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Inward and Outward Spillover Effects of One Unit's Treatment on Network Neighbors under Partial Interference

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that counting a neighbor's treatment effect from the treated sender's side (outward) and from the affected receiver's side (inward) generally gives different estimands, and characterizes exactly when they coincide.

desk verdict A clear, correct formalization of two averaging perspectives for spillover effects; the central theorem and equivalence conditions are new and should be refereed. read the letter →

arxiv 2506.06615 v1 pith:MKDR725U submitted 2025-06-07 stat.ME

Fei Fang , Edoardo M Airoldi , Laura Forastiere This is my paper
classification stat.ME MSC 62D20
keywords causalinferenceinterferencesocialnetworksoutwardspillovereffectinwardpartialHorvitz-Thompsonestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

One person's treatment can change the outcomes of the people around them, but how that spillover is measured depends on whose perspective you count from. This paper defines two estimands under partial interference—treatment only affects outcomes within the same cluster: the outward spillover effect, the average effect of one person's treatment on their out-neighbors' outcomes, and the inward spillover effect, the average effect on a person's own outcome when one of their in-neighbors is treated. The paper proves that these two estimands generally differ, even in undirected networks, and identifies a precise degree-weighted condition, equation (2), that decides exactly when they differ. It also gives three structural conditions under which the two coincide, derives unbiased Horvitz-Thompson estimators with conservative variance formulas, and shows that when the estimands coincide, the estimators' relative efficiency depends on graph structure and can be reversed by reversing star-graph edges.

What carries the argument

The load-bearing object is the pairwise spillover effect $\tau_{i_k j_k}(\alpha)=\bar Y_{i_k}(Z_{j_k}=1,\alpha)-\bar Y_{i_k}(Z_{j_k}=0,\alpha)$, where $\bar Y$ averages the receiver's potential outcome over a hypothetical treatment assignment $\alpha$ applied to all other units in the cluster. The argument turns on the contrast between the two averaging weights—$1/(N^{out}|N^{out}_{j_k}|)$ in the outward estimand versus $1/(N^{in}|N^{in}_{i_k}|)$ in the inward estimand—so the difference between the estimands is a weighted sum over ordered sender–receiver pairs. In inference, the same weights appear inside Horvitz-Thompson estimators via $W_{j_k}(Z_k)=P_\alpha(Z_{-j_k})/P_\beta(Z_k)$, which reweights the realized design $\beta$ to the hypothetical design $\alpha$.

What would settle it

Take an undirected star graph with one center and four leaves, with pairwise spillover effects +1 from center to each leaf and −1 from each leaf to the center, and any assignment mechanism $\alpha$; the definitions give $\tau^{out}(\alpha)=-3/5$ and $\tau^{in}(\alpha)=3/5$, matching the nonzero status of equation (2). A single instance where the equality or difference of the two estimands disagreed with the zero or nonzero status of that weighted sum would refute Theorem 1.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a precise algebraic identity for the difference between the two spillover estimands. Let $\tau_{i_k j_k}(\alpha)$ be the pairwise spillover effect on unit $i$ when neighbor $j$ is treated, averaged over a hypothetical assignment $\alpha$ of the rest of the cluster. The outward effect weights each such pair by $1/(N^{out}|N^{out}_{j_k}|)$, averaging from the sender's perspective; the inward effect weights by $1/(N^{in}|N^{in}_{i_k}|)$, averaging from the receiver's perspective. Theorem 1 states that $\tau^{out}(\alpha)\neq\tau^{in}(\alpha)$ if and only if the degree-weighted sum over all sender–receiver pairs in equation (2) is nonzero. The paper further shows three sufficient conditions for equality: cluster-specific homogeneous pairwise effects on undirected graphs, globally homogeneous pairwise effects, or a constant degree-ratio condition; and it shows that even when the estimands coincide, the Horvitz-Thompson estimators can differ in variance.

Load-bearing premise

The comparison rests on the analyst's choice of the hypothetical assignment mechanism $\alpha$ that averages over the rest of each cluster; if that choice does not describe the policy scenario, the estimands and all equality conditions answer a different question.

