Higher-order organization of complex networks

Resolving a network of hubs Graphs are a pervasive tool for modeling and analyzing network data throughout the sciences. Benson et al. developed an algorithmic framework for studying how complex networks are organized by higher-order connectivity patterns (see the Perspective by Pržulj and Malod-Dognin). Motifs in transportation networks reveal hubs and geographical elements not readily achievable by other methods. A motif previously suggested as important for neuronal networks is part of a “rich club” of subnetworks. Science, this issue p. 163; see also p. 123 A mathematical framework for clustering reveals organizational features of a variety of networks. Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be captured at the level of individual nodes and edges. However, higher-order organization of complex networks—at the level of small network subgraphs—remains largely unknown. Here, we develop a generalized framework for clustering networks on the basis of higher-order connectivity patterns. This framework provides mathematical guarantees on the optimality of obtained clusters and scales to networks with billions of edges. The framework reveals higher-order organization in a number of networks, including information propagation units in neuronal networks and hub structure in transportation networks. Results show that networks exhibit rich higher-order organizational structures that are exposed by clustering based on higher-order connectivity patterns.