Hypergraph neural networks (HGNNs) effectively model complex high-order relationships in domains like protein interactions and social networks by connecting multiple vertices through hyperedges, enhancing modeling capabilities, and reducing information loss. Developing foundation models for hypergraphs is challenging due to their distinct data, which includes both vertex features and intricate structural information. We present Hyper-FM, a Hypergraph Foundation Model for multi-domain knowledge extraction, featuring Hierarchical High-Order Neighbor Guided Vertex Knowledge Embedding for vertex feature representation and Hierarchical Multi-Hypergraph Guided Structural Knowledge Extraction for structural information. Additionally, we curate 10 text-attributed hypergraph datasets to advance research between HGNNs and LLMs. Experiments on these datasets show that Hyper-FM outperforms baseline methods by approximately 13.3\%, validating our approach. Furthermore, we propose the first scaling law for hypergraph foundation models, demonstrating that increasing domain diversity significantly enhances performance, unlike merely augmenting vertex and hyperedge counts. This underscores the critical role of domain diversity in scaling hypergraph models.

Fundus fluorescein angiography (FFA) is critical for diagnosing retinal vascular diseases, but beginners often struggle with image interpretation. This study develops FFA Sora, a text-to-video model that converts FFA reports into dynamic videos via a Wavelet-Flow Variational Autoencoder (WF-VAE) and a diffusion transformer (DiT). Trained on an anonymized dataset, FFA Sora accurately simulates disease features from the input text, as confirmed by objective metrics: Frechet Video Distance (FVD) = 329.78, Learned Perceptual Image Patch Similarity (LPIPS) = 0.48, and Visual-question-answering Score (VQAScore) = 0.61. Specific evaluations showed acceptable alignment between the generated videos and textual prompts, with BERTScore of 0.35. Additionally, the model demonstrated strong privacy-preserving performance in retrieval evaluations, achieving an average Recall@K of 0.073. Human assessments indicated satisfactory visual quality, with an average score of 1.570(scale: 1 = best, 5 = worst). This model addresses privacy concerns associated with sharing large-scale FFA data and enhances medical education.

Existing benchmarks like NLGraph and GraphQA evaluate LLMs on graphs by focusing mainly on pairwise relationships, overlooking the high-order correlations found in real-world data. Hypergraphs, which can model complex beyondpairwise relationships, offer a more robust framework but are still underexplored in the context of LLMs. To address this gap, we introduce LLM4Hypergraph, the first comprehensive benchmark comprising 21,500 problems across eight loworder, five high-order, and two isomorphism tasks, utilizing both synthetic and real-world hypergraphs from citation networks and protein structures. We evaluate six prominent LLMs, including GPT-4o, demonstrating our benchmark's effectiveness in identifying model strengths and weaknesses. Our specialized prompting framework incorporates seven hypergraph languages and introduces two novel techniques, Hyper-BAG and Hyper-COT, which enhance high-order reasoning and achieve an average 4% (up to 9%) performance improvement on structure classification tasks. This work establishes a foundational testbed for integrating hypergraph computational capabilities into LLMs, advancing their comprehension. Large Language Models (LLMs) (Vaswani, 2017; Devlin, 2018; Brown, 2020; Ouyang et al., 2022) have made significant strides in domains such as dialogue systems (Bubeck et al., 2023) and image understanding (Zhao et al., 2023). However, they often produce untruthful or unsupported content, known as hallucinations (Wang et al., 2023). To mitigate this, Retrieval-Augmented Generation (RAG) (Vu et al., 2023) enhances prompts with relevant, factual, and up-to-date information (Khandelwal et al., 2019), thereby grounding outputs more effectively. RAG typically retrieves structured data with complex relational dependencies (Guu et al., 2020), such as social networks or molecular structures, which are efficiently represented as graphs. Graph representations capture intricate interdependencies and provide a concise encapsulation of data relationships. This has spurred research to improve LLMs' understanding of graph-structured data (Guo et al., 2023), leading to benchmarks like NLGraph (Wang et al., 2024), GraphQA (Fatemi et al., 2023), and LLM4DyG (Zhang et al., 2023). These benchmarks evaluate and enhance LLMs' capabilities in handling graph-related tasks, promoting the integration of graph-based representations in large language models. However, real-world data often involve complex correlations beyond simple pairwise relationships (Zhou et al., 2006). For example, sentences within a document sharing common keywords may exhibit high-order correlations that traditional graph models fail to capture (PM et al., 2017). In multimodal scenarios (Kim et al., 2020; Feng et al., 2023), interactions across different data types further increase correlation complexity, exceeding the capabilities of conventional graphs, which are limited to pairwise correlations.

