  Linear regression on network-linked observations has been an essential tool
in modeling the relationship between response and covariates with additional
network structures. Previous methods either lack inference tools or rely on
restrictive assumptions on social effects and usually assume that networks are
observed without errors. This paper proposes a regression model with
nonparametric network effects. The model does not assume that the relational
data or network structure is exactly observed and can be provably robust to
network perturbations. Asymptotic inference framework is established under a
general requirement of the network observational errors, and the robustness of
this method is studied in the specific setting when the errors come from random
network models. We discover a phase-transition phenomenon of the inference
validity concerning the network density when no prior knowledge of the network
model is available while also showing a significant improvement achieved by
knowing the network model. Simulation studies are conducted to verify these
theoretical results and demonstrate the advantage of the proposed method over
existing work in terms of accuracy and computational efficiency under different
data-generating models. The method is then applied to middle school students'
network data to study the effectiveness of educational workshops in reducing
school conflicts.
