Whenever possible, the efficacy of a new treatment, such as a drug or behavioral intervention, is investigated by randomly assigning some individuals to a treatment condition and others to a control condition, and comparing the outcomes between the two groups. Often, when the treatment aims to slow an infectious disease, groups or clusters of individuals are assigned en masse to each treatment arm. The structure of interactions within and between clusters can reduce the power of the trial, i.e. the probability of correctly detecting a real treatment effect. We investigate the relationships among power, within-cluster structure, between-cluster mixing, and infectivity by simulating an infectious process on a collection of clusters. We demonstrate that current power calculations may be conservative for low levels of between-cluster mixing, but failing to account for moderate or high amounts can result in severely underpowered studies. Power also depends on within-cluster network structure for certain kinds of infectious spreading. Infections that spread opportunistically through very highly connected individuals have unpredictable infectious breakouts, which makes it harder to distinguish between random variation and real treatment effects. Our approach can be used before conducting a trial to assess power using network information if it is available, and we demonstrate how empirical data can inform the extent of between-cluster mixing.