Hypergraph Neural Networks (HGNNs) have recently attracted much attention and exhibited satisfactory performance due to their superiority in high-order correlation modeling. However, it is noticed that the high-order modeling capability of hypergraph also brings increased computation complexity, which hinders its practical industrial deployment. In practice, we find that one key barrier to the efficient deployment of HGNNs is the high-order structural dependencies during inference. In this paper, we propose to bridge the gap between the HGNNs and inference-efficient Multi-Layer Perceptron (MLPs) to eliminate the hypergraph dependency of HGNNs and thus reduce computational complexity as well as improve inference speed. Experiments on eight hypergraph datasets demonstrate that even without hypergraph dependency, the proposed LightHGNNs can still achieve competitive or even better performance than HGNNs and outperform vanilla MLPs by 16.3 on average. Extensive experiments on three graph datasets further show the average best performance of our LightHGNNs compared with all other methods. Experiments on synthetic hypergraphs with 5.5w vertices indicate LightHGNNs can run 100 faster than HGNNs, showcasing their ability for latency-sensitive deployments. Compared to the graph with pair-wise correlation, the hypergraph is composed of degree-free hyperedges, which have an inherent superior modeling ability to represent those more complex high-order correlations. However, for large-scale industrial applications, especially for those big-data, small-memory, and high-speed demand environments, the Multi-Layer Perceptrons (MLPs) remain the primary workhorse. The main reason for such an academic-industrial gap for HGNNs is the dependence on the hypergraph structure in inference, which requires large memories in practice.

The isomorphism problem is a fundamental problem in network analysis, which involves capturing both low-order and high-order structural information. In terms of extracting low-order structural information, graph isomorphism algorithms analyze the structural equivalence to reduce the solver space dimension, which demonstrates its power in many applications, such as protein design, chemical pathways, and community detection. For the more commonly occurring high-order relationships in real-life scenarios, the problem of hypergraph isomorphism, which effectively captures these high-order structural relationships, cannot be straightforwardly addressed using graph isomorphism methods. Besides, the existing hypergraph kernel methods may suffer from high memory consumption or inaccurate sub-structure identification, thus yielding sub-optimal performance. In this paper, to address the abovementioned problems, we first propose the hypergraph Weisfiler-Lehman test algorithm for the hypergraph isomorphism test problem by generalizing the Weisfiler-Lehman test algorithm from graphs to hypergraphs. Secondly, based on the presented algorithm, we propose a general hypergraph Weisfieler-Lehman kernel framework and implement two instances, which are Hypergraph Weisfeiler-Lehamn Subtree Kernel and Hypergraph Weisfeiler-Lehamn Hyperedge Kernel. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, we found that our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures.

We study the problem of best-item identification from choice-based feedback. In this problem, a company sequentially and adaptively shows display sets to a population of customers and collects their choices. The objective is to identify the most preferred item with the least number of samples and at a high confidence level. We propose an elimination-based algorithm, namely Nested Elimination (NE), which is inspired by the nested structure implied by the information-theoretic lower bound. NE is simple in structure, easy to implement, and has a strong theoretical guarantee for sample complexity. Specifically, NE utilizes an innovative elimination criterion and circumvents the need to solve any complex combinatorial optimization problem. We provide an instance-specific and non-asymptotic bound on the expected sample complexity of NE. We also show NE achieves high-order worst-case asymptotic optimality. Finally, numerical experiments from both synthetic and real data corroborate our theoretical findings.

We consider a preference learning setting where every participant chooses an ordered list of $k$ most preferred items among a displayed set of candidates. (The set can be different for every participant.) We identify a distance-based ranking model for the population's preferences and their (ranked) choice behavior. The ranking model resembles the Mallows model but uses a new distance function called Reverse Major Index (RMJ). We find that despite the need to sum over all permutations, the RMJ-based ranking distribution aggregates into (ranked) choice probabilities with simple closed-form expression. We develop effective methods to estimate the model parameters and showcase their generalization power using real data, especially when there is a limited variety of display sets.

In this paper, we present a hypergraph neural networks (HGNN) framework for data representation learning, which can encode high-order data correlation in a hypergraph structure. Confronting the challenges of learning representation for complex data in real practice, we propose to incorporate such data structure in a hypergraph, which is more flexible on data modeling, especially when dealing with complex data. In this method, a hyperedge convolution operation is designed to handle the data correlation during representation learning. In this way, traditional hypergraph learning procedure can be conducted using hyperedge convolution operations efficiently. HGNN is able to learn the hidden layer representation considering the high-order data structure, which is a general framework considering the complex data correlations. We have conducted experiments on citation network classification and visual object recognition tasks and compared HGNN with graph convolutional networks and other traditional methods. Experimental results demonstrate that the proposed HGNN method outperforms recent state-of-the-art methods. We can also reveal from the results that the proposed HGNN is superior when dealing with multi-modal data compared with existing methods.