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 • Rank=M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. P=[1/n] n1 be a vector. • d is a n1 matrix. • di=1 if page i has no out links. Otherwise, di=0. • D=ppT, and E=p[1] 1n • M’=(1-)(M+D) +E. • Rank=M’Rank=(1-)(M+D) Rank +P.

5. Topic sensitive Page Rank • Let Tj be the set of URLs in the ODP category cj. • P=vj, where • Vj,i =1/|Tj| if page j points to page i. Otherwisr vj,i=0. • 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(cj|q)•r j,d , • P(cj|q) 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, Iu—number 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, Ou—number 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 (CNSR’04), 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.

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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 X=x1x2...xm, another sequence Z=z1z2...zk is a subsequence of X if there exists a strictly increasing sequence i1i2...ik of indices of X such that for all j=1,2,...k, we have xij=zj. • Example 1: If X=abcdefg, Z=abdg is a subsequence of X. X=abcdefg,Z=ab 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: X=abcdefg and Y=aaadgfd. Z=adf is a common subsequence of X and Y. X=abc defg Y=aaaadgfd Z=a 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 X=x1x2...xm, and Y=y1y2...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 X=x1x2...xm, and Y=y1y2...yn be two sequences, and • Z=z1z2...zk be any LCS of X and Y. • 1. If xm=yn, then zk=xm=yn and Z[1..k-1] is an LCS of X[1..m-1] and Y[1..n-1]. • 2. If xmyn, then zkxm implies that Z is an LCS of X[1..m-1] and Y. • 2. If xmyn, then zkyn implies that Z is an LCS of X and Y[1..n-1].

26. The recursive equation • Let c[i,j] be the length of an LCS of X[1...i] and X[1...j]. • c[i,j] can be computed as follows: 0 if i=0 or j=0, c[i,j]= c[i-1,j-1]+1 if i,j>0 and xi=yj, max{c[i,j-1],c[i-1,j]} if i,j>0 and xiyj. Computing the length of an LCS • There are nm c[i,j]’s. So we can compute them in a specific order.

27. The algorithm to compute an LCS • 1. for i=1 to m do • 2. c[i,0]=0; • 3. for j=0 to n do • 4. c[0,j]=0; • 5. for i=1 to m do • 6. for j=1 to n do • 7. { • 8. if x[I] ==y[j] then • 9. c[i,j]=c[i-1,j-1]=1; • 10 b[i,j]=1; • 11. elseif c[i-1,j]>=c[i,j-1] then • 12. c[i,j]=c[i-1,j] • 13. b[i,j]=2; • 14. else c[i,j]=c[i,j-1] • 15. b[i,j]=3; • 14 }

28. Example 3: X=BDCABA and Y=ABCBDAB.

29. Constructing an LCS (back-tracking) • We can find an LCS using b[i,j]’s. • We start with b[n,m] and track back to some cell b[0,i] or b[i,0]. • The algorithm to construct an LCS 1. i=m 2. j=n; 3. if i==0 or j==0 then exit; 4. if b[i,j]==1 then { i=i-1; j=j-1; print “xi”; } 5. if b[i,j]==2 i=i-1 6. if b[i,j]==3 j=j-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: X=abc and Y=abb. Both abbc and abcb are the shortest common supersequences for X and Y.

31. Recursive Equation: • Let c[i,j] be the length of an LCS of X[1...i] and X[1...j]. • c[i,j] can be computed as follows: j if i=0 i if j=0, c[i,j]= c[i-1,j-1]+1 if i,j>0 and xi=yj, min{c[i,j-1]+1,c[i-1,j]+1} if i,j>0 and xiyj.