[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83235-en":3,"doc-seo-83235-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83235,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Forced condensation and anti-condensation on heavy-tailed networks","Studies a driven selection mechanism on a fixed heavy-tailed network where, at each step, new mass is injected and its direction is recomputed using a power-normalization feedback rule. A primitive mixing matrix transports the combined mass, and the feedback exponent θ determines whether larger or smaller coordinates are favored. After scaling out deterministic growth, the long-run injection profile is described by a nonlinear Perron–Frobenius fixed point on the simplex. Hilbert’s projective metric characterizes stability, showing distinct roles of degree tilt, participation-ratio transitions, and few-node localization, supported by finite network numerics.","arXiv :2607 .07399v1 [physics .soc-ph] 8 Jul 2026  \nForced condensation and anti-condensation on heavy-tailed networks  \nAshwin Bhattathiripada , Vipin P. Veetila  \na Economics Area, Indian Institute of Management Kozhikode, Kozhikode, Kerala 673570,  \nIndia  \nAbstract  \nWe study a driven selection mechanism on a fixed heavy-tailed network. At each step fresh mass is injected, its direction is recomputed from the current mass profile by a power-normalization rule, and the combined mass is transported by a primitive mixing matrix. The exponent θ controls the feedback. Positive values give more weight to larger coordinates, while negative values favor smaller ones. When θ = 0, the injected mass is distributed uniformly. After deterministic growth of the total mass is scaled out, the long-run injection profile is characterized by a nonlinear Perron–Frobenius fixed point on the simplex. Hilbert’s projective metric gives a simple way to understand the stability of the system. The discounted network response brings positive profiles closer together, while the escort map scales their projective distance by |θ| . On heavy-tailed networks this fixed point separates three effects that are often conflated: response or degree tilt, anomalous inverse-participation-ratio scaling, and genuine few-node localization. Positive feedback selects high-response nodes and, when response follows degree, a hubdirected branch. Negative feedback selects low-response nodes and typically produces a broad peripheral cloud unless the lower tail of the response field is itself thin. Numerical experiments on finite power-law networks support these results. They show convergence, illustrate when the forcing rate becomes unimportant because mixing is sufficiently fast, and confirm both the sign law and the crossover in the participation ratio. This mechanism is different from both conserved-mass condensation and graph growth. Instead, feedback selects a non-equilibrium profile on a fixed, heterogeneous network.  \nKeywords: non-equilibrium statistical mechanics, complex networks, condensation, heavy-tailed networks, nonlinear Perron–Frobenius theory, Hilbert metric  \nEmail addresses: [ashwinbtt@gmail.com](ashwinbtt@gmail.com) ; [vipin@iimk.ac.in](vipin@iimk.ac.in).  \n1. Introduction  \nCondensation is one of the simplest ways in which a many-component system can become very unequal. In a zero-range or mass-transport process, a large share of the conserved mass can collect at a single site. In a growing network, a node with a fitness advantage can acquire a finite fraction of all links. Similar concentration can arise in multiplicative economic models, where wealth may become concentrated among a small number of agents [1] . These examples differ in their details, but they share a common warning: ina heterogeneous system, the interesting object is often not the total amount of mass, but the normalized profile selected by the dynamics.  \nThis paper studies a deliberately spare version of that question. The network is fixed. It is not grown by preferential attachment during the dynamics. The total mass is not conserved either. Fresh mass is injected at every step, the injection direction is chosen from the current mass profile, and the network then mixes the result. The feedback rule is a power law. If θ > 0, nodes that already carry more mass receive a larger share of the next injection. If θ \u003C 0, the rule is reversed and the next injection favors smaller coordinates. The case θ = 0 is the neutral baseline.  \nThe model was chosen to isolate a specific mechanism. Condensation in complex networks is usually discussed through conserved stochastic transport, zero-range processes, balls-in-boxes models, eigenvector localization, or graph growth with preferential attachment and fitness [2 , 3 , 4 , 5 , 6 , 7 , 8] . Those mechanisms are important, but they mix several ingredients. Here the substrate is fixed and the total mass grows in a trivial deterministic way. All non","cbCaifL7nUcutKoX","https://ap.wps.com/l/cbCaifL7nUcutKoX","pdf",1070608,2,1,43,"English","en",105,"# Introduction\n## Model setup and feedback rule\n## Fixed point and projective-metric stability\n## Heavy-tailed effects: degree tilt, participation ratio, localization\n## Contrast with standard condensation mechanisms","[{\"question\":\"How does the feedback exponent θ control condensation or anti-condensation on the network?\",\"answer\":\"For θ\\u003e0, injection favors nodes with larger coordinates, leading to positive feedback toward high-response regions. For θ\\u003c0, injection favors smaller coordinates, producing anti-condensation and typically broader peripheral mass placement.\"},{\"question\":\"What determines the long-run injection profile after deterministic mass growth is removed?\",\"answer\":\"After scaling out deterministic growth, the system’s long-run injection profile is characterized by a nonlinear Perron–Frobenius fixed point on the simplex.\"},{\"question\":\"Which stability tool is used to analyze convergence and what is its main outcome?\",\"answer\":\"Hilbert’s projective metric is used; the network resolvent contracts projective distances while the escort (power-normalization) map rescales them by |θ|, yielding a contractive fixed-point regime for |θ|\\u003c1.\"}]",1784186121,108,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"forced-condensation-and-anti-condensation-on-heavy-tailed-networks","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/forced-condensation-and-anti-condensation-on-heavy-tailed-networks/83235/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the feedback exponent θ control condensation or anti-condensation on the network?","Question",{"text":75,"@type":76},"For θ>0, injection favors nodes with larger coordinates, leading to positive feedback toward high-response regions. For θ\u003C0, injection favors smaller coordinates, producing anti-condensation and typically broader peripheral mass placement.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What determines the long-run injection profile after deterministic mass growth is removed?",{"text":80,"@type":76},"After scaling out deterministic growth, the system’s long-run injection profile is characterized by a nonlinear Perron–Frobenius fixed point on the simplex.",{"name":82,"@type":73,"acceptedAnswer":83},"Which stability tool is used to analyze convergence and what is its main outcome?",{"text":84,"@type":76},"Hilbert’s projective metric is used; the network resolvent contracts projective distances while the escort (power-normalization) map rescales them by |θ|, yielding a contractive fixed-point regime for |θ|\u003C1.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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"]