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We define notions of a weak homotopy for finite hypergraphs and an exponential hypergraph with a right adjoint to the categorical product of finite hypergraphs, and investigate a connection between the weak homotopy and the exponential hypergraph. Then we discuss a Hom construction associated to a pair of finite hypergraphs and prove that the homotopy of hypergraph homomorphisms, defined in (Grigor’Yan et al. in Topol Appl 267:106877, 2019), could be characterized by properties of the Hom construction. In addition, we establish some properties of Hom constructions involving the categorical product of finite hypergraphs. As an application we show that the homotopy of d -colorings of a simple hypergraph H could be characterized by properties of Hom constructions associated to the maximum simple hypergraph and H .

We propose high-order hypergraph walks as a framework to generalize graph-based network science techniques to hypergraphs. Edge incidence in hypergraphs is quantitative, yielding hypergraph walks with both length and width. Graph methods which then generalize to hypergraphs include connected component analyses, graph distance-based metrics such as closeness centrality, and motif-based measures such as clustering coefficients. We apply high-order analogs of these methods to real world hypernetworks, and show they reveal nuanced and interpretable structure that cannot be detected by graph-based methods. Lastly, we apply three generative models to the data and find that basic hypergraph properties, such as density and degree distributions, do not necessarily control these new structural measurements. Our work demonstrates how analyses of hypergraph-structured data are richer when utilizing tools tailored to capture hypergraph-native phenomena, and suggests one possible avenue towards that end.

Relations among multiple entities are prevalent in many fields, and hypergraphs are widely used to represent such group relations. Hence, machine learning on hypergraphs has received considerable attention, and especially much effort has been made in neural network architectures for hypergraphs (a.k.a., hypergraph neural networks). However, existing studies mostly focused on small datasets for a few single-entity-level downstream tasks and overlooked scalability issues, although most real-world group relations are large-scale. In this work, we propose new tasks, datasets, and scalable training methods for addressing these limitations. First, we introduce two pair-level hypergraph-learning tasks to formulate a wide range of real-world problems. Then, we build and publicly release two large-scale hypergraph datasets with tens of millions of nodes, rich features, and labels. After that, we propose PCL, a scalable learning method for hypergraph neural networks. To tackle scalability issues, PCL splits a given hypergraph into partitions and trains a neural network via contrastive learning. Our extensive experiments demonstrate that hypergraph neural networks can be trained for large-scale hypergraphs by PCL while outperforming 16 baseline models. Specifically, the performance is comparable, or surprisingly even better than that achieved by training hypergraph neural networks on the entire hypergraphs without partitioning.

Soft set theory is the most developed tool for demonstrating uncertain, vague, not clearly defined objects in a parametric manner. Bipolar uncertainty incorporates a significant role in apprehending discrete and applied mathematical modeling and decision analysis of various physical systems. Graphical and algebraic structures can be studied more precisely when bipolar parametric linguistic properties are to be dealt with, emphasizing the need of a bipolar mathematical approach with soft set theory. In this research paper, we apply the powerful technique of bipolar fuzzy soft sets to hypergraphs and present a novel framework of bipolar fuzzy soft hypergraphs. We elaborate various methods for the construction of bipolar fuzzy soft hypergraphs. We discuss the concept of linearity in bipolar fuzzy soft hypergraphs and study isomorphism properties of bipolar fuzzy soft line graphs of bipolar fuzzy soft hypergraphs, dual and 2-section of bipolar fuzzy soft hypergraphs. We present an application of bipolar fuzzy soft information for analyzing chat conversations of pedophiles and detecting online child grooming cases.

Hypergraphs naturally represent group interactions, which are omnipresent in many domains: collaborations of researchers, co-purchases of items, and joint interactions of proteins, to name a few. In this work, we propose tools for answering the following questions in a systematic manner: (Q1) what are the structural design principles of real-world hypergraphs? (Q2) how can we compare local structures of hypergraphs of different sizes? (Q3) how can we identify domains from which hypergraphs are? We first define hypergraph motifs (h-motifs), which describe the overlapping patterns of three connected hyperedges. Then, we define the significance of each h-motif in a hypergraph as its occurrences relative to those in properly randomized hypergraphs. Lastly, we define the characteristic profile (CP) as the vector of the normalized significance of every h-motif. Regarding Q1, we find that h-motifs ’ occurrences in 11 real-world hypergraphs from 5 domains are clearly distinguished from those of randomized hypergraphs. In addition, we demonstrate that CPs capture local structural patterns unique to each domain, thus comparing CPs of hypergraphs addresses Q2 and Q3. The concept of CP is naturally extended to represent the connectivity pattern of each node or hyperedge as a vector, which proves useful in node classification and hyperedge prediction. Our algorithmic contribution is to propose MoCHy, a family of parallel algorithms for counting h-motifs ’ occurrences in a hypergraph. We theoretically analyze their speed and accuracy and show empirically that the advanced approximate version MoCHy-A + is up to 25 × more accurate and 32 × faster than the basic approximate and exact versions, respectively. Furthermore, we explore ternary hypergraph motifs that extends h-motifs by taking into account not only the presence but also the cardinality of intersections among hyperedges. This extension proves beneficial for all previously mentioned applications.