Editorial extensions

If this is right

  • Whenever the graph is non-regular and pairwise spillover effects are heterogeneous, estimates of outward and inward spillover effects will generally answer different questions, so studies should state which perspective they target.
  • Under Conditions 1, 2, or 3 the two estimands coincide, so either Horvitz-Thompson estimator is unbiased for both; the variance formulas then determine which estimator is preferable for a given graph.
  • The gap between the conservative variance bound and the true variance grows with the magnitude of pairwise spillover effects and with cluster size, and shrinks as the number of clusters grows, so confidence intervals may be needlessly wide in clustered designs with strong interference.
  • For star graphs, the direction of the edges decides efficiency: outward-facing stars make the outward estimator less variable, while inward-facing stars make the inward estimator less variable.
  • Asymptotic normality requires many clusters of bounded size; with a few large clusters the normal approximation and the associated confidence intervals are not justified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The degree-weight decomposition in the proof suggests a ready extension: cluster-weighted estimands, exposure-mapped variants, or conditional effects with continuous covariates should obey the same algebra as long as the averaging weights factor into sender and receiver components.
  • In policy terms, the outward effect identifies whom to treat to generate benefits for others, whereas the inward effect identifies whom to protect; the paper does not tell the analyst which perspective a vaccination or information campaign should adopt, but it makes the consequences of that choice explicit.
  • Because the estimands are indexed by the hypothetical assignment $\alpha$, the same experimental data can be used to evaluate several policy scenarios as long as each $\alpha$ is covered by the realized design; this decouples the scientific estimand from the actual randomization scheme.
  • Condition 3 offers a purely graphical route to equivalence that does not require homogeneous effects, so it can be checked from degrees alone; in empirical networks this is the most practical sufficient condition.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The manuscript defines two average spillover estimands under partial interference in clustered networks: the outward effect, which averages a sender's effect on its out-neighbors' outcomes, and the inward effect, which averages a receiver's outcome change caused by the treatment of one of its in-neighbors. Both estimands marginalize over a hypothetical assignment mechanism α within the rest of the cluster. The central result (Theorem 1) gives a necessary and sufficient condition for the two estimands to differ, expressed as a degree-weighted sum of pairwise spillover effects; Conditions 1–3 supply interpretable sufficient conditions for equality. The paper then proposes Horvitz-Thompson estimators for both estimands, proves unbiasedness, asymptotic normality, conservative variance bounds, and consistent variance estimators, and compares the conservative and true variances analytically and by simulation, including outward and inward star graphs. The theoretical development is carried out through direct algebra, with proofs in the appendix.

Significance. If the results hold, the paper provides a useful clarification of when sender- and receiver-perspective spillover effects coincide, which is directly relevant to choosing estimands in network experiments. Theorem 1 is a clean algebraic characterization, the sufficient conditions are interpretable (homogeneous pairwise effects or degree-ratio balance), and the worked examples make the distinction concrete. The inference section extends standard Horvitz-Thompson weighting to these estimands and gives explicit conservative variances, with simulations validating the analytical comparisons. A notable strength is that the central theorem is derived from definitions without fitting any parameter to data, and the proofs are checkable from the displayed algebra. The dependence of both estimands on the analyst-chosen α is a design choice rather than an internal inconsistency, though it is a limitation the authors could state more prominently. The main substantive gap is the absence of an explicit cross-cluster independence assumption for the design-based inference results.

