Recent genome sequencing studies have shown that the somatic mutations that drive cancer development are distributed across a large number of genes. This mutational heterogeneity complicates efforts to distinguish functional mutations from sporadic, passenger mutations. Since cancer mutations are hypothesized to target a relatively small number of cellular signaling and regulatory pathways, a common approach is to assess whether known pathways are enriched for mutated genes. However, restricting attention to known pathways will not reveal novel cancer genes or pathways. An alterative strategy is to examine mutated genes in the context of genome-scale interaction networks that include both well characterized pathways and additional gene interactions measured through various approaches. We introduce a computational framework for de novo identification of subnetworks in a large gene interaction network that are mutated in a significant number of patients. This framework includes two major features. First, we introduce a diffusion process on the interaction network to define a local neighborhood of "influence" for each mutated gene in the network. Second, we derive a two-stage multiple hypothesis test to bound the false discovery rate (FDR) associated with the identified subnetworks. We test these algorithms on a large human protein-protein interaction network using mutation data from two recent studies: glioblastoma samples from The Cancer Genome Atlas and lung adenocarcinoma samples from the Tumor Sequencing Project. We successfully recover pathways that are known to be important in these cancers, such as the p53 pathway. We also identify additional pathways, such as the Notch signaling pathway, that have been implicated in other cancers but not previously reported as mutated in these samples. Our approach is the first, to our knowledge, to demonstrate a computationally efficient strategy for de novo identification of statistically significant mutated subnetworks. We anticipate that our approach will find increasing use as cancer genome studies increase in size and scope.