[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124295-en":3,"doc-seo-124295-105":30,"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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},124295,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","A machine learning approach to predicting dynamical observables from network structure","Estimating outcomes of dynamical processes from structural features remains a key unsolved challenge in network science. The study proposes machine-learning algorithms to link network topology to dynamics despite nonlinearities, correlations, and feedback between structure and evolution. It predicts epidemic outbreak size from a single infected node and quantifies synchronization in Kuramoto oscillator systems. Topological feature analysis identifies k-core for epidemics and betweenness centrality and accessibility for synchronization, with ranked metric importance and accuracy surpassing prior work.","Downloaded from [https://royalsocietypublishing.org/ on 25 February 2025](https://royalsocietypublishing.org/ on 25 February 2025)  \n[royalsocietypublishing.org/journal/rspa](royalsocietypublishing.org/journal/rspa)  \nCite this article: Rodrigues FA, Peron T, Connaughton C, Kurths J, Moreno Y. 2025 A machine learning approach to predicting dynamical observables from network structure. Proc. R. Soc. A 481: 20240435 .  \n[https://doi.org/10.1098/rspa.2024.0435](https://doi.org/10.1098/rspa.2024.0435)  \nReceived: 16 June 2024  \nAccepted: 02 October 2024  \nSubject Category:  \nPhysics  \nSubject Areas:  \ncomplexity  \nKeywords:  \ncomplex networks, machine learning, dynamical processes  \nAuthor for correspondence:  \nFrancisco A. Rodrigues  \n[e-mail: francisco@icmc.usp.br](e-mail: francisco@icmc.usp.br)  \nA machine learning approach to predicting dynamical observables from network structure  \nFrancisco A. Rodrigues1,2 , Thomas Peron1 , Colm Connaughton2,3 , Jürgen Kurths4,5 and Yamir Moreno6,7,8  \n1 Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, Brazil  \n2 Mathematics Institute, University of Warwick, Gibbet Hill Road,  \nCoventry CV4 7AL, UK  \n3 London Mathematical Laboratory, London, UK  \n4 Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany 5 Department of Physics, Humboldt University, 12489 Berlin, Germany 6 Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain  \n7 Department of Theoretical Physics, University of Zaragoza, 50018 Zaragoza, Spain 8 CENTAI Institute, Turin, Italy  \n FAR, 0000-0002-0145-5571; YM, 0000-0002-0895-1893  \nEstimating the outcome of a given dynamical process from structural features is a key unsolved challenge in network science. This goal is hampered by difficulties associated with nonlinearities, correlations and feedbacks between the structure and dynamics of complex systems. In this work, we develop an approach based on machine learning algorithms that provides an important step towards understanding the relationship between the structure and dynamics of networks. In particular, it allows us to estimate from the network structure the outbreak size of a disease starting from a single node, as well as the degree of synchronicity of a system made up of Kuramoto oscillators. We show which topological features of the network are key for this estimation and provide a ranking of the importance of network metrics with much higher accuracy than previously done. For epidemic propagation, the k-core plays a fundamental role, while for synchronization, the betweenness centrality and accessibility are the measures most related to the state of an oscillator.  \n© 2025 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License [http://creativecommons.org/licenses/](http://creativecommons.org/licenses/)[ ](http://creativecommons.org/licenses/)[by/4.0/](by/4.0/), [which permits unrestricted use](which permits unrestricted use), [provided the original author and](provided the original author and)[ ](provided the original author and)[source are credited.](source are credited.)  \nDownloaded from [https://royalsocietypublishing.org/ on 25 February 2025](https://royalsocietypublishing.org/ on 25 February 2025)  \nFor all the networks, we find that random forests can predict the outbreak size or synchronization state with high accuracy, indicating that the network structure plays a fundamental role in the spreading process. Our approach is general and can be applied to almost any dynamic process running on complex networks. Also, our work is an important step towards applying machine learning methods to unravel dynamical patterns that emerge in complex networked systems.  \n1. Introduction  \nModern network science has been successful at showing that accounting properly for the interaction patterns of a system’s components is crucial for describing its functionality","cbCaiqzcCRlkHgY7","https://ap.wps.com/l/cbCaiqzcCRlkHgY7","pdf",5778949,1,12,"English","en",105,"# Introduction\n## Structure–dynamics prediction challenge\n## Network properties influencing dynamical processes\n## Study objectives","[{\"question\":\"What is the main problem addressed in the paper?\",\"answer\":\"The paper addresses the challenge of estimating dynamical outcomes from structural features of complex networks.\"},{\"question\":\"How does the method predict epidemic spreading outcomes?\",\"answer\":\"Using machine learning, it estimates outbreak size starting from infection at a single node and identifies key structural features supporting this estimation.\"},{\"question\":\"Which network metrics are most related to synchronization of Kuramoto oscillators?\",\"answer\":\"The results indicate betweenness centrality and accessibility have the strongest relationship with the oscillators’ synchronization state.\"}]","A machine learning approach to predicting dynamical observables from network structure | PDF",1785821432,30,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-machine-learning-approach-to-predicting-dynamical-observables-from-network-structure","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-machine-learning-approach-to-predicting-dynamical-observables-from-network-structure/124295/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main problem addressed in the paper?","Question",{"text":75,"@type":76},"The paper addresses the challenge of estimating dynamical outcomes from structural features of complex networks.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method predict epidemic spreading outcomes?",{"text":80,"@type":76},"Using machine learning, it estimates outbreak size starting from infection at a single node and identifies key structural features supporting this estimation.",{"name":82,"@type":73,"acceptedAnswer":83},"Which network metrics are most related to synchronization of Kuramoto oscillators?",{"text":84,"@type":76},"The results indicate betweenness centrality and accessibility have the strongest relationship with the oscillators’ synchronization state.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":29,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]