Editing Markov chain (section)

===Internet applications===
[[File:PageRank_with_Markov_Chain.png|right|thumb|A state diagram that represents the PageRank algorithm with a transitional probability of M, or <math>\frac{\alpha}{k_i} + \frac{1-\alpha}{N}</math>.]]
The [[PageRank]] of a webpage as used by [[Google]] is defined by a Markov chain.<ref>{{US patent|6285999}}</ref><ref name="BrijP.2016">{{cite book|url=https://books.google.com/books?id=Ctk6DAAAQBAJ&pg=PA448|title=Handbook of Research on Modern Cryptographic Solutions for Computer and Cyber Security|author1=Gupta, Brij|author2=Agrawal, Dharma P.|author3=Yamaguchi, Shingo|date=16 May 2016|publisher=IGI Global|isbn=978-1-5225-0106-0|pages=448–}}</ref><ref name="LangvilleMeyer2006">{{cite journal|last1=Langville|first1=Amy N.|last2=Meyer|first2=Carl D.|year=2006|title=A Reordering for the PageRank Problem|url=http://meyer.math.ncsu.edu/Meyer/PS_Files/ReorderingPageRank.pdf |journal=SIAM Journal on Scientific Computing|volume=27|issue=6|pages=2112–2113|citeseerx=10.1.1.58.8652|doi=10.1137/040607551 |bibcode=2006SJSC...27.2112L }}</ref> It is the probability to be at page <math>i</math> in the stationary distribution on the following Markov chain on all (known) webpages. If <math>N</math> is the number of known webpages, and a page <math>i</math> has <math>k_i</math> links to it then it has transition probability <math>\frac{\alpha}{k_i} + \frac{1-\alpha}{N}</math> for all pages that are linked to and <math>\frac{1-\alpha}{N}</math> for all pages that are not linked to. The parameter <math>\alpha</math> is taken to be about 0.15.<ref name="pagerank">{{cite tech report |author1= Page, Lawrence |author2=Brin, Sergey |author3=Motwani, Rajeev |author4=Winograd, Terry |title= The PageRank Citation Ranking: Bringing Order to the Web |year= 1999 |citeseerx=10.1.1.31.1768}}</ref>

Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.{{cn|date=January 2025}}