Web Markov Skeleton Processes and Applications

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Web Markov Skeleton Processes and Applications

• Zhi-Ming Ma
• 10 June, 2013,
  St.Petersburg
• Email mazm_at_amt.ac.cn
• http//www.amt.ac.cn/member/mazhiming/index.html

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• Y. Liu, Z. M. Ma, C. Zhou
• Web Markov Skeleton Processes and Their
  Applications, Tohoku Math J. 63 (2011), 665-695
• Y. Liu, Z. M. Ma, C. Zhou
• Further Study on Web Markov Skeleton
  Processes, in Stochastic Analysis and
  Applications to Finance,World Scientific,2012
• C. Zhou Some Results on Mirror Semi-Markov
  Processes, manuscript

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Web Markov Skeleton Process
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Simple WMSP
Many simple WMSPs are
Non-Markov Processes
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Mirror Semi-Markov Process
Mirror Semi-Markov Process is not a Hou-Lius
Markov Skeleton Process, i.e. it does not satisfy
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Multivariate Point Process
associated with WMSP
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Let
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Consequently
where
Define
We can prove that
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where
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Time-homogeneous mirror semi-Markov processes
are all independent of n
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More property of of time homogeneity
Renewal Theory
Contribution probability
Staying times and first entry times
Limit distribution for
semi-Markov process
Limit distribution for mirror
semi-Markov processes
Reconstruction of Mirror Semi-Markov
Processes
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Why it is called a Web Markov Skeleton
Process?
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A simple Markov Skeleton Process
Page Rank, a ranking algorithm used by the
Google search engine.
1998, Sergey Brin and Larry Page ,
Stanford University

• From probabilistic point of view,
• PageRank is the stationary distribution
  of a Markov chain.

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Markov chain describing surfing behavior
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Markov chain describing surfing behavior
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• Web surfers usually have two basic ways to
  access web pages
• with probability a, they visit a web page by
  clicking a hyperlink.
• 2. with probability 1-a, they visit a web page
  by inputting its URL address.

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Weak points of PageRank

• Using only static web graph structure
• Reflecting only the will of web managers,
• but ignore the will of users e.g. the staying
  time of users on a web.
• Can not effectively against spam and junk pages.

BrowseRankSIGIR.ppt
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Data Mining
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Browsing Process

• Markov property

• Time-homogeneity

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Computation of the Stationary Distribution

• Stationary distribution
• is the mean of the staying time on page i.
• The more important a page is, the longer
  staying time on it is.
• is the mean of the first re-visit time at
  page i. The more important a page is, the smaller
  the re-visit time is, and the larger the visit
  frequency is.

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BrowseRank Letting Web Users Vote for Page
Importance

• Yuting Liu,
• Bin Gao, Tie-Yan Liu, Ying Zhang,
• Zhiming Ma, Shuyuan He, and Hang Li
• July 23, 2008, Singapore
• the 31st Annual International ACM SIGIR
  Conference on Research Development on
  Information Retrieval.

Best student paper !
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• Browse Rank the next PageRank
• says Microsoft

• jerbrowser.wmv

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• Browsing Processes will be a
• Basic Mathematical Tool in
• Internet Information Retrieval
• Beyond
• --General fromework of Browsing Processes?
• --How about inhomogenous process?
• --Marked point process
• --Mobile Web not really Markovian

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ExtBrowseRank and semi-Markov
processes
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MobileRank and Mirror Semi-Markov
Processes
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MobileRank and Mirror Semi-Markov
Processes
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Web Markov Skeleton Process

• 10 B. Gao, T. Liu, Z. M. Ma, T. Wang, and H. Li
• A general markov framework for page importance
  computation, In proceedings of CIKM '2009,
• 11 B. Gao, T. Liu, Y. Liu, T. Wang, Z. M. Ma
  and H. LI
• Page Importance Computation based on Markov
  Processes, Information Retrieval
• online first
• lthttp//www.springerlink.com/conten
  t/7mr7526x21671131

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Research on Random Complex Networks and
Information Retrieval In recent years
we have been involved in the research direction
of Random Complex Netowrks and Information
Retrieval. Below are some of the related outputs
by our group (in collaboration with Microsoft
Research Asia)
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More property of time homogeneity
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Theorem LMZ 2011a
for all n
Theorem LMZ 2011b General case
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The statistical properties of a time homogeneous
mirror semi-Markov process is completely
determined by
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Reconstruction of Mirror Semi-Markov
Processes
Theorem LMZ 2011b
We can construct
such that
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uniformly
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Limit distribution for
semi-Markov process
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Limit distribution for mirror
semi-Markov processes
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Staying times and first entry times
Staying time on the state j
Distribution
Expectation
Distribution
Expectation
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Contribution probability
from state i to state j
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Renewal Theory
Proposition
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Renewal Equation LMZ2011a
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Renewal functional
Below are the resuls on the renewal functional
LMZ2011a
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Thank you !
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Time Homogeneous WMSP
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More property of of time homogeneity
Theorem LMZ 2011b
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Reconstruction of WMSP
LMZ2011b
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Ranking Websites, a Probabilistic View
Internet Mathematics, Volume 3 (2007), Issue 3

• Ying Bao, Gang Feng, Tie-Yan Liu, Zhi-Ming Ma,
  and Ying Wang

AggregateRank Bring Order to
Web Sites 29th Annual International Conference
on Research Development on Information
Retrieval (SIGIR06). G.Feng, T.Y. Liu, Ying
Wang, Y.Bao, Z.M.Ma et al
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