[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126868-en":3,"doc-seo-126868-105":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},126868,1099523885336,"Violet","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Instability in Complex Oscillator Networks - Limitations and Potentials of Network Measures and Machine Learning","Instability in complex oscillator networks - Limitations and Potentials of Network Measures and Machine Learning investigates how network science structure measures relate to stability across non-linear and linear oscillator models, and across real-world power grid topologies and dynamics. The study shows that the measure–stability relationship is highly sensitive to the underlying ensemble: small changes in network properties can even invert the correlation. It evaluates stability prediction using both graph neural networks and combinations of many network measures, finding they can work within one ensemble but fail when the ensemble changes, limiting causal interpretation.","arXiv :2402 . 17500v2 [nlin .AO] 17 Jul 2026  \nInstability in Complex Oscillator Networks: Limitations and Potentials of Network Measures and Machine Learning  \nChristian Nauck 1,2*†, Michael Lindner 1,3†, Nora Molkenthin 1 , Jürgen Kurths 1 , Eckehard Schöll3 , Jörg Raisch2 , Frank Hellmann 1* 1* RD IV Complexity Science, Potsdam Institute for Climate Impact Research,  \nTelegrafenberg A31, Potsdam, 14473, Brandenburg, Germany.  \n2 Control Systems, Technical University of Berlin, Einsteinufer 17, Berlin, 10587, Germany.  \n3Department of Digital Transformation in Energy Systems, Institute of Energy Technology, Technical University of Berlin, Einsteinufer 25 (TA 8), 10587,  \nGermany.  \n*Corresponding author(s). E-mail(s): [nauck@pik-potsdam.de](nauck@pik-potsdam.de) ;  \n[hellmann@pik-potsdam.de](hellmann@pik-potsdam.de) ;  \n†These authors contributed equally to this work.  \nKeywords: Complex Systems, Network Science, Machine Learning, Graph Neural Networks,  \nComplex Oscillators, structure function relationship for non-linear stability  \nA central question of network science is how functional properties of systems emerge from their structure. For networked dynamical systems, structure is typically captured through network measures. We investigate the relationship between these measures and stability metrics across non-linear and linear oscillators, as well as real-world power grid topologies and dynamics. We find that this relationship is highly sensitive to the underlying ensemble: minor changes in the networks considered, such as going from mean degree 6 to mean degree 8, can invert the correlation between a network measure and stability. We also investigate network measures as inputs for machine learning, as well as Graph Neural Networks (GNNs) as predictors of stability. Both GNNs and the non-linear combination of many network measures  \n1  \ncan accurately predict stability within a given ensemble, yet both can fail when the ensemble changes. We conclude that neither approach reliably identifies the underlying structural causes of instability.  \nIntroduction  \nNetworks of coupled oscillators are indispensable for modeling natural and human-made systems. In fields as diverse as biology, neuroscience, ecology, physics, and engineering, important systems including the heart, the brain, food webs, coupled lasers, chemical reactions, power grids, and even firefly populations, are described as oscillators on complex networks [1–6] . The function of these systems is shaped in considerable parts by their connectivity, which is described by the network’s topology.  \nThe paradigmatic model used to understand networks of oscillatory systems is the Kuramoto model [7, 8] and its variants. These consist of linear oscillators with a non-linear coupling, and feature extremely rich collective dynamical behavior, such as chimera states, frequency clusters, isolated desynchronization, and spatial chaos [1, 2, 4–6] . When including amplitude dynamics, one obtains Stuart-Landau oscillators. When combined with specific non-linear couplings, these have been shown to provide realistic models for the dynamics of renewable power grids [9, 10] . A complementary class of models are given by diffusively coupled non-linear oscillators, with van der Pol oscillators being a prominent example. Whereas the rich behavior and multistability in the Kuramoto setting are driven by the non-linear coupling [11] van der Pol oscillators also feature complex, multistable local dynamics.  \nThe collective phenomena in all these systems are rooted in synchronization [3] . In some contexts, such as the brain, complete synchronization indicates dysfunction such as epilepsy. In others, such as power grids, it is a fundamental prerequisite for the system to work. Thus, a central question in the fields of networked complex systems is how robust the synchronous state is, either with the aim to design controls to disrupt it, e.g. deep brain stimulation [12], orto create a syste","cbCaioBtzMasSPgJ","https://ap.wps.com/l/cbCaioBtzMasSPgJ","pdf",3906999,1,57,"English","en",105,"# Introduction\n## Structure vs. function in networked oscillator systems\n## Network measures and stability metrics\n## Graph neural networks for predicting instability","[{\"question\":\"What is the main question addressed by this work?\",\"answer\":\"How functional properties, specifically stability and instability, emerge from network structure using network measures across different oscillator settings and real-world power grid topologies.\"},{\"question\":\"Why can correlations between network measures and stability change?\",\"answer\":\"Because the relationship is highly sensitive to the ensemble; even minor changes in network properties can invert the correlation between a measure and stability.\"},{\"question\":\"How well do graph neural networks predict stability in this study?\",\"answer\":\"They can accurately predict stability within a given ensemble, but they may fail when the ensemble changes, reducing reliable generalization.\"}]","Instability in Complex Oscillator Networks - Limitations and Potentials of Network Measures and Machine Learning | PDF",1785935313,144,{"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},"instability-in-complex-oscillator-networks-limitations-and-potentials-of-network-measures-and-machine-learning","",{"@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/instability-in-complex-oscillator-networks-limitations-and-potentials-of-network-measures-and-machine-learning/126868/",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","2026-08-05",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 question addressed by this work?","Question",{"text":75,"@type":76},"How functional properties, specifically stability and instability, emerge from network structure using network measures across different oscillator settings and real-world power grid topologies.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why can correlations between network measures and stability change?",{"text":80,"@type":76},"Because the relationship is highly sensitive to the ensemble; even minor changes in network properties can invert the correlation between a measure and stability.",{"name":82,"@type":73,"acceptedAnswer":83},"How well do graph neural networks predict stability in this study?",{"text":84,"@type":76},"They can accurately predict stability within a given ensemble, but they may fail when the ensemble changes, reducing reliable generalization.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]