Topic-Sensitive PageRank

1
Topic-Sensitive PageRank

• Reference Taher H. Haveliwala, Topic-Sensitive
  PageRank A context-sensitive ranking algorithm
  for websearch, IEEE Trans. On Knowledge and Data
  Engineering, vol. 15, No. 4, PP. 784-796.
• The original PageRank
• purely based on the hyperlinks of web pages.
• Contents are not considered.
• A vector of PageRank is used for all web pages.
• Topic-Sensitive PageRank
• For each topic, a vector of PageRank is created.
• Each page has several PageRank values. One for
  each topic.

2
Creating a Page Rank vector for each topic

• How to select topics?
• Using a small set of topics is important for low
  computation cost and quick response time.
• Open Directory http//www.dmoz.org
• 16 top level topics

3
Original PageRank

• RankM?Rank
• Rank is a vector, one element for each web page.
• M is a n?n matric
• If there is a link from page j to page I, then
• Mi,j 1/Nj, where Nj is the number of
  out-links of page j.

4
Another Version

• Let n be the total number of web pages.
• P1/n n?1 be a vector.
• d is a n?1 matrix.
• di1 if page i has no out links. Otherwise,
  di0.
• Dp?pT, and Ep?1 1?n
• M(1-?)(MD) ?E.
• RankM?Rank(1-?)(MD) ?Rank ?P.

5
Topic sensitive Page Rank

• Let Tj be the set of URLs in the ODP category cj.
  
• Pvj, where
• Vj,i 1/Tj if page j points to page i.
  Otherwisr vj,i0.
• The pageRank vector for topic cj is PR(?, vj).
• Compute the pageRAnk for all pages related to
  topic cj as if for the original PageRank by
  considering Tj.

6
The Retrieval Score

• Let r j, d be the PageRank of document d given by
  the rank PR(? , vj).
• Sq,d ? j P(cjq)r j,d ,
• P(cjq) is the score that topic is related to q.

7
Similarity Measures for Induced Rankings Another
Version

• Let ?1 and ?2 be two rankings of documents.
• OSim(?1,?2) indicates the degree of overlap
  between the top k URLs of the two rankings.
• OSim (?1,?2) A?B/k.
• KSimn (?1,?2)
• (u, v) ?1 and ?2 agree on order of (u,
  v)/(U)(U-1)
• Let ? be the true ranking given by user. To
  compare ?1 and ?2, we can use OSim (?1,?2) or
  KSimn (?1,?2) .

8
Weighted PageRank

• Assign larger rank values to more important
  pages.
• Each outlink page gets its value proportional to
  it popularity.
• W in (v, u) is the weight of link(v, u)
  calculated based on the number of inlinks of page
  u and the number of inlinks of all reference
  pages of page v.
• w in (v, u) Iu/?p?R(v) Ip,
• Iunumber of inlinks of page u.
• R(v)-the set of all pages that v points to.

9
Weighted PageRank

• w out (v, u) Ou/?p?R(v) Op,
• Ounumber of outlinks of page u.
• Let B(u) be the set of pages that points to v.
• PR(u)(1-d) d ? v?B(u) PR(v) w in (v, u) w out
  (v, u)
• Reference Wenpu Xing and Ali Ghorbani, Weighted
  PageRank Algorithm, Proceedings of the 2nd Annual
  Conference on Communication Networks and Services
  Research (CNSR04), 2004.

10
Choices of Search Engines

• Many search engines exist to compete for users
• The results are not necessarily the same
• Different users prefer different search engines
• Search results may, in the future, be biased
  towards paid advertisements.

11
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20
MetaSearch Engine

• Metasearch Engines are designed to increase the
  coverage of web by forwarding users queries to
  multiple search engines
• Users requests are sent to multiple search
  engines such as AlltheWeb, Google, MSN.
• Then the results from the individual search
  engine are combined into a single result set to
  present to users.

