[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-detail-657263-en":59,"doc-seo-657263-105":81},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":5,"data":60},{"doc_id":61,"user_id":62,"nickname":63,"user_avatar":64,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":66,"doc_content":67,"file_id":68,"file_url":69,"file_type":70,"file_size":71,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":72,"language":73,"language_code":74,"site_id":75,"html_lang":74,"table_of_contents":76,"faqs":77,"seo_title":78,"seo_description":66,"update_tm":79,"read_time":80},657263,2336477982702,"Marglet","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Compositional System Dynamics: The Higher Mathematics Underlying System Dynamics Diagrams & Practice","This work establishes a robust mathematical foundation for compositional System Dynamics modeling using category theory. System Dynamics diagrams—stock & flow, system structure, and causal loop diagrams—are formulated as categorical constructs that enable scalable, transparent, and systematic reasoning. Encoding diagrams as data via attributed C-sets and categorical tools such as structured cospans, pushouts, pullbacks, and functor mappings supports modular composition and structure-preserving translations between syntax and semantics, including alternative stochastic transition dynamics beyond ODEs.","arXiv :2509 . 18475v1 [ ee ss . SY] 22 Sep 2025  \nCompositional System Dynamics: The Higher Mathematics Underlying System Dynamics Diagrams & Practice  \nXiaoyan Li∗ Evan Patterson Patricia L. Mabry Nathaniel D. Osgood  \nAbstract  \nThis work establishes a robust mathematical foundation for compositional System Dynamics modeling, leveraging category theory to formalize and enhance the representation, analysis, and composition of system models. Here, System Dynamics diagrams, such as stock & flow diagrams, system structure diagrams, and causal loop diagrams, are formulated as categorical constructs, enabling scalable, transparent, and systematic reasoning. By encoding these diagrams as data using attributed C-sets and utilizing advanced categorical tools like structured cospans, pushouts, pullbacks, and functor mappings, the framework supports modular composition, stratification, and seamless mapping between syntax and semantics.  \nThe approach underwrites traditional practice with firm mathematical structure, facilitates the identification of certain forms of pathways and feedback loops, the detection of simple patterns within complex diagrams, common structure between diagrams, and structurepreserving mappings between diverse diagram types. Additionally, this framework supports alternative semantics, such as stochastic transition dynamics, extending beyond traditional ordinary differential equation (ODE) representations. Applications in compositional modeling, modularity, and team-based collaboration demonstrate the practical advantages of this advanced framework.  \nFuture directions include integrating dimensional annotations, supporting hybrid and agent-based modeling paradigms, and expanding the framework’s applicability to global and local temporal reasoning through temporal sheaves. By revealing and formalizing the hidden mathematical structure of System Dynamics diagrams, this work empowers practitioners to tackle complex systems with clarity, scalability, and rigor.  \nKeywords. stock & flow diagrams, causal loop diagrams, system structure diagrams, category theory, composition, stratification, structure-preserving maps, System Dynamics modeling  \n1 Introduction  \nCompared to other simulation traditions, System Dynamics is distinguished by its diverse repertoire of diagrams, their accessibility to project stakeholders, and their central role in which they facilitate insights into system behavior. A great deal of the value delivered by System Dynamics modeling project is realized by the accessibility and varied abstraction levels associated with such diagrams, and the different dynamic patterns that they help explain.  \nWhile creation, refinement of and thinking around such diagrams—whether stock & flow models, system structure diagrams, and causal loop diagrams—constitutes the central activity of many modeling projects, there has been surprisingly little examination of deeper mathematical underlying System Dynamics diagrams, the formal relationships of different diagrams to eachother, and formalization of the relationships between diagram types. This work highlights the clear mathematical underpinnings for each such diagram type and which relate those types to each other. In the process of doing so, it illustrates the compelling benefits for thinking and tool use to be secured by a truly compositional form of System Dynamics, particularly  \n∗ Corresponding author: [xiaoyan.li2@uleth.ca](xiaoyan.li2@uleth.ca).  \nas it relates to scalability, transparency, performance, clarity, cohesiveness, productivity, and versatility of System Dynamics modeling. While deferring more detailed treatment to our recently published manuscripts on the mathematical structure of System Dynamics within the mathematics literature, this paper introduces key mathematical concepts that enable these advancements, illustrating their practical affordances and advantages. Additionally, we discuss ongoing and future work to extend this foundational appro","cbCaiixpOmFfM9dY","https://ap.wps.com/l/cbCaiixpOmFfM9dY","pdf",8427456,41,"English","en",105,"# 1 Introduction\n## 1.1 Motivation\n## 1.1.1 Example Use Cases","[{\"question\":\"What is the core contribution of this work for System Dynamics modeling?\",\"answer\":\"It provides a compositional mathematical foundation for System Dynamics diagrams using category theory, enabling formal representation, analysis, and composition of system models.\"},{\"question\":\"How does the framework represent System Dynamics diagrams?\",\"answer\":\"Stock \\u0026 flow diagrams, system structure diagrams, and causal loop diagrams are encoded as categorical constructs and then represented as data using attributed C-sets and categorical mappings.\"},{\"question\":\"What types of model behavior can the approach support beyond standard ODE semantics?\",\"answer\":\"It supports alternative semantics, including stochastic transition dynamics, extending beyond traditional ordinary differential equation (ODE) representations.\"}]","Compositional System Dynamics: The Higher Mathematics Underlying System Dynamics Diagrams & Practice | PDF",1792976466,103,{"code":4,"msg":82,"data":83},"ok",{"site_id":75,"language":74,"slug":84,"title":65,"keywords":85,"description":66,"schema_data":86,"social_meta":140,"head_meta":142,"extra_data":144,"updated_unix":145},"compositional-system-dynamics-the-higher-mathematics-underlying-system-dynamics-diagrams-practice","",{"@graph":87,"@context":139},[88,102,122],{"@type":89,"itemListElement":90},"BreadcrumbList",[91,95,97,100],{"item":92,"name":93,"@type":94,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":96,"name":9,"@type":94,"position":14},"https://docshare.wps.com/document/",{"item":98,"name":40,"@type":94,"position":99},"https://docshare.wps.com/document/research-report/",3,{"item":101,"name":65,"@type":94,"position":19},"https://docshare.wps.com/document/compositional-system-dynamics-the-higher-mathematics-underlying-system-dynamics-diagrams-practice/657263/",{"url":101,"name":65,"@type":103,"image":104,"author":109,"headline":65,"publisher":111,"fileFormat":114,"inLanguage":74,"description":66,"dateModified":115,"datePublished":116,"encodingFormat":114,"isAccessibleForFree":117,"interactionStatistic":118},"DigitalDocument",{"url":105,"@type":106,"width":107,"height":108},"https://docshare.wps.com/thumbnails/compositional-system-dynamics-the-higher-mathematics-underlying-system-dynamics-diagrams-practice/657263.png","ImageObject",300,407,{"name":63,"@type":110},"Person",{"url":92,"name":112,"@type":113},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":119,"interactionType":120,"userInteractionCount":24},"InteractionCounter",{"@type":121},"ViewAction",{"@type":123,"mainEntity":124},"FAQPage",[125,131,135],{"name":126,"@type":127,"acceptedAnswer":128},"What is the core contribution of this work for System Dynamics modeling?","Question",{"text":129,"@type":130},"It provides a compositional mathematical foundation for System Dynamics diagrams using category theory, enabling formal representation, analysis, and composition of system models.","Answer",{"name":132,"@type":127,"acceptedAnswer":133},"How does the framework represent System Dynamics diagrams?",{"text":134,"@type":130},"Stock & flow diagrams, system structure diagrams, and causal loop diagrams are encoded as categorical constructs and then represented as data using attributed C-sets and categorical mappings.",{"name":136,"@type":127,"acceptedAnswer":137},"What types of model behavior can the approach support beyond standard ODE semantics?",{"text":138,"@type":130},"It supports alternative semantics, including stochastic transition dynamics, extending beyond traditional ordinary differential equation (ODE) representations.","https://schema.org",{"og:url":101,"og:type":141,"og:title":65,"og:site_name":112,"og:description":66},"article",{"robots":143,"canonical":101},"index,follow",{"doc_id":61,"site_id":75},1793096723]