Protein-protein interactions (PPIs) mediate the transmission and regulation of oncogenic signals that are essential to cellular proliferation and survival, and thus represent potential targets for anti-cancer therapeutic discovery. Despite their significance, there is no method to experimentally disrupt and interrogate the essentiality of individual endogenous PPIs. The ability to computationally predict or infer PPI essentiality would help prioritize PPIs for drug discovery and help advance understanding of cancer biology. Here we introduce a computational method (MEDICI) to predict PPI essentiality by combining gene knockdown studies with network models of protein interaction pathways in an analytic framework. Our method uses network topology to model how gene silencing can disrupt PPIs, relating the unknown essentialities of individual PPIs to experimentally observed protein essentialities. This model is then deconvolved to recover the unknown essentialities of individual PPIs. We demonstrate the validity of our approach via prediction of sensitivities to compounds based on PPI essentiality and differences in essentiality based on genetic mutations. We further show that lung cancer patients have improved overall survival when specific PPIs are no longer present, suggesting that these PPIs may be potentially new targets for therapeutic development. Software is freely available at https://github.com/cooperlab/MEDICI. Datasets are available at /programs/ctd2/data-portal.