Network is a popular format for encoding structured information in applications ranging from spatial economics to neuroimaging studies. Discovering features of local processes and structures plays a key role in understanding and interpreting the overall state of complex networks. For example, the absence or inhibition of interaction in the protein-protein network impacts the expression levels of protein pathways, which determines the presence or absence of disease; the existence of structural network fragments is significant for functional behavior in the neural system.

In this thesis, we will show that, through various regularization approaches, we can discover local substructures that affect global states or properties of network instances, or efficiently learn coherent models over networks that are robust to missing or corrupted edge weights. Second, with increases in both the amount and the modalities of neuroimaging data, there is a need for models that integrate diverse functional and structural data and that can identify plausible patterns in the complex brain architecture. We will discuss how to model both the structure and function of brain connectivities, while we place hard network constraints driven by prior knowledge and model assumptions.