major comments (1)
  1. [Section 4; proof of Proposition 3, around equation (27)] The proof of Proposition 3 uses cov(V_{j k_1}, V_{i k_2}) = 0 for k_1 ≠ k_2, and the proof of Proposition 2 requires each U_{j k} to be correlated with at most a constant number of units. These facts follow only if treatment assignment is independent (or at least uncorrelated) across clusters, an assumption that is never stated. Assumptions 3–6 do not imply it, and Section 4 explicitly allows assignment to be 'dependent' without restricting the dependence to within clusters. Without a cross-cluster independence assumption, V^c(τ̂_out) and V^c(τ̂_in) need not be conservative and the stated CLT in Proposition 2 may fail. Please add an explicit assumption that treatment vectors are independent across clusters (while allowing arbitrary within-cluster dependence), and propagate it to Propositions 6 and 7, which inherit this requirement.
minor comments (5)
  1. [Definition 5] In the display for τ̂_in(α), the second weight is written as \tilde W^0_{jk}(Z_k), but it should be \tilde W^0_{ik}(Z_k); the same subscript slip appears in equation (13) of Appendix E.
  2. [Definitions 2 and 3] The sets N_out_k and N_in_k are written as subsets of N (all units) rather than N_k (the k-th cluster). Taken literally, N_out and N_in would count each unit K times; these definitions should read N_out_k ⊆ N_k and N_in_k ⊆ N_k.
  3. [Table 1 and Appendix C.1] The quantity H_jk is defined as (N_out·|N_out_jk|)^{-1} − (N_in·|N_in_ik|)^{-1}, but the numerical values listed in Table 1 and the expressions in Appendix C.1 use the product (A−B)(A+B) = A^2 − B^2. Please align the label H_jk with the quantity actually used in the subsequent calculations.
  4. [Section 5, discussion after Proposition 5] The statement that the conservative-variance bias 'grows as |N_out_k| increases' should be qualified, because N_out and N_in also change with the graph size; as written, the sentence can be misread as a monotone claim independent of the denominators appearing in Proposition 5.
  5. [Section 2, Definitions 2–3] The dependence of both estimands on α is clear from the definitions, but the paper would benefit from an explicit sentence in the discussion advising that α should be chosen to match the policy scenario of interest, since otherwise the equality or inequality results may not answer the intended scientific question.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 and the estimator properties follow by direct algebra from the definitions and stated assumptions, with no fitted parameter presented as a prediction.

full rationale

The paper's central results are derived self-containedly. Theorem 1 follows from Definitions 2 and 3 by the summation reordering shown in equations (17) and (18), so it is a genuine algebraic characterization rather than a restatement of an input. Conditions 1, 2, and 3 are proven from the same definitions in Appendix F.1. The unbiasedness of the Horvitz-Thompson estimators in Proposition 1 is proven directly from the definition of the weight W_jk(Z_k) and Assumptions 1 and 3, and the CLT in Proposition 2 uses the boundedness and dependence conditions from Assumptions 4, 5, and 6. The variance propositions are analytical comparisons of the derived variance expressions. The paper does cite Wang et al. (2024) for the conservative variance construction and CLT lemma, but Wang et al. (2024) is not authored by any of the present authors, so this is external support rather than a self-citation chain. Moreover, the borrowings are methodological tools, not the central estimand comparison, which stands on its own algebra. No fitted input is renamed as a prediction, and no estimand is defined in terms of another estimand in a way that forces the stated equivalence. The only concerns visible in the manuscript are typographical: Table 1 labels H_jk as a first-order reciprocal difference while the calculations actually use the product of that difference and the corresponding sum, and Definition 5 has a subscripts slip in the notation for the control weight. These are correctness or presentation issues, not circularity, and they do not affect the theorem or the qualitative efficiency comparisons because the displayed numerical values are the ones used in the calculations.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's central claims rest on standard partial-interference assumptions plus technical regularity conditions. No new physical entities or fitted free parameters are introduced; the estimands are defined directly from potential outcomes and the assignment mechanisms, which are assumed known or specified by the design.

assumptions (7)
  • domain assumption Assumption 1: Partial interference. Potential outcomes depend only on the treatment vector within the same cluster.
    This is the fundamental framework that defines the estimands and is stated in Section 2.
  • domain assumption Assumption 2: No fully isolated clusters. Each cluster has at least one unit with an in-neighbor.
    Ensures the spillover effects are well-defined and the asymptotic framework has N_out and N_in growing with K.
  • domain assumption Assumption 3: Overlap between hypothetical and realized treatment assignments. Any treatment vector with positive hypothetical probability also has positive realized probability.
    Needed for the Horvitz-Thompson estimators to be unbiased (Proposition 1).
  • domain assumption Assumption 4: Positivity of realized treatments. The probability of any feasible treatment vector is uniformly bounded below by a positive constant.
    Used to bound the weights W_jk and establish consistency and asymptotic normality.
  • domain assumption Assumption 5: Constant cluster size. n_k = O(1) for all clusters.
    The asymptotic theory relies on the number of clusters growing while cluster size stays bounded; this limits applicability to large clusters.
  • domain assumption Assumption 6: Bounded potential outcomes. There exists a constant C such that |Y_ik(z_k)| <= C for all units and treatment vectors.
    Technical condition for finite moments and the central limit theorem.
  • standard math Standard probability and measure-theoretic background for design-based inference, including the CLT lemma from Ogburn et al. (2022) and Wang et al. (2024).
    The paper relies on existing lemmas for asymptotic normality of bounded correlated random variables.