21
Longest common subsequence

• Definition 1 Given a sequence Xx1x2...xm,
  another sequence Zz1z2...zk is a subsequence of
  X if there exists a strictly increasing sequence
  i1i2...ik of indices of X such that for all
  j1,2,...k, we have xijzj.
• Example 1 If Xabcdefg, Zabdg is a subsequence
  of X. Xabcdefg,Zab d g

22

• Definition 2 Given two sequences X and Y, a
  sequence Z is a common subsequence of X and Y if
  Z is a subsequence of both X and Y.
• Example 2 Xabcdefg and Yaaadgfd. Zadf is a
  common subsequence of X and Y.
• Xabc defg
• Yaaaadgfd
• Za d f

23

• Definition 3 A longest common subsequence of X
  and Y is a common subsequence of X and Y with the
  longest length. (The length of a sequence is the
  number of letters in the seuqence.)
• Longest common subsequence may not be unique.
• Example abcd
• acbd
• Both acd and abd are LCS.

24
Longest common subsequence problem

• Input Two sequences Xx1x2...xm, and
• Yy1y2...yn.
• Output a longest common subsequence of X and Y.
• A brute-force approach
• Suppose that m?n. Try all subsequence of X
  (There are 2m subsequence of X), test if such a
  subsequence is also a subsequence of Y, and
  select the one with the longest length.

25
Charactering a longest common subsequence

• Theorem (Optimal substructure of an LCS)
• Let Xx1x2...xm, and Yy1y2...yn be two
  sequences, and
• Zz1z2...zk be any LCS of X and Y.
• 1. If xmyn, then zkxmyn and Z1..k-1 is an
  LCS of X1..m-1 and Y1..n-1.
• 2. If xm ?yn, then zk?xm implies that Z is an LCS
  of X1..m-1 and Y.
• 2. If xm ?yn, then zk?yn implies that Z is an LCS
  of X and Y1..n-1.

26
The recursive equation

• Let ci,j be the length of an LCS of X1...i
  and X1...j.
• ci,j can be computed as follows
• 0
  if i0 or j0,
• ci,j ci-1,j-11 if
  i,jgt0 and xiyj,
• maxci,j-1,ci-1,j if i,jgt0
  and xi?yj.
• Computing the length of an LCS
• There are n?m ci,js. So we can compute them in
  a specific order.

27
The algorithm to compute an LCS

• 1. for i1 to m do
• 2. ci,00
• 3. for j0 to n do
• 4. c0,j0
• 5. for i1 to m do
• 6. for j1 to n do
• 7.
• 8. if xI yj then
• 9. ci,jci-1,j-11
• 10 bi,j1
• 11. else if ci-1,jgtci,j-1 then
• 12. ci,jci-1,j
• 13. bi,j2
• 14. else ci,jci,j-1
• 15. bi,j3
• 14

28

• Example 3 XBDCABA and YABCBDAB.

29
Constructing an LCS (back-tracking)

• We can find an LCS using bi,js.
• We start with bn,m and track back to some cell
  b0,i or bi,0.
• The algorithm to construct an LCS
• 1. im
• 2. jn
• 3. if i0 or j0 then exit
• 4. if bi,j1 then
• ii-1
• jj-1
• print xi
• 5. if bi,j2 ii-1
• 6. if bi,j3 jj-1
• 7. Goto Step 3.
• The time complexity O(nm).

30
Shortest common supersequence

• Definition Let X and Y be two sequences. A
  sequence Z is a supersequence of X and Y if both
  X and Y are subsequence of Z.
• Shortest common supersequence problem
• Input Two sequences X and Y.
• Output a shortest common supersequence of X and
  Y.
• Example Xabc and Yabb. Both abbc and abcb are
  the shortest common supersequences for X and Y.

31

• Recursive Equation
• Let ci,j be the length of an LCS of X1...i
  and X1...j.
• ci,j can be computed as follows
• j
  if i0
• i
  if j0,
• ci,j ci-1,j-11 if
  i,jgt0 and xiyj,
• minci,j-11,ci-1,j1 if
  i,jgt0 and xi?yj.

32
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