Causal structure learning has long been the central task of inferring causal insights from data. Despite the abundance of real-world processes exhibiting higher-order mechanisms, however, an explicit treatment of interactions in causal discovery has received little attention. In this work, we focus on extending the causal additive model (CAM) to additive models with higher-order interactions. This second level of modularity we introduce to the structure learning problem is most easily represented by a directed acyclic hypergraph which extends the DAG. We introduce the necessary definitions and theoretical tools to handle the novel structure we introduce and then provide identifiability results for the hyper DAG, extending the typical Markov equivalence classes. We next provide insights into why learning the more complex hypergraph structure may actually lead to better empirical results. In particular, more restrictive assumptions like CAM correspond to easier-to-learn hyper DAGs and better finite sample complexity. We finally develop an extension of the greedy CAM algorithm which can handle the more complex hyper DAG search space and demonstrate its empirical usefulness in synthetic experiments.

Learning complex network dynamics is fundamental to understanding, modelling and controlling real-world complex systems. There are two main problems in the task of predicting the dynamic evolution of complex networks: on the one hand, existing methods usually use simple graphs to describe the relationships in complex networks; however, this approach can only capture pairwise relationships, while there may be rich non-pairwise structured relationships in the network. First-order GNNs have difficulty in capturing dynamic non-pairwise relationships. On the other hand, theoretical prediction models lack accuracy and data-driven prediction models lack interpretability. To address the above problems, this paper proposes a higher-order network dynamics identification method for long-term dynamic prediction of complex networks. Firstly, to address the problem that traditional graph machine learning can only deal with pairwise relations, dynamic hypergraph learning is introduced to capture the higher-order non-pairwise relations among complex networks and improve the accuracy of complex network modelling. Then, a dual-driven dynamic prediction module for physical data is proposed. The Koopman operator theory is introduced to transform the nonlinear dynamical differential equations for the dynamic evolution of complex networks into linear systems for solving. Meanwhile, the physical information neural differential equation method is utilised to ensure that the dynamic evolution conforms to the physical laws. The dual-drive dynamic prediction module ensures both accuracy and interpretability of the prediction. Validated on public datasets and self-built industrial chain network datasets, the experimental results show that the method in this paper has good prediction accuracy and long-term prediction performance.

-- In graph signal processing, learning the weighted connections between nodes from a set of sample signals is a fundamental task when the underlying relationships are not known a priori. This task is typically addressed by finding a graph Laplacian on which the observed signals are smooth. With the extension of graphs to hypergraphs - where edges can connect more than two nodes - graph learning methods have similarly been generalized to hypergraphs. However, the absence of a unified framework for calculating total variation has led to divergent definitions of smoothness and, consequently, differing approaches to hyperedge recovery. We confront this challenge through generalization of several previously proposed hypergraph total variations, subsequently allowing ease of substitution into a vector based optimization. T o this end, we propose a novel hypergraph learning method that recovers a hypergraph topology from time-series signals based on a smoothness prior . Our approach, designated as Hypergraph Structure Learning with Smoothness (HSLS), addresses key limitations in prior works, such as hyperedge selection and convergence issues, by formulating the problem as a convex optimization solved via a forward-backward-forward algorithm, ensuring guaranteed convergence. Additionally, we introduce a process that simultaneously limits the span of the hyperedge search and maintains a valid hyperedge selection set. In doing so, our method becomes scalable in increasingly complex network structures. The experimental results demonstrate improved performance, in terms of accuracy, over other state-of-the-art hypergraph inference methods; furthermore, we empirically show our method to be robust to total variation terms, biased towards global smoothness, and scalable to larger hypergraphs. YPERGRAPHS are considered as generalized graphs that capture higher order relationships [1]. While graphs encode pairwise relationships between nodes through edges, the higher order nature of hypergraphs extends node relations to allow an arbitrary number of nodes to be connected by a hyperedge. Figure 1 contains a sample hypergraph displaying these higher order connections where nodes are considered workers and hyperedges connect workers who are collaborating on a project. B. T. Brown, H. Zhang, and D. L. Lau are with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY 40506, USA. G. R. Arce is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA. This work was partially supported by the National Science Foundation under grants 1815992 and 1816003 and the AFOSR award FA9550-22-1-0362.

Binary (0-1) integer programming (BIP) is pivotal in scientific domains requiring discrete decision-making. As the advance of AI computing, recent works explore neural network-based solvers for integer linear programming (ILP) problems. Yet, they lack scalability for tackling nonlinear challenges. To handle nonlinearities, state-of-the-art Branch-and-Cut solvers employ linear relaxations, leading to exponential growth in auxiliary variables and severe computation limitations. To overcome these limitations, we propose BIPNN (Binary Integer Programming Neural Network), an unsupervised learning framework to solve nonlinear BIP problems via hypergraph neural networks (HyperGNN). Specifically, BIPNN reformulates BIPs-constrained, discrete, and nonlinear (sin, log, exp) optimization problems-into unconstrained, differentiable, and polynomial loss functions. The reformulation stems from the observation of a precise one-to-one mapping between polynomial BIP objectives and hypergraph structures, enabling the unsupervised training of HyperGNN to optimize BIP problems in an end-to-end manner. On this basis, we propose a GPU-accelerated and continuous-annealing-enhanced training pipeline for BIPNN. The pipeline enables BIPNN to optimize large-scale nonlinear terms in BIPs fully in parallel via straightforward gradient descent, thus significantly reducing the training cost while ensuring the generation of discrete, high-quality solutions. Extensive experiments on synthetic and real-world datasets highlight the superiority of our approach.

