Experimental determination of protein function is resource-consuming. As an alternative, computational prediction of protein function has received attention. In this context, protein structural classification (PSC) can help, by allowing for determining structural classes of currently unclassified proteins based on their features, and then relying on the fact that proteins with similar structures have similar functions. Existing PSC approaches rely on sequence-based or direct 3-dimensional (3D) structure-based protein features. In contrast, we first model 3D structures of proteins as protein structure networks (PSNs). Then, we use network-based features for PSC. We propose the use of graphlets, state-of-the-art features in many research areas of network science, in the task of PSC. Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from weighted PSNs. When evaluated on a large set of ∼9, 400 CATH and ∼12, 800 SCOP protein domains (spanning 36 PSN sets), the best of our proposed approaches are superior to existing PSC approaches in terms of accuracy, with comparable running times.