[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84219-en":3,"doc-seo-84219-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84219,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Hypergraph Neural Stochastic Diffusion: An SDE Framework for Uncertainty Estimation","Hypergraph neural networks model higher-order relations well, yet their predictive uncertainty is less studied. Uncertainty in hypergraphs stems not only from noisy attributes and ambiguous labels, but also from variations in node–hyperedge incidence structures and complex higher-order dependencies. The work introduces Hypergraph Neural Stochastic Diffusion (HyperNSD), casting hypergraph representations as stochastic processes on incidence structures. A learnable drift captures deterministic diffusion, while a learnable stochastic forcing characterizes structural ambiguity and representation noise, enabling intrinsic uncertainty quantification via trajectory variability.","Hypergraph Neural Stochastic Diffusion: An SDE Framework for Uncertainty Estimation  \nZhiheng Zhou, Mengyao Zhou, Dengyi Zhao, Xingqin Qi, Guiying Yan  \narXiv :2607 .07330v 1 [ cs .LG] 8 Jul 2026  \nThis work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.  \nAbstract—Hypergraph neural networks have shown powerful capability in modeling higher-order relations, yet their predictive uncertainty remains underexplored. Unlike pairwise graphs, uncertainty in hypergraphs arises not only from noisy attributes and ambiguous labels, but also from variations in node–hyperedge incidence structures and complex higher-order dependencies. Existing approaches mainly estimate uncertainty from final predictions or rely on computationally expensive ensembles and Bayesian inference, limiting their ability to capture uncertainty evolution during representation learning. In this paper, we propose Hypergraph Neural Stochastic Diffusion (HyperNSD), a stochastic differential equation framework for uncertainty estimation on hypergraphs. HyperNSD models hypergraph representations as stochastic processes evolving over node–hyperedge incidence structures. A learnable drift function captures deterministic higher-order diffusion dynamics, while alearnable stochastic forcing function characterizes structural ambiguity and representation noise. Predictive uncertainty is directly quantified through the variability of stochastic representation trajectories, providing an intrinsic uncertainty measure beyond post-hoc confidence scores. We formulate HyperNSD with neural drift and diffusion networks, enabling joint learning of prediction and uncertainty propagation. Theoretical analyses establish wellposedness, perturbation stability, permutation equivariance, and numerical convergence of the proposed stochastic dynamics. Experiments on multiple hypergraph benchmarks demonstrate that HyperNSD achieves reliable uncertainty estimation for outof-distribution and misclassification detection while preserving competitive prediction accuracy. These results provide a principled stochastic-dynamical framework for trustworthy higherorder representation learning.  \nIndex Terms—Hypergraph Neural Networks, Uncertainty Estimation, Hypergraph Neural Stochastic Diffusion  \nI. INTRODUCTION  \nHypergraphs provide a natural representation for complex systems involving higher-order relations, where one interaction may simultaneously connect more than two entities.  \nThis work was supported by the National Natural Science Foundation of China (No.12231018 and No.12471330), the Shandong Provincial Natural Science Foundation (No.ZR2025MS71) and the Postdoctoral Innovation Program of Shandong Province (No.SDCX-ZG-202603012) . (Corresponding author: Xingqin Qi, Guiying Yan.)  \nZhiheng Zhou, Dengyi Zhao and Xingqin Qi are with the School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China (e-mail: [zhouzhiheng@amss.ac.cn](zhouzhiheng@amss.ac.cn); [zhaodengyi@mail.sdu.edu.cn](zhaodengyi@mail.sdu.edu.cn); qix  \n[ingqin@sdu.edu.cn](ingqin@sdu.edu.cn)).  \nMengyao Zhou and Guiying Yan are with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences and also with the University of Chinese Academy of Sciences, Beijing 100190, China (e-mail: [zhoumengyao@amss.ac.cn](zhoumengyao@amss.ac.cn); [yangy@amss.ac.cn](yangy@amss.ac.cn)).  \nUnlike ordinary pairwise graphs, hypergraphs explicitly encode group-wise dependencies through hyperedges and are therefore well suited for modeling relational patterns in social networks, biological systems, recommendation, and brain network analysis [1]–[4] . Motivated by this expressive capability, hypergraph neural networks (HGNNs) have been developed to extend graph neural message passing from pairwise edges to node–hyperedge incidence structures [5]–[9] . By aggregating information between nodes and hyperedges, t","cbCaimdgnU6KEjyr","https://ap.wps.com/l/cbCaimdgnU6KEjyr","pdf",5702800,1,26,"English","en",105,"# I. INTRODUCTION\n## Higher-order relations in hypergraphs\n## Predictive reliability and deterministic HGNN behavior\n## Challenges of uncertainty estimation in hypergraphs","[{\"question\":\"Why is uncertainty estimation more challenging for hypergraph neural networks than for ordinary graph models?\",\"answer\":\"In hypergraphs, uncertainty depends not only on node attributes, labels, or pairwise edges, but also on the node–hyperedge incidence structure. Changing a hyperedge can simultaneously affect aggregation contexts for all incident nodes, so uncertainty must be modeled during higher-order message passing rather than only at the output.\"},{\"question\":\"What does HyperNSD model, and how is uncertainty quantified in the framework?\",\"answer\":\"HyperNSD models hypergraph representations as stochastic processes evolving over node–hyperedge incidence structures. Predictive uncertainty is quantified intrinsically from the variability of stochastic representation trajectories rather than relying on post-hoc confidence scores.\"},{\"question\":\"How are deterministic dynamics and randomness represented in HyperNSD?\",\"answer\":\"HyperNSD uses a learnable drift function to represent deterministic higher-order diffusion dynamics and a learnable stochastic forcing function to capture structural ambiguity and representation noise.\"}]",1784194100,66,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"hypergraph-neural-stochastic-diffusion-an-sde-framework-for-uncertainty-estimation","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/hypergraph-neural-stochastic-diffusion-an-sde-framework-for-uncertainty-estimation/84219/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is uncertainty estimation more challenging for hypergraph neural networks than for ordinary graph models?","Question",{"text":75,"@type":76},"In hypergraphs, uncertainty depends not only on node attributes, labels, or pairwise edges, but also on the node–hyperedge incidence structure. Changing a hyperedge can simultaneously affect aggregation contexts for all incident nodes, so uncertainty must be modeled during higher-order message passing rather than only at the output.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does HyperNSD model, and how is uncertainty quantified in the framework?",{"text":80,"@type":76},"HyperNSD models hypergraph representations as stochastic processes evolving over node–hyperedge incidence structures. Predictive uncertainty is quantified intrinsically from the variability of stochastic representation trajectories rather than relying on post-hoc confidence scores.",{"name":82,"@type":73,"acceptedAnswer":83},"How are deterministic dynamics and randomness represented in HyperNSD?",{"text":84,"@type":76},"HyperNSD uses a learnable drift function to represent deterministic higher-order diffusion dynamics and a learnable stochastic forcing function to capture structural ambiguity and representation noise.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]