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higher-order organization

Higher-order organization refers to the structural arrangement and connectivity patterns in complex networks that are governed by interactions among groups of multiple elements rather than simple pairwise connections between individual nodes. Unlike traditional network analysis that focuses on single vertices and standard edges, higher-order organization is characterized by multi-node structures, such as recurring subgraphs, network motifs, and hypergraphs, which form the functional building blocks of larger systems. By capturing how these multi-element configurations assemble and cluster throughout a network, higher-order organization reveals functional units, hierarchical modules, and collective dynamics in complex physical, biological, social, and technological systems that remain undetectable through pairwise analyses alone.

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Higher-order organization of complex networks

Higher-order organization of complex networks

Austin R. Benson, David F. Gleich, Jure Leskovec

OrganizationsPurdue UniversityStanford University

Why you should read this

Develops a scalable, mathematically rigorous framework for clustering complex networks based on higher-order subgraph patterns rather than simple edges, exposing functional modular structures in massive biological, neural, and transportation systems.

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 based on 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.

Added

2026-09-25