In many application domains, such as computational biology, the goal of graphical model structure learning is to uncover discrete relationships between entities. For example, in our problem of interest concerning HIV vaccine design, we want to infer which HIV peptides interact with which immune system molecules (HLA molecules). For problems of this nature, we are interested in determining the number of non-spurious arcs in a learned graphical model. We describe both a Bayesian and frequentist approach to this problem. In the Bayesian approach, we use the posterior distribution over model structures to compute the expected number of true arcs in a learned model. In the frequentist approach, we develop a method based on the concept of the False Discovery Rate. On synthetic data sets generated from models similar to the ones learned, we find that both the Bayesian and frequentist approaches yield accurate estimates of the number of non-spurious arcs. In addition, we speculate that the frequentist approach, which is non-parametric, may outperform the parametric Bayesian approach in situations where the models learned are less representative of the data. Finally, we apply the frequentist approach to our problem of HIV vaccine design.