[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81526-en":3,"doc-seo-81526-105":29,"detail-sidebar-cat-0-en-105":94},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},81526,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Cascades on Networks with Functional Structure","Study of a directed multiplex Watts threshold model derives cascade size, single-seed cascade probability, and cascade conditions. The work then proposes constrained multiplex network models intended to capture networks with functional, complementary structure. Results show that the specific functional-structure choice reshapes the model’s phase transitions in multiple ways, revealing how complementarity assumptions modify complex contagion outcomes and cascade thresholds on realistic multiplex settings.","arXiv :2505 .24631v2 [nlin .AO] 10 Jul 2026  \nCascades on Networks with Functional Structure  \nChristian Kluge∗and Christian Kuehn∗†‡  \nJuly 13, 2026  \nAbstract  \nWe consider a version of the Watts threshold model on directed multiplex configuration model networks, and present a detailed analysis of the cascade size, single-seed cascade probability and cascade condition. We then introduce a smaller class of network models that we call constrained multiplex networks, which is designed to represent networks with so-called functional or complementary structure. We find that the particular choice of functional structure affects the phase transitions of the cascade model in a variety of ways.  \n1 Introduction  \nComplex contagion on networks has been an active area of research in the past twenty years, with applications ranging from the spreading of behavior in social networks [1] to systemic risk in financial networks [2] . The Watts Threshold Model (WTM) [3] -also called the Linear Threshold Model or Watts’ Cascade Model - has been a classic in this area. It is one of a variety of spreading processes used to examine the fragility or robustness of networks. For a recent overview of this broader context, we refer the reader to [4] . In the WTM, nodes are activated once a sufficiently large fraction of their neighbors are active. The precise value of “sufficiently large” is given by a threshold fraction ϕ ∈ [0 , 1], which may be random and different for each node. The WTM is often presented as modeling the spreading of beliefs or behaviors in a social network. In this interpretation, nodes correspond to individuals and the “active” state means the individual has adopted the belief or behavior. Initially, the model was analyzedon treelike configuration-model networks [5], but its analysis and that of similar models has also been carried out for clustered random networks [6, 7, 8, 9], which is natural as social networks feature high clustering [10] . Separate from the WTMs interpretation as a social process, it has also found use as a simple, analytically tractable model of default cascades among banks [11, 12, 13], with  \n∗ Technical University of Munich, School of Computation Information and Technology, Department of Mathematics, Garching b. M¨unchen, Germany  \n†Munich Data Science Institute, Garching b. M¨unchen, Germany  \n‡Complexity Science Hub Vienna, Vienna, Austria  \nthe goal of better understanding financial systemic risk. In this context, nodes are banks and “active” means the bank has defaulted.  \nIn recent years, there have been efforts to extend the modeling of systemic risk beyond the financial sector to the real economy [14, 15], sparked by the availability of detailed, firm-level supply chain data [16] . We believe that the WTM can-in appropriately modified form-play a role in this process. This is because stylized, analytically tractable models allow us to develop an intuitive understanding for the mechanisms at play, even if they lack the quantitative conclusions that more complex agent-based models can provide. This would require analyzing a variant of the WTM on networks capable of representing supply chains, which are inherently directed and multiplex, as one needs to distinguish suppliers and customers, as well as different goods.  \nThe structure of supply chain networks is quite interesting in its own right, as it has recently been proposed that they feature so-called “complementaritydriven” or “functional” structure [17] . This relatively new concept originates from work on protein interaction networks [18] . It is in some sense the opposite of the principle of homophily-the tendency of nodes to connect to others that are similar to themselves-, which is well-known from social networks [19] . By contrast, in networks driven by complementarity, nodes connect to others that are different and in some way compatible. In the case of supply chains, this means that we expect a link to connect a producer of some good to","cbCaidnR0PcS2fyA","https://ap.wps.com/l/cbCaidnR0PcS2fyA","pdf",565830,1,41,"English","en",105,"# Abstract\n# Introduction\n## Complex contagion and the Watts Threshold Model\n## Multiplex extensions and the or-rule\n## Motivation from supply-chain and complementarity\n## Paper contributions and scope","[{\"question\":\"What model variant is analyzed in the document?\",\"answer\":\"A version of the Watts threshold model is studied on directed multiplex configuration model networks, using an or-rule activation mechanism across layers.\"},{\"question\":\"Which quantities related to cascades are analyzed?\",\"answer\":\"The analysis provides detailed results for cascade size, single-seed cascade probability, and the cascade condition.\"},{\"question\":\"What are constrained multiplex networks and why are they introduced?\",\"answer\":\"Constrained multiplex networks are a smaller class of models designed to represent functional or complementary structure, capturing how multiplex structure affects contagion dynamics.\"},{\"question\":\"How does functional structure influence the cascade model?\",\"answer\":\"The particular choice of functional structure changes the phase transitions of the cascade model in several ways, indicating that complementarity assumptions alter cascade behavior.\"}]",1784174021,103,{"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":89,"head_meta":91,"extra_data":93,"updated_unix":27},"cascades-on-networks-with-functional-structure","",{"@graph":35,"@context":88},[36,53,67],{"@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/cascades-on-networks-with-functional-structure/81526/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80,84],{"name":71,"@type":72,"acceptedAnswer":73},"What model variant is analyzed in the document?","Question",{"text":74,"@type":75},"A version of the Watts threshold model is studied on directed multiplex configuration model networks, using an or-rule activation mechanism across layers.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Which quantities related to cascades are analyzed?",{"text":79,"@type":75},"The analysis provides detailed results for cascade size, single-seed cascade probability, and the cascade condition.",{"name":81,"@type":72,"acceptedAnswer":82},"What are constrained multiplex networks and why are they introduced?",{"text":83,"@type":75},"Constrained multiplex networks are a smaller class of models designed to represent functional or complementary structure, capturing how multiplex structure affects contagion dynamics.",{"name":85,"@type":72,"acceptedAnswer":86},"How does functional structure influence the cascade model?",{"text":87,"@type":75},"The particular choice of functional structure changes the phase transitions of the cascade model in several ways, indicating that complementarity assumptions alter cascade behavior.","https://schema.org",{"og:url":51,"og:type":90,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":92,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":95},[96,100,104,108,113,118,123,126,131,134,138],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Story & 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