When people interact, one individual's outcome can be affected by others' treatment status across multiple networks (for example, online and face-to-face ties). Estimating the spillover effect for a specific network is essential to understand how influence travels, but unbiased estimation typically requires observing all relevant networks — an assumption often violated in practice.

🔍 When Multiple Networks Matter

• Spillover effects may operate through more than one network (e.g., Twitter and offline contacts).
• To attribute spillovers to a specific network, the analysis must isolate that network's effect from other, possibly unobserved, networks.

🧪 A Robust Problem, Not Ordinary Omitted-Variable Bias

• Bias from unobserved networks is shown to persist even when treatment assignment is randomized.
• That bias also persists when the unobserved networks and the observed network of interest are independently generated — distinguishing this problem from conventional omitted variable bias.

📊 Tools Offered: Parametric and Nonparametric Sensitivity Analysis

• Parametric sensitivity methods that quantify how much unobserved network influence would change estimated spillover effects under explicit assumptions.
• Nonparametric sensitivity methods that bound possible bias without relying on strong parametric forms.
• These tools enable researchers to assess the potential influence of unobserved networks on causal conclusions about network-specific spillovers.

📈 Concrete Demonstrations

• A simulation study grounded in a real-world Twitter network illustrates how unobserved ties alter estimates.
• An empirical application uses data from a network field experiment in China to show the methods in practice.

🧭 Why This Matters

• Failure to account for unobserved networks can lead to persistent bias in spillover estimates even in randomized designs.
• The proposed sensitivity analyses provide a practical way to evaluate and report how robust network-specific causal claims are to missing network information.