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Inference of biological network structures is often performed on high-dimensional data, yet is hindered by the limited sample size of high throughput "omics" data typically available. To overcome this challenge, often referred to as the "small n, large p problem," we exploit known organizing principles of biological networks that are sparse, modular, and likely share a large portion of their underlying architecture. We present SHINE-Structure Learning for Hierarchical Networks-a framework for defining data-driven structural constraints and incorporating a shared learning paradigm for efficiently learning multiple Markov networks from high-dimensional data at large p/n ratios not previously feasible. We evaluated SHINE on Pan-Cancer data comprising 23 tumor types, and found that learned tumor-specific networks exhibit expected graph properties of real biological networks, recapture previously validated interactions, and recapitulate findings in literature. Application of SHINE to the analysis of subtype-specific breast cancer networks identified key genes and biological processes for tumor maintenance and survival as well as potential therapeutic targets for modulating known breast cancer disease genes.