1. 7CCSMWAL Algorithmic Issues in the WWW
Lecture HITS

2. HITS HITS: Important (non-Google) method of ranking web pages
The method has many attractions, but so far has not had such a successful implementation as PageRank
HITS classifies the pages in 2 ways, based on in-degree and out-degree
Uses/used a similar algorithm? Teoma (Ask Jeeves), Twitter
As of 2010 Ask.com referred to the Teoma algorithm as the ExpertRank algorithm

3. Some documentation https://en.wikipedia.org/wiki/HITS_algorithm
The next one is very ‘famous’
Basically what search engines do is mainly secret
Why would that be?

4. ASK

5. HITS (Hypertext Induced Topic Search)
Similar to PageRank, but uses both in-links and out-links to create two popularity scores for each page
HITS defines hubs and authorities
A hub is a page with many out-links
An authority is a page with many in-links
A page can be both a hub and an authority
Hub
Auth

6. Hubs and Authorities
“Good authorities are pointed to by good hubs and good hubs point to good authorities”
Measures of goodness for each page Pi
Authority score (or weight) xi
Hub score (or weight) yi
The definitions
where E is the set of all hyperlinks
Hub i: (i, j) refers to hyperlinks from Pi to Pj
Auth i: (j, i) refers to the hyperlink from Pj to Pi

7. Suppose the hub and authority scores of
Example P1
Suppose the hub and authority scores of
P1, P2, P3, P4 are P2 P5
Pages P1 P2 P3 P4
Authority score xi
3.5
1.2
4.2
1.0
Hub score yi
2.1
0.3
5.2
1.1 P3 P4
The authority score of P5 is = 7.6
Because Hubs P(1),P(2),P(3) point to P(5)
The hub score of P5 is = 4.5
Because P5 points to Authorities P(1), P(4);

8. Iterative Approach
Similar to the pagerank computation, the authority and hub scores are computed iteratively, followed by normalization (scores add to 1)
Let x(k)i and y(k)i be the authority and hub scores, respectively, at iteration k
For instance, in each iteration, we start by updating the authority scores and then the hub scores

9. Score Computation in Matrix Form
Adjacency matrix L
Lij = 1, if there exists a link from Pi to Pj
Lij = 0, otherwise.
Example 1 3 2 4

10. Score Computation in Matrix Form
Let x(k) be the column vector of the authority scores
Let y(k) be the column vector of the hub scores
Then we have y(k) = L x(k) because for each i
y(k)i = j = 1 to n Lij * x(k)j =  (i,j) in E x(k)j

11. Score Computation in Matrix Form
Transpose of a nm matrix M is a mn matrix MT where MTij = Mji
Example

12. Score Computation in Matrix Form
We have x(k) = LT y(k-1) because for each i
x(k)i = j = 1 to n LTij * y(k-1)j =  (j,i) in E y(k-1)j
Note: x authority score of page, y hub score of page
Note if edge (i,j) then LTij =1

13. HITS Score Computation in Matrix Form
Let x(k) be the column vector of the authority scores
Let y(k) be the column vector of the hub scores
Let L be the adjacency matrix of the network
HITS Algorithm
Initialize: y(0) = e (column vector of all ones)
Until convergence, do x(k) = LT y(k-1)
y(k) = L x(k)
Normalize* x(k) and y(k)
k = k+1
* To avoid the values getting too large. Need to add to 1

14. Example

15. Graph

16. results
hub authority Pagerank99 PageRank85 betweenness

17. R program require(igraph) #read in graph mygraph # see the data
mygraph<-read.table("compare-the-measures.csv",sep=",")
mygraph # see the data
#turn data into a graph
mygraph<-graph.data.frame(mygraph, directed=T)
plot(mygraph,vertex.color="white") #draw the graph
#HITS (Kleinberg scores)
H<-hub.score(mygraph)$vector
A<-authority.score(mygraph)$vector
#normalize HITS
#H=H/max(H)
#A=A/max(A)
H=H/sum(H)
A=A/sum(A)
#pagerank with 'no damping' and 'Google damping' P1<-page.rank(mygraph,damping= )$vector P2<-page.rank(mygraph,damping=0.85)$vector P1=P1/sum(P1) P2=P2/sum(P2) #betweenness centrality B<-betweenness(mygraph) B=B/sum(B) answer=data.frame(hub=H,authority=A,Pagerank99=P1, PageRank85=P2,betweenness=B) format(round(answer, 2), nsmall = 3)

18. HITS Score Computation in Matrix Form
Let x(k) be the column vector of the authority scores
Let y(k) be the column vector of the hub scores
Let L be the adjacency matrix of the network
HITS Algorithm
Initialize: y(0) = e (column vector of all ones)
Until convergence, do x(k) = LT y(k-1)
y(k) = L x(k)
Normalize* x(k) and y(k)
k = k+1
* To avoid the values getting too large. Need to add to 1

19. Score Computation in Matrix Form
In step 2 of the algorithm, the two equations
x(k) = LT y(k-1)
y(k) = L x(k)
can be simplified by substitution to
x(k) = LT L x(k-1)
y(k) = L LT y(k-1)
The equations become very similar to that of the pagerank computation: (k) = (k-1)H
In HITS
LTL is called the authority matrix
L LT is called the hub matrix

20. HITS Implementation Two main steps
A neighbourhood graph N related to the query terms is built (Query Dependent)
The authority and hub scores for each page in N are computed Two ranked lists: the most authoritative pages and most “hubby” pages
We focus on the first step as the second step has been explained

21. Neighbourhood Graph N Formation
Initialized with all pages containing references to the query terms
Make use of the content index (inverted file)
Expand the graph N by adding vertices (from the Web graph) that link either to or from vertices in N
Relevant pages without the query terms can be added. E.g., with query term “car”, pages containing “automobile” may be added. Some what arbitrary process.
N can become very large if a page containing the query terms has huge in-degree or out-degree In practice, a limit, say 100, is applied to the expansion from in-links or out-links of a page with the query terms

22. Score Computation with N
Once N is built, the adjacency matrix L corresponding to N is formed
The number of pages in N (and size of L) is much smaller than the total number of pages on the Web
It incurs much smaller cost in computing the authority and hub scores, when compared with PageRank method
In fact, we do not need to iterate both equations because when a stable authority vector x is obtained we can apply y = Lx to get the stable hub vector y

23. Normalization (Total = 1)
To limit the values of the authority and hub scores, and ensure the convergence
Normalization step
x(k)  x(k) / m(x(k))
where m(x(k)) can be the sum of individual values in x(k), i.e., x(k)1 + x(k) x(k)n
n is the number of pages in N (not the whole Web graph)
Similarly for y: y(k)  y(k) / m(y(k))

24. Example
Suppose P1 and P6 contain the query terms and the neighbourhood graph around P1 and P6 includes P2, P3, P5 and P10
The adjacency matrix L 3 10 1 6 2 5

25. Example (cont) The respective authority and hub matrices Authority
matrix 3 10 1 6 2 5
Hub
matrix

26. For the first 20 iterations
Example (cont)
x(0) = (1,1,1,1,1,1)T
x(1) = LT L x(0) = (1,0,4,2,4,0)
Normalize x(1), each element is divided by the sum of x(1), which is 11  (1/11,0/11,4/11,2/11,4/11,0/11)  (.0909, 0, .3636, .1818, .3636, 0)
x(2) = LTL x(1)
...
For the first 20 iterations

27. Example (cont)
By the HITS algorithm, the stable authority scores x and hub scores y are
x = ( )T
y = ( )T
Labels=(1, 2, 3, 5, 6, 10)
Ties may occur and be broken by any tie-breaking strategy, e.g., by page number
Authority ranking: P6, P3, P5, P1, P2, P10
Hub ranking: P1, P3, P6, P10, P2, P5
P6 is the most authoritative page and P1 is the best hub

28. Modification
Similar to PageRank method, we can also incorporate teleporting by modifying the authority and hub matrices
Authority matrix:  LT L + (1 – )(1/n) I
Hub matrix:  L LT + (1 – )(1/n) I
where  is a number between 0 & 1 (similar to  in PageRank)
For the example with  =0.95, the modification obtains the authority and hub scores
x = ( )T
y = ( )T
Note that the rankings remain the same in this example

29. Strength of HITS Provide two ranked lists
The most authoritative pages: for more in-depth information
The most “hubby” pages: for a portal to related information in a broad search
Size of the problem is much smaller than that of the PageRank method
Intuitively, hubs with a high score carry a lot of traffic, good place for advertising etc

30. Weakness of HITS Query-dependent
For each query, a neighbourhood graph must be built in query time
It is easy to make HITS query-independent by computing the authority and hub vectors, x and y, using the adjacency matrix of the entire web graph. Then, it leads to the size problem that PageRank method is facing

31. Weakness of HITS Susceptibility to spamming
Adding more out-links to a page can easily increase the hub score
Authority score and hub score are interdependent. An authority score will increase as a hub score increases
Bharat & Henzinger (1998) proposed to normalize links in two situations
If k links from Host 1 to the same page in Host 2, each link has a weight of 1/k in computing the authority score of Pi
I.e., each page Pj in Host 1 that points to Pi contributes only yj/k to xi
(1)
k links
...
... Pi
...
Host 1
Host 2

32. Weakness of HITS
If h links from the same page of Host 1 to the some pages in Host 2, each link has a weight of 1/h in computing the hub score of Pi
i.e., each page Pj in Host 2 that is pointed to from Pi contributes only xj/h to yi
(2)
h links
... Pi
...
...
Host 1
Host 2

33. Weakness of HITS Topic drift
Very authoritative yet off-topic page can be included in the neighbourhood graph, if the page is linked to a page containing the query terms
E.g., query for “Jaguar” (wild cat) may return homepages of car manufacturers or lists of car manufacturers
Neighbourhood Graph N which is built may not relate to properly to the query

34. Weakness of HITS
Bharat & Henzinger suggest a solution that weights the authority and hub scores of the pages in N by a measure of relevancy to the query
Let S(Q,Pi) be the “relevancy score”, (a number between 0 and 1), of page Pi to the query
Can be computed using traditional information retrieval techniques (see later)
For any hyperlink (Pi, Pj),
Pi contributes S(Q,Pi) * yi score to the authority score of Pj, where yi is the hub score of Pi
Pj contributes S(Q,Pj) * xj score to the hub score of Pj, where xj is the authority score of Pj