Complex Networks are popular means for studying a wide variety of systems across the social and natural sciences. Recent technological advances allow for a description of these systems on an unprecedented scale. However, due to the immense size and complexity of the resulting networks, efficient evaluation remains a data-analytic challenge. In a recent series of articles (Weber, Saucan, Jost; J. Complex Networks 2017, 2018), we developed geometric tools for efficiently analyzing the structure and evolution of complex networks. The core component of our theory, a discrete Ricci curvature, translates central tools from differential geometry to the discrete realm. With these tools, we extend the commonly used node-based approach to include edge-based information such as edge weights and directionality for a more comprehensive network characterization.The analysis of a wide range of complex networks suggests connections between curvature and higher order network structure. Our results identify important structural features, including long-range connections of high curvature acting as bridges between major network components. Thus, curvature identifies the network’s core structure on which expensive network hypothesis testing and further network analysis becomes more feasible.