<p>Research on graph representation learning (GRL) has made major strides over the past decade, with widespread applications in domains such as e-commerce, personalization, fraud & abuse, life sciences, and social network analysis. Despite its widespread success, fundamental questions on practices employed in modern day GRL have remained unanswered. Unraveling and advancing two such fundamental questions on the practices in modern day GRL forms the overarching theme of my thesis.</p> <p>The first part of my thesis deals with the mathematical foundations of GRL. GRL is used to solve tasks such as node classification, link prediction, clustering, graph classification, and so on, albeit with seemingly different frameworks (e.g. Graph neural networks for node/graph classification, (implicit) matrix factorization for link prediction/ clustering, etc.). The existence of very distinct frameworks for different graph tasks has puzzled researchers and practitioners alike. In my thesis, using group theory, I provide a theoretical blueprint that connects these seemingly different frameworks, bridging methods like matrix factorization and graph neural networks. With this renewed understanding, I then provide guidelines to better realize the full capabilities of these methods in a multitude of tasks.</p> <p>The second part of my thesis deals with cases where modeling real-world objects as a graph is an oversimplified description of the underlying data. Specifically, I look at two such objects (i) modeling hypergraphs (where edges encompass two or more vertices) and (ii) using GRL for predicting protein properties. Towards (i) hypergraphs, I develop a hypergraph neural network which takes advantage of the inherent sparsity of real world hypergraphs, without unduly sacrificing on its ability to distinguish non isomorphic hypergraphs. The designed hypergraph neural network is then leveraged to learn expressive representations of hyperedges for two tasks, namely hyperedge classification and hyperedge expansion. Experiments show that using our network results in improved performance over the current approach of converting the hypergraph into a dyadic graph and using (dyadic) GRL frameworks. Towards (ii) proteins, I introduce the concept of conditional invariances and leverage it to model the inherent flexibility present in proteins. Using conditional invariances, I provide a new framework for GRL which can capture protein-dependent conformations and ensures that all viable conformers of a protein obtain the same representation. Experiments show that endowing existing GRL models with my framework shows noticeable improvements on multiple different protein datasets and tasks.</p>