Minimax rates for the linear-in-means model reveal an identifiability-estimability gap

The linear-in-means model is widely used to study peer influence in social networks. We consider estimation in the linear-in-means model when a randomized treatment is applied to nodes in a network. We show that even when peer effects are identified, they may not be estimable at standard rates, due to near-perfect collinearity. We prove a minimax lower bound on estimation error and show that estimation becomes more difficult as networks grow denser. In sufficiently dense networks, consistent estimation of peer effects is impossible. To address this challenge, we investigate network-dependent treatment assignment. Using random dot product graphs, we show that treatments depending on network structure can prevent asymptotic collinearity when there is sufficient degree heterogeneity. However, such dependence is not a panacea, as different dependence structures must be individually evaluated for estimability. These results suggest caution when using the linear-in-means model to estimate peer effects and highlight the importance of explicitly modeling the relationship between treatments and network structure.