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 t

opic.

Creating a Page Rank vector for each topic

• How to select topics?– Using a small set of topics is important for low co

mputation cost and quick response time.– Open Directory: http://www.dmoz.org

• 16 top level topics

Original PageRank

• Rank=MRank– Rank is a vector, one element for each web p

age.– M is a nn 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.

Another Version

• Let n be the total number of web pages.

P=[1/n] n1 be a vector.

• d is a n1 matrix. – di=1 if page i has no out links. Otherwise, di=0.

• D=ppT, and E=p[1] 1n

• M’=(1-)(M+D) +E.• Rank=M’Rank=(1-)(M+D) Rank +P.

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.

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.

Similarity Measures for Induced Rankings Another Version

• Let 1 and 2 be two rankings of documents.• OSim(1,2) indicates the degree of overlap b

etween the top k URLs of the two rankings. • OSim (1,2) =|AB|/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) .

Weighted PageRank

• Assign larger rank values to more important pages.• Each outlink page gets its value proportional to it popul

arity.

• 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/pR(v) Ip,

Iu—number of inlinks of page u.

R(v)-the set of all pages that v points to.

Weighted PageRank

• w out (v, u) =Ou/pR(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 vB(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.

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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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.

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 x ij=zj.

• Example 1: If X=abcdefg, Z=abdg is a subsequence of X. X=abcdefg,Z=ab d g

• 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

• 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.

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 mn. 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.

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 zkxm implies that Z is an LCS

of X[1..m-1] and Y.• 2. If xm yn, then zkyn implies that Z is an LCS

of X and Y[1..n-1].

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 x i=yj, max{c[i,j-1],c[i-1,j]} if i,j>0 and x iyj.

Computing the length of an LCS• There are nm c[i,j]’s. So we can compute them

in a specific order.

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. else if 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 }

Example 3: X=BDCABA and Y=ABCBDAB.

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=m2. 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-16. if b[i,j]==3 j=j-17. Goto Step 3.

• The time complexity: O(nm).

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.

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 xiyj.