Real world hypergraphs have several noteworthy features. First, there may be severe degree heterogeneity (i.e., the degree of one node is many times higher than that of another). Second, the overall sparsity levels may vary significantly from one hypergraph to another. Last, a node may have mixed-memberships across multiple communities (i.e., nonzero weights on more than one

Hypergraph neural networks (HGNN) have recently become attractive and received significant attention due to their excellent performance in various domains. However, most existing HGNNs rely on first-order approximations of hypergraph connectivity patterns, which ignores important high-order information. To address this issue, we propose a novel adjacency-tensor-based \textbf{T}ensorized \textbf{H}ypergraph \textbf{N}eural \textbf{N}etwork (THNN). THNN is a faithful hypergraph modeling framework through high-order outer product feature message passing and is a natural tensor extension of the adjacency-matrix-based graph neural networks. The proposed THNN is equivalent to a high-order polynomial regression scheme, which enables THNN with the ability to efficiently extract high-order information from uniform hypergraphs. Moreover, in consideration of the exponential complexity of directly processing high-order outer product features, we propose using a partially symmetric CP decomposition approach to reduce model complexity to a linear degree. Additionally, we propose two simple yet effective extensions of our method for non-uniform hypergraphs commonly found in real-world applications. Results from experiments on two widely used {hypergraph datasets for 3-D visual object classification} show the model's promising performance.

Network data has attracted tremendous attention in recent years, and most conventional networks focus on pairwise interactions between two vertices. However, real-life network data may display more complex structures, and multi-way interactions among vertices arise naturally. In this article, we propose a novel method for detecting community structure in general hypergraph networks, uniform or non-uniform. The proposed method introduces a null vertex to augment a non-uniform hypergraph into a uniform multi-hypergraph, and then embeds the multi-hypergraph in a low-dimensional vector space such that vertices within the same community are close to each other. The resultant optimization task can be efficiently tackled by an alternative updating scheme. The asymptotic consistencies of the proposed method are established in terms of both community detection and hypergraph estimation, which are also supported by numerical experiments on some synthetic and real-life hypergraph networks.