The hierarchical network model (HNM) is a framework introduced by Sweet, Thomas, and Junker for modeling interventions and other covariate effects on ensembles of social networks, such as what would be found in randomized controlled trials in education research. In this article, we develop calculations for the power to detect an intervention effect using the hierarchical latent space model, an important subfamily of HNMs. We derive basic convergence results and asymptotic bounds on power, showing that standard error for the treatment effect is inversely proportional to the product of the number of ties and the number of networks; a result rather different from the usual effect of cluster size in hierarchical linear models, for example. We explore these results with a simulation study and suggest a tentative approach to power for practical applications.