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The talk highlights routed planar networks, including scale-invariance and comparisons between real-world subnetwork topologies and explicit scale-invariant models, while asking about efficient enumeration and canonical one-parameter families of distributions. A second topic explores fringe-tree combinatorics in a phylogenetic random tree model, including whether the model reproduces observed fringe behavior and matches data-driven clade proportions.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/obscure-results-and-open-problems-some-of-my-favorites-things-i-would-like-some-smart-young-person-to-work-on/295256/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/obscure-results-and-open-problems-some-of-my-favorites-things-i-would-like-some-smart-young-person-to-work-on/295256.png","ImageObject",300,407,{"name":92,"@type":93},"Felix Montgomery","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does “scale-invariance” mean in the routed planar network discussion?","Question",{"text":112,"@type":113},"Scale-invariance means the observed subnetwork statistics in a randomly positioned window do not depend on the window’s scale, such as whether the side length is 10 km or 100 km.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What open questions are raised for routed network topologies?",{"text":117,"@type":113},"The talk asks about efficient coding/enumeration of different topologies, the existence of a canonical one-parameter family of distributions, and how to model rare real-world topologies that do not appear in limited data.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the purpose of the phylogenetic fringe-tree model in Topic 2?",{"text":121,"@type":113},"The model for a random phylogenetic tree has no parameters and gives explicit numerical predictions designed to mimic the uneven way large splits occur; the open question is whether it also mimics the fringe behavior seen in data.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},295256,1789715681,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144,"preview_page_count":4},549768064778,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Obscure results and open problems: some of my  \nfavorites.  \nDavid Aldous  \n28 January 2025  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nSub-title: Things I would like some smart young person to work on.  Probability distributions on routed planar networks.  \n Combinatorics of fringe trees in a phylogenetic model.  \n Nearest neighbor of nearest neighbor, on the complete network.  \nAre you smarter than an AI?  \n The Nearest Unvisited Vertex walk.  \n Find a qualitatively realistic model for a subway network.  A layer model for random DAGs  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nTopic 1: Probability distributions on routed planar networks.  \nq  \nq  \nq  \nq  \nq  \nq  \nq  \n7 points in a window.  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nScale-invariance means: doing this within a randomly positioned window, the statistics of the subnetwork observed don't depend on the scale, i.e. don't depend on whether the side length is 10 km or 100 km.  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nAs undergraduate project we have looked at real-world subnetwork topologies (for k = 4 vertices, roughly at corners of a square) .  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nand listed all 71 topologies on 4 vertices – different conventions from usual planar graph theory. Could compare distributions over these topologies in real-world spatial networks and in models of spatial networks.  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nOpen problems about “routed networks”  \n Efficient naming (coding) of the different topologies?  Enumeration? [cf. Catherine Greenhill talk Thursday]  \n Is there a (mathematically) canonical one-parameter family of probability distributions on these topologies?  \n (In our small data study) empirical frequencies of topologies are very non-uniform. Can one make a model where some of the  \nreal-world-rare topologies don’t occur?  \n We have 2 explicit models of scale-invariant networks; calculate/simulate the distribution for these models.  \nReferences:  \n[https://escholarship.org/content/qt1g44k4dk/qt1g44k4dk.pdf](https://escholarship.org/content/qt1g44k4dk/qt1g44k4dk.pdf) (the 71 topologies) .  \n[https://arxiv.org/abs/1204.0817](https://arxiv.org/abs/1204.0817) and [https://arxiv.org/abs/2407.07887](https://arxiv.org/abs/2407.07887)[ ](https://arxiv.org/abs/2407.07887)(scale-invariant network models) .  \n[All links are live on these slides posted on my home page]  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nTopic 2: Combinatorics of fringe trees in a phylogenetic model.  \n26 DAVID J. ALDOUS AND SVANTE JANSON  \nBottoa:vilaAdlgorausm showiOnsorgesueneltcnfop77earprolems: some of my favorites.   \n Broad ongoing project with Svante Janson and Boris Pittel: average age ≈ 76.  \n We have a model for a random phylogenetic tree on n leaves (Svante talk tomorrow)  \n Has zero parameters, so gives explicit numerical predictions.  \n Model designed to mimic the (very uneven) way that large splits occur in real phylogenetic trees.  \n But does it happen to also mimic the fringe behavior?  \nDavid Aldous Obscure results and open problems: some of my favorites.  \nTHE CRITICAL BETA-SPLITTING RANDOM TREE III 27  \nFigure 7 . Proportions of leaves in clades of a given shape, for each shape with 2 − 6 leaves in the fringe tree. The top number is from our model, the bottom number [···] from our small data set.  \nDavid Aldous Obscure results and open problems: some of my favorites.","cbCaiqTZMzb9kYTv","https://ap.wps.com/l/cbCaiqTZMzb9kYTv","pdf",2118476,36,"English","# Obscure results and open problems: some of my favorites\n## Topic 1: Probability distributions on routed planar networks\n## Open problems about “routed networks”\n## Topic 2: Combinatorics of fringe trees in a phylogenetic model","[{\"question\":\"What does “scale-invariance” mean in the routed planar network discussion?\",\"answer\":\"Scale-invariance means the observed subnetwork statistics in a randomly positioned window do not depend on the window’s scale, such as whether the side length is 10 km or 100 km.\"},{\"question\":\"What open questions are raised for routed network topologies?\",\"answer\":\"The talk asks about efficient coding/enumeration of different topologies, the existence of a canonical one-parameter family of distributions, and how to model rare real-world topologies that do not appear in limited data.\"},{\"question\":\"What is the purpose of the phylogenetic fringe-tree model in Topic 2?\",\"answer\":\"The model for a random phylogenetic tree has no parameters and gives explicit numerical predictions designed to mimic the uneven way large splits occur; the open question is whether it also mimics the fringe behavior seen in data.\"}]","Obscure results and open problems - some of my favorites - Things I would like some smart young person to work on | PDF",91]