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

Fei Fang; Edoardo M. Airoldi; Laura Forastiere

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arXiv · 2506.06615

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

Fei Fang·Edoardo M. Airoldi·Laura Forastiere

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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. 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 provide a necessary and sufficient condition for when outward and inward spillover effects differ--even in undirected networks--and show that they are equivalent under specific conditions. 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.

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Fei Fang, Edoardo M. Airoldi, Laura Forastiere. 2026-09-06. Inward and Outward Spillover Effects of One Unit's Treatment on Network Neighbors under Partial Interference. https://arxiv.org/abs/2506.06615

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