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Pith. "Pith review of Inward and Outward Spillover Effects of One Unit's Treatment on Network Neighbors under Partial Interference." pith.science (2026). https://pith.science/paper/MKDR725U

@misc{pith2026250606615,
  author       = {Pith},
  title        = {Pith review of: Inward and Outward Spillover Effects of One Unit's Treatment on Network Neighbors under Partial Interference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MKDR725U}},
  note         = {Machine review of arXiv:2506.06615}
}
read the original abstract

In settings where interference is present, direct effects are commonly defined as the average effect of a unit's treatment on their own outcome while fixing the treatment status or probability among interfering units, and spillover effects measure the average effect of a change in the latter while the individual's treatment status is kept fixed. Here, we define the average causal effect of a unit's treatment status on the outcome of their network neighbors, while fixing the treatment probability in the remaining interference set. We propose two different weighting schemes defining two causal effects: i) the outward spillover effect, which represents the average effect of a unit's treatment on their neighbors' potential outcomes, and ii) the inward spillover effect, which represents the impact of each neighbor's treatment on an individual's own potential outcome. We prove that outward and inward spillover effects generally differ, even in an undirected network. However, under specific conditions these two causal estimands become equivalent. We provide numerous examples illustrating the conditions for equivalence or discrepancy of the two spillover effects. We then compare their Horvitz-Thompson estimators, examining their relative variance under various graph structures and structural assumptions on potential outcomes.

Figures

Figures reproduced from arXiv: 2506.06615 by the authors.

Figure 1
Figure 1. The figure shows an undirected star graph , with edges representing the presence of links [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. The figure shows a regular graph, with edges representing the presence of links between [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Directed graph, with directed edges representing the presence and direction of links [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Undirected graph for K = 2 clusters, with edges representing the presence of links between nodes. The value assigned to each edge from jk to ik is the pairwise spillover effect τik,jk(α), for any α. Then we have τ out(α) = 1 5 ·  1 4 · (1 · 4) + (1 1 · 1) · 4  + 1 5 …
Figure 5
Figure 5. Figure 5: Directed graph for K = 2 clusters, with directed edges representing the presence and direction of links between nodes. The value assigned to each edge from jk to ik is the pairwise spillover effect τik,jk(α), for any α. Then we have τ out(α) = 1 4 + 1 · 1 3 · (1 · 3)…
Figure 6
Figure 6. Figure 6: Directed graph, with directed edges representing the presence and direction of links [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: This figure depicts a realization of an outward star graph where unit 1 has 9 outward edges. 1 2 3 4 5 6 7 8 9 10 [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]
Figure 9
Figure 9. Figure 9: Directed graph for K = 2 clusters, with directed edges representing the presence and direction of links between nodes. The structural model of potential outcome models is specified as: Yik(zk) = β0 + β1zik + β2k X jk∈N in ik g(Xjk) · zjk + ϵik, where β2k varies by clus…
Figure 10
Figure 10. Figure 10: Undirected graph for K = 2 clusters, with edges representing the presence of links between nodes. The structural model for potential outcomes for all ik, with k = 1, 2 and i = 1, . . . , 10, is defined as follows: Yik(zjk) = β0 + β1zik + X jk∈Nik g(Xjk) · zjk + ϵik, w…
Figure 11
Figure 11. Figure 11: Directed graph for K = 1 cluster, with directed edges representing the presence and direction of links between nodes. where X11 = X41 = F, X21 = X31 = M. The structural model for potential outcomes for all ik, with k = 1 and i = 1, . . . , 4, is defined as follows: Yi…

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