Sparse graphical models can capture uncertainty of interconnected systems while promoting parsimony and simplicity - two attributes that can be utilized to identify the topology and control processes defined on networks. This thesis advocates such models in the context of learning the structure of gene-regulatory networks, for which it is argued that single nucleotide polymorphisms can be seen as perturbation data that are critical to identify edge directionality. Applied to the immune-related gene network, these models facilitate the discovery of new regulation pathways. Learning gene-regulating interactions is critical not only to understand how cells differentiate and behave, but also to decipher mechanisms triggering diseases with a genetic component. The impact here is on the development of a new generation of drugs designed to target specific genes. In particular, the genetic interactions of an uncharacterized chemical compound are identified by comparing its effect on the fitness of Saccharomyces cerevisiae (yeast) to that of double-deletion knockouts. As drug targeting is limited by expensive and time-involving laboratory tests, a judicious design of experiments is instrumental in order to reduce the required number of diagnostic mutant strains. During in-vitro experiments with 82 test-drugs, an orderly data reduction of 30% was shown possible without altering the identification of the primary chemical-genetic interactions. Sparsity in wireless cognitive networks emerges due to the geographical distribution of sources, and also due to the scarcity of the radio frequency spectrum used for transmission. In this context, sparsity is leveraged for mapping the interference temperature across space while identifying unoccupied frequency bands. This is achieved by a novel so-terms nonparametric basis pursuit (NBP) method, which entails a basis expansion model with coefficients belonging to a function space. The spatial awareness markedly impacts spectral efficiency, especially when cognitive radios collaborate to reach consensus in a decentralized manner. Tested in a simulated communication setting, NBP captures successfully both shadowing as well as path-loss effects. In additional tests with real-field RF measurements, the spectrum maps reveal the frequency bands utilized for transmission and also reveal the position of the sources. Finally, a blind NBP alternative is introduced to yield a Bayesian nuclear-norm regularization approach for matrix completion. In this context, it becomes possible to incorporate prior covariance information which enables smoothing and prediction. Blind NBP can be further applied to impute missing entries of third- or higher-order data arrays (tensors). These attracted features of blind NBP are illustrated for network flow prediction and imputation of missing entries in three-way ribonucleic-acid (RNA) sequencing arrays and magnetic-resonance-imaging (MRI) tensors.