Cooperative multi-agent reinforcement learning faces significant challenges in effectively organizing agent relationships and facilitating information exchange, particularly when agents need to adapt their coordination patterns dynamically. This paper presents a novel framework that integrates dynamic spectral clustering with hypergraph neural networks to enable adaptive group formation and efficient information processing in multi-agent systems. The proposed framework dynamically constructs and updates hypergraph structures through spectral clustering on agents' state histories, enabling higher-order relationships to emerge naturally from agent interactions. The hypergraph structure is enhanced with attention mechanisms for selective information processing, providing an expressive and efficient way to model complex agent relationships. This architecture can be implemented in both value-based and policy-based paradigms through a unified objective combining task performance with structural regularization. Extensive experiments on challenging cooperative tasks demonstrate that our method significantly outperforms state-of-the-art approaches in both sample efficiency and final performance.

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.

With the rapid development of online social networks and the inadequacies in content moderation mechanisms, the detection of fake news has emerged as a pressing concern for the public. Various methods have been proposed for fake news detection, including text-based approaches as well as a series of graph-based approaches. However, the deceptive nature of fake news renders text-based approaches less effective. Propagation tree-based methods focus on the propagation process of individual news, capturing pairwise relationships but lacking the capability to capture high-order complex relationships. Large heterogeneous graph-based approaches necessitate the incorporation of substantial additional information beyond news text and user data, while hypergraph-based approaches rely on predefined hypergraph structures. To tackle these issues, we propose a novel dynamic hypergraph-based multi-view fake news detection model (DHy-MFND) that learns news embeddings across three distinct views: text-level, propagation tree-level, and hypergraph-level. By employing hypergraph structures to model complex high-order relationships among multiple news pieces and introducing dynamic hypergraph structure learning, we optimize predefined hypergraph structures while learning news embeddings. Additionally, we introduce contrastive learning to capture authenticity-relevant embeddings across different views. Extensive experiments on two benchmark datasets demonstrate the effectiveness of our proposed DHy-MFND compared with a broad range of competing baselines.

Rating is a typical user explicit feedback that visually reflects how much a user likes a related item. The (rating) matrix completion is essentially a rating prediction process, which is also a significant problem in recommender systems. Recently, graph neural networks (GNNs) have been widely used in matrix completion, which captures users' preferences over items by formulating a rating matrix as a bipartite graph. However, existing methods are susceptible due to data sparsity and long-tail distribution in real-world scenarios. Moreover, the messaging mechanism of GNNs makes it difficult to capture high-order correlations and constraints between nodes, which are essentially useful in recommendation tasks. To tackle these challenges, we propose a Multi-Channel Hypergraph Contrastive Learning framework for matrix completion, named MHCL. Specifically, MHCL adaptively learns hypergraph structures to capture high-order correlations between nodes and jointly captures local and global collaborative relationships through attention-based cross-view aggregation. Additionally, to consider the magnitude and order information of ratings, we treat different rating subgraphs as different channels, encourage alignment between adjacent ratings, and further achieve the mutual enhancement between different ratings through multi-channel cross-rating contrastive learning. Extensive experiments on five public datasets demonstrate that the proposed method significantly outperforms the current state-of-the-art approaches.

Multivariate time series anomaly detection technology plays an important role in many fields including aerospace, water treatment, cloud service providers, etc. Excellent anomaly detection models can greatly improve work efficiency and avoid major economic losses. However, with the development of technology, the increasing size and complexity of data, and the lack of labels for relevant abnormal data, it is becoming increasingly challenging to perform effective and accurate anomaly detection in high-dimensional and complex data sets. In this paper, we propose a hypergraph based spatiotemporal graph convolutional neural network model STGCN_Hyper, which explicitly captures high-order, multi-hop correlations between multiple variables through a hypergraph based dynamic graph structure learning module. On this basis, we further use the hypergraph based spatiotemporal graph convolutional network to utilize the learned hypergraph structure to effectively propagate and aggregate one-hop and multi-hop related node information in the convolutional network, thereby obtaining rich spatial information. Furthermore, through the multi-scale TCN dilated convolution module, the STGCN_hyper model can also capture the dependencies of features at different scales in the temporal dimension. An unsupervised anomaly detector based on PCA and GMM is also integrated into the STGCN_hyper model. Through the anomaly score of the detector, the model can detect the anomalies in an unsupervised way. Experimental results on multiple time series datasets show that our model can flexibly learn the multi-scale time series features in the data and the dependencies between features, and outperforms most existing baseline models in terms of precision, recall, F1-score on anomaly detection tasks. Our code is available on: https://git.ecdf.ed.ac.uk/msc-23-24/s2044819

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.