[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657631-105":59,"doc-detail-657631-en":130},{"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":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","google-pagerank-markov-chains-introduction","Google Pagerank - Markov Chains - Introduction","","Google PageRank explains how to rank webpages by modeling a random web surfer moving across links over the long run. The method addresses key issues such as pages with no outbound links and the surfer’s occasional random jumps to any page, which is incorporated through a damping factor. The approach is formalized using a transition matrix and the steady-state probability (probability eigenvector) that provides the ranking for each page. The document then generalizes to an n-page network and discusses scalability and related computational exercises.",{"@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/google-pagerank-markov-chains-introduction/657631/",{"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/google-pagerank-markov-chains-introduction/657631.png","ImageObject",300,407,{"name":92,"@type":93},"cksankcs","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does PageRank solve for Google?","Question",{"text":112,"@type":113},"PageRank ranks webpages using link-based criteria so that pages linked by more important pages receive higher scores.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the PageRank random surfer differ from always following links?",{"text":117,"@type":113},"At each step the surfer follows outgoing links with high probability, but with a smaller probability it jumps to a random page, preventing the process from being just link-following.",{"name":119,"@type":110,"acceptedAnswer":120},"How is PageRank computed in the general n-page case?",{"text":121,"@type":113},"A transition matrix is built using outbound-link probabilities when links exist and uniform probabilities when a page has no outbound links; page ranks come from the steady-state probability eigenvector of this matrix.","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},657631,1793020286,{"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":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},3985747882755,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Google Pagerank  \nJustin Wyss-Gallifent July 21, 2021  \n7.1 Introduction .............................. 2  \n7.2 Relationship to Markov Chains ................... 2  \n7.3 General Pagerank Matrix ...................... 5  \n7.4 Scalability ............................... 5  \n7.5 Matlab ................................. 6  \n7.6 Exercises ............................... 7  \n7.1 Introduction  \nOne of the principal requirements that Google has to deal with is ranking webpages. A web page should be ranked higher by some sort of criteria. So how can we go about doing this? Given a web page, the basic idea might be to look at how many pages are linking to this page; The more the better. However then we have to appreciate how important those pages are, since being linked to by a useless page is not as important as being linked to by an important one, and so the problem goes back and back.  \nNote the term \\Pagerank\" comes from the name of Larry Page, one of the founders of Google, not because it's related to web pages.  \n7.2 Relationship to Markov Chains  \nLet's examine a very basic internet, see how it connects to Markov Chains, and how the Google Pagerank method works.  \nExample 7.1 . Suppose the internet consists of only four pages, P1 , P2 , P3 , P4 , linked as follows, where, for example, an arrow from P1 to P3 indicates a link from P1 to P3 .  \nNotice that this looks very much like the diagram of a Markov Chain. If that's the case why don't we just assign probability values to the directions like we did with population diagrams? For example if a web page has two outbound links we could assign each a probability value of 0:5 and so on? If we did this then the steady-state vector would correspond to where a web surfer would end up in the long term, and this seems like a reasonable way to assign value to webpages.  \nOne obvious problem is that a web page may have no outbound links. If that's the case we wouldn't know what to assign for the probabilities.  \nAnother obvious problem is that this method doesn't really act like a web surfer. Web surfers don't just follow links, they also jump to other pages independently of where they were.  \nThe Google Pagerank (GP) algorithm takes the following approach:  \n(1) We assume that a Random Websurfer (RW) starts at some page.  \n(2) If the page has outbound links then there is an 85% chance that WR chooses one of those links and those links are equally likely. There is a 15% chance that RW chooses a page at random from all possible pages.  \n(3) If the page has no outbound links then there is a 100% chance that RW chooses a page at random from all possible pages.  \n(4) RW will continue to do this forever.  \nAfter reading this it becomes fairly clear that this is exactly a Markov Chain. The picture above is not an exact representation of the movement of RW because we need to take into account the 15% chance that RW randomly chooses a page. We could connect every page to every other page in the diagram but that would be a bit silly so instead we just recognize that the connections are there.  \nExample 7.1 Revisited.  \n• There is a weight of 0:15=4 connecting Pi to Pj for all i; j.  \n• There is an additional weight of 0:85(1=n) connecting Pi to Pj provided that Pi links to Pj and that Pi links to n pages total.  \nThis is particularly easy to see in terms of two separate matrices:  \nT =  \n0:15 4  \n2  \n6 6  \n4  \n1 1  \n1 1  \n2  \nT = 6  \n4  \n0:0375  \n0:4625  \n0:4625 0:0375  \n1 1  \n1 1  \n1 1  \n1 1  \n1 1  \n1 1  \n3 2  \n7 + 0:85 6 5 4  \n0  \n1=2  \n1=2  \n0  \n0 0  \n1 0  \n0 0  \n0 1  \n0 0  \n1 0  \n3  \n7 7  \n5  \n0:0375  \n0:0375 0:8875 0:0375  \n0:0375  \n0:0375  \n0:0375 0:8875  \n0:0375  \n0:0375 0:8875 0:0375  \n3  \n7 7  \n5  \nCheck your sanity-this should be a transition matrix. Not only that but it's a regular transition matrix because the 􀀌rst part of the sum forces all entries of T 1 to be nonzero. Consequently it obeys our theorem, having an eigenvalue of 􀀕 = 1 and a corresponding probability eigenvector.  \nThe cor","cbCaitgsDmle6IJf","https://ap.wps.com/l/cbCaitgsDmle6IJf","pdf",177023,"English","# 7.1 Introduction\n## Ranking webpages and link importance\n# 7.2 Relationship to Markov Chains\n## Random web surfer model and damping factor\n## Example 7.1 and ranking interpretation\n## Handling pages with no outbound links (Example 7.2)\n# 7.3 General Pagerank Matrix\n## Transition matrix construction for n pages\n# 7.4 Scalability\n## Eigenvalue properties and computational considerations\n# 7.5 Matlab\n## Implementation guidance\n# 7.6 Exercises\n## Practice problems","[{\"question\":\"What problem does PageRank solve for Google?\",\"answer\":\"PageRank ranks webpages using link-based criteria so that pages linked by more important pages receive higher scores.\"},{\"question\":\"How does the PageRank random surfer differ from always following links?\",\"answer\":\"At each step the surfer follows outgoing links with high probability, but with a smaller probability it jumps to a random page, preventing the process from being just link-following.\"},{\"question\":\"How is PageRank computed in the general n-page case?\",\"answer\":\"A transition matrix is built using outbound-link probabilities when links exist and uniform probabilities when a page has no outbound links; page ranks come from the steady-state probability eigenvector of this matrix.\"}]","Google Pagerank - Markov Chains - Introduction | PDF",1792985681,25]