How Much and When Do We Need Higher-order Information in Hypergraphs? A Case Study on Hyperedge Prediction

Hypergraphs provide a natural way of representing group relations, whose complexity motivates an extensive array of prior work to adopt some form of abstraction and simplification of higher-order interactions. However, the following question has yet to be addressed: How much abstraction of group interactions is sufficient in solving a hypergraph task, and how different such results become across datasets? This question, if properly answered, provides a useful engineering guideline on how to trade off between complexity and accuracy of solving a downstream task. To this end, we propose a method of incrementally representing group interactions using a notion of n-projected graph whose accumulation contains information on up to n-way interactions, and quantify the accuracy of solving a task as n grows for various datasets. As a downstream task, we consider hyperedge prediction, an extension of link prediction, which is a canonical task for evaluating graph models. Through experiments on 15 real-world datasets, we draw the following messages: (a) Diminishing returns: small n is enough to achieve accuracy comparable with near-perfect approximations, (b) Troubleshooter: as the task becomes more challenging, larger n brings more benefit, and (c) Irreducibility: datasets whose pairwise interactions do not tell much about higher-order interactions lose much accuracy when reduced to pairwise abstractions

Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex $v$ has a weight $\gamma_e(v)$ for each incident hyperedge $e$ that describes the contribution of $v$ to the hyperedge $e$. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.Comment: Accepted to ICML 201

Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering

We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVW-based splitting functions, we convert hypergraphs with EDVW into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVW. For submodular hypergraphs with EDVW-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVW