Linear regression on network-linked observations has been essential in modeling the relationships between responses and covariates with additional network structures. Many approaches either lack inference tools or rely on restrictive assumptions of social effects. More importantly, these methods usually assume that networks are error-free. I introduce a regression model with nonparametric network effects based on subspace assumptions. This model does not assume the network structure to be precisely observed and is provably robust to network observational errors. An inference framework is established under the general requirement of network observational errors, and corresponding robustness is studied in detail when observational errors arise from random network models. Results reveal a phase-transition phenomenon of inference validity in relation to network density when no prior knowledge of the network model is available. I also show that significant improvements can be achieved when the network model is known. I then briefly discuss an ensemble network estimation strategy, network mixing, which can improve the adaptivity of the proposed method. The regression model is applied to investigate social impacts on students' perceptions of school safety based on observed friendship relations. It enables reliable analysis thanks to the nonparametric network effects and the robustness to network observational errors.