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.

翻译：线性均值模型被广泛用于研究社交网络中的同伴影响。我们考虑在网络节点接受随机化处理时，线性均值模型中的估计问题。研究表明，即使同伴效应在理论上可识别，由于近乎完美的共线性，它们可能无法以标准速率进行估计。我们证明了估计误差的最小最大下界，并指出随着网络密度增加，估计难度会相应增大。在足够稠密的网络中，同伴效应的一致估计变得不可行。为应对这一挑战，我们探究了依赖于网络结构的处理分配机制。利用随机点积图，我们证明当网络存在足够的度异质性时，依赖于网络结构的处理分配能够避免渐近共线性问题。然而，这种依赖性并非万能解决方案，不同的依赖结构必须单独评估其可估计性。这些结果表明，在使用线性均值模型估计同伴效应时需保持谨慎，并强调了显式建模处理与网络结构之间关系的重要性。