^{-1/2}\bm{y}$, the optimization variable should be $\check{\bm{y}}$, not $\bm{b}$. It would be helpful to rewrite the maximization variable to avoid ambiguity about what is being optimized.

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