[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85002-en":3,"doc-seo-85002-105":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},85002,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Stability of Flow Models for Graph Signals","Generating signals on graphs requires permutation-equivariant models that remain stable under relative structural perturbations. While Graph Neural Networks (GNNs) have established favorable stability behavior, the propagation of structural errors through continuous generative flow dynamics is insufficiently understood. This work studies continuous normalized flow models parameterized by GNNs, proving permutation equivariance for both continuous-time ODE dynamics and discrete samplers, and deriving explicit stability bounds on generated probability distributions. A regularized flow-matching training strategy penalizes the vector field’s spatial Lipschitz constant, improving robustness to structural noise without reducing output quality.","Stability of Flow Models for Graph Signals  \nMartin Schmidt and Gonzalo Mateos  \narXiv :2607 .07510v1 [ ee ss . SP] 8 Jul 2026  \nAbstract—Generating signals on graphs requires permutationequivariant models that exhibit stability with respect to relative structural perturbations. While favorable stability properties of Graph Neural Networks (GNNs) have been well documented, it is unclear how structural errors propagate through the dynamics of continuous generative flow models that are gaining traction for graph signal generation. In this paper, we analyze continuous normalized flow models parameterized by GNNs and show that permutation equivariance is preserved for both the resulting continuous-time ordinary differential equations and their discrete numerical approximations used as graph signal samplers. Our primary contribution is to derive explicit stability bounds on the generated probability distributions, which quantify how relative graph perturbations affect the final sampled signals. Motivated by these theoretical bounds, we introduce a stability-promoting regularized flow matching strategy that actively penalizes the spatial Lipschitz constant of the vector field during model training. Experiments using synthetic smooth signals on stochastic block model graphs and real-world fMRI signals on brain connectomes demonstrate that this bound-oriented approach yields generative models that are more robust to structural noise, without sacrificing output quality.  \nIndex Terms—Graph signal processing, flow models, permutation equivariance, stability, graph neural networks.  \nI. INTRODUCTION  \nGenerative models are powerful tools for learning un  \nderlying data distributions from finite samples. Recent advances in continuous-time generative modeling, including flow matching [1], diffusion models [2], and neural ordinary differential equations (ODEs) [3], have achieved remarkable success generating realistic images, audio and video. However, their application to non-Euclidean domains is still developing. Specifically, while considerable effort has gone into generating graph topologies [4]–[9], significantly less attention has been paid to generating signals supported on the nodes of a given graph structure [10]–[13], despite their ubiquity in real-world applications [14] . Such signals arise in diverse contexts, including neural activity over structural brain connectomes [15], traffic flows on transportation networks [16], sensor measurements in distributed systems [17], and power injections in electrical grids [18] . Learning to faithfully generate such signals can then be useful for various reasons, e.g., to produce synthetic samples in low-data regimes, to serve as implicit priors in inverse or reconstruction problems, and to enable simulation of realistic system behavior. For instance, recent works have leveraged graph signal diffusion models for probabilistic forecasting of stock prices to capture  \nThis work was supported in part by NSF under Grant ECCS 2231036 .(Corresponding author: Gonzalo Mateos.)  \nMartin Schmidt and Gonzalo Mateos are with the Dept. of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627, USA ([e-mails: mschmi21@ur.rochester.edu](e-mails: mschmi21@ur.rochester.edu); [gmateosb@ece.rochester.edu](gmateosb@ece.rochester.edu)).  \nuncertainties and tail events [11], modeling correlated useritem interactions for collaborative filtering in recommender systems [12], and optimizing wireless resource allocation [19] . While said application-driven impetus has fueled exciting architectural advances [10]–[13], the present paper studies fundamental equivariance and stability properties of flow models for graph signal generation. In addition to addressing theoretical questions left unanswered by prior work, the upshot of our analysis has practical implications to model training.  \nPermutation equivariance and stability. Because highquality data can be both limited and expensive, neural n","cbCaigDidbKjjo4C","https://ap.wps.com/l/cbCaigDidbKjjo4C","pdf",391940,1,13,"English","en",105,"# Introduction\n## Permutation equivariance and stability\n## Innovations in context","[{\"question\":\"What problem does the paper address in graph signal generation?\",\"answer\":\"The paper addresses how structural perturbations in a graph affect signals generated by continuous-time generative flow models, and how such errors propagate through the generative dynamics.\"},{\"question\":\"How does the paper establish permutation equivariance for flow models?\",\"answer\":\"It shows that permutation equivariance is preserved for both the resulting continuous-time ODEs and the discrete numerical approximations used to sample graph signals.\"},{\"question\":\"What training method improves robustness in the proposed approach?\",\"answer\":\"The paper introduces a stability-promoting regularized flow matching strategy that penalizes the spatial Lipschitz constant of the vector field during 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problem does the paper address in graph signal generation?","Question",{"text":75,"@type":76},"The paper addresses how structural perturbations in a graph affect signals generated by continuous-time generative flow models, and how such errors propagate through the generative dynamics.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper establish permutation equivariance for flow models?",{"text":80,"@type":76},"It shows that permutation equivariance is preserved for both the resulting continuous-time ODEs and the discrete numerical approximations used to sample graph signals.",{"name":82,"@type":73,"acceptedAnswer":83},"What training method improves robustness in the proposed approach?",{"text":84,"@type":76},"The paper introduces a stability-promoting regularized flow matching strategy that penalizes the spatial Lipschitz constant of the vector field during 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