Let R be a commutative ring with nonzero identity and k≥2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R,k), the set of all k-zero-divisors of R, and the hyperedges of the form a1,a2,a3,…,ak, where a1,a2,a3,…,ak are k distinct elements in Z(R,k), which means (i) a1a2a3⋯ak=0 and (ii) the products of all elements of any (k−1) subsets of a1,a2,a3,…,ak are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite σ-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite σ-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique.

Traditional Hypergraph Neural Networks (HGNNs) often assume that hypergraph structures are perfectly constructed, yet real-world hypergraphs are typically corrupted by noise, missing data, or irrelevant information, limiting the effectiveness of hypergraph learning. To address this challenge, we propose SHSL, a novel Self-supervised Hypergraph Structure Learning framework that jointly explores and optimizes hypergraph structures without external labels. SHSL consists of two key components: a self-organizing initialization module that constructs latent hypergraph representations, and a differentiable optimization module that refines hypergraphs through gradient-based learning. These modules collaboratively capture high-order dependencies to enhance hypergraph representations. Furthermore, SHSL introduces a dual learning mechanism to simultaneously guide structure exploration and optimization within a unified framework. Experiments on six public datasets demonstrate that SHSL outperforms state-of-the-art baselines, achieving Accuracy improvements of 1.36% 32.37% and 2.23% 27.54% on hypergraph exploration and optimization tasks, and 1.19% 8.4% on non-hypergraph datasets. Robustness evaluations further validate SHSL’s effectiveness under noisy and incomplete scenarios, highlighting its practical applicability. The implementation of SHSL and all experimental codes are publicly available at: https://github.com/MingyuanLi88888/SHSL.

Accurately predicting peptide secondary structures remains a challenging task due to the lack of discriminative information in short peptides. In this study, PHAT is proposed, a deep hypergraph learning framework for the prediction of peptide secondary structures and the exploration of downstream tasks. The framework includes a novel interpretable deep hypergraph multi‐head attention network that uses residue‐based reasoning for structure prediction. The algorithm can incorporate sequential semantic information from large‐scale biological corpus and structural semantic information from multi‐scale structural segmentation, leading to better accuracy and interpretability even with extremely short peptides. The interpretable models are able to highlight the reasoning of structural feature representations and the classification of secondary substructures. The importance of secondary structures in peptide tertiary structure reconstruction and downstream functional analysis is further demonstrated, highlighting the versatility of our models. To facilitate the use of the model, an online server is established which is accessible via http://inner.wei‐group.net/PHAT/. The work is expected to assist in the design of functional peptides and contribute to the advancement of structural biology research. Accurately predicting peptide secondary structures remains a challenging task due to the lack of discriminative information in short peptides. Based on transfer learning and hypergraph algorithm, sequential semantic information can be incorporated from large‐scale biological corpus and structural. semantic information from multi‐scale structural segmentation, leading to better accuracy and interpretability even with extremely short peptides.

Migraine is a common chronic neurological disorder that lacks objective imaging biomarkers, while resting-state functional magnetic resonance imaging (rs-fMRI) can be used to extract potential biomarkers. Recently, graph neural networks (GNNs) have gained significant popularity in the classification of brain disorders because of their powerful ability to model brains. Hypergraph neural networks (HGNNs), a branch of GNNs, are particularly effective in capturing high-order neighborhood information. In this paper, we proposed a hypergraph neural network model, incorporating hypergraph dual attention mechanism and hypergraph pooling strategy (APHGNN), for migraine classification derived from the preprocessed rs-fMRI data. First, we constructed hypergraphs from functional connectivity matrices based on the preprocessed rs-fMRI data. Then, we designed three network layers: in the hypergraph dual attention layer, we introduced attention mechanism in both the hyperedge feature aggregation phase and the node feature aggregation phase, making full use of both node and hyperedge information to update node features; in the hypergraph pooling layer, we employed a node selection-based pooling strategy to score and filter nodes, retaining key node features; in the readout layer, we calculated the average and maximum values of the key node features, concatenated and aggregated them, and used the resulting vectors for classification. The experimental results demonstrate that our model outperforms other baseline methods in classification performance and exhibits good generalization. Additionally, the key brain regions extracted through the hypergraph pooling strategy can serve as potential biomarkers for migraine, providing valuable insights for migraine diagnosis.