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Complex systems frequently exhibit multi-way, rather than pairwise, interactions. These group interactions cannot be faithfully modeled as collections of pairwise interactions using graphs and instead require hypergraphs. However, methods that analyze hypergraphs directly, rather than via lossy graph reductions, remain limited. Hypergraph motifs hold promise in this regard, as motif patterns serve as building blocks for larger group interactions which are inexpressible by graphs. Recent work has focused on categorizing and counting hypergraph motifs based on the existence of nodes in hyperedge intersection regions. Here, we argue that the relative sizes of hyperedge intersections within motifs contain varied and valuable information. We propose a suite of efficient algorithms for finding top-k triplets of hyperedges based on optimizing the sizes of these intersection patterns. This formulation uncovers interesting local patterns of interaction, finding hyperedge triplets that either (1) are the least similar with each other, (2) have the highest pairwise but not groupwise correlation, or (3) are the most similar with each other. We formalize this as a combinatorial optimization problem and design efficient algorithms based on filtering hyperedges. Our comprehensive experimental evaluation shows that the resulting hyperedge triplets yield insightful information on real-world hypergraphs. Our approach is also orders of magnitude faster than a naive baseline implementation. 

Modern graph or network datasets often contain rich structure that goes beyond simple pairwise connections between nodes. This calls for complex representations that can capture, for instance, edges of different types as well as so-called “higher-order interactions” that involve more than two nodes at a time. However, we have fewer rigorous methods that can provide insight from such representations. Here, we develop a computational framework for the problem of clustering hypergraphs with categorical edge labels — or different interaction types — where clusters corresponds to groups of nodes that frequently participate in the same type of interaction. Our methodology is based on a combinatorial objective function that is related to correlation clustering on graphs but enables the design of much more efficient algorithms that also seamlessly generalize to hypergraphs. When there are only two label types, our objective can be optimized in polynomial time, using an algorithm based on minimum cuts. Minimizing our objective becomes NP-hard with more than two label types, but we develop fast approximation algorithms based on linear programming relaxations that have theoretical cluster quality guarantees. We demonstrate the efficacy of our algorithms and the scope of the model through problems in edge-label community detection, clustering with temporal data, and exploratory data analysis.