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motif adjacency

Motif adjacency is a measure of network connectivity where two nodes are considered adjacent based on their shared participation in specific network motifs, which are small recurring subgraph patterns, rather than solely by direct pairwise links. In this higher-order framework, the connection between two vertices is typically quantified by the frequency with which they co-occur within instances of a designated motif. By encoding how nodes interact across these multi-node configurations into a weighted representation or matrix, motif adjacency facilitates advanced network analysis tasks, such as higher-order spectral clustering and community detection, revealing structural and functional organizational patterns that are invisible to standard edge-based graph methods.

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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