1. Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
Ranking
Link-based Ranking
(2° generation)
Reading 21

2. The Web as a Directed Graph
Sec. 21.1
The Web as a Directed Graph
Page A
hyperlink
Page B
Anchor
Assumption 1: A hyperlink between pages denotes author perceived relevance (quality signal)
Assumption 2: The text in the anchor of the hyperlink describes the target page (textual context)

3. Sec
Indexing anchor text
When indexing a document D, include anchor text from links pointing to D.
Armonk, NY-based computer
giant IBM announced today
Big Blue today announced
record profits for the quarter
Joe’s computer hardware links
Sun HP
IBM
Can score anchor text with weight depending on the authority of the anchor page’s website

4. Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
Random Walks
Paolo Ferragina
Dipartimento di Informatica
Università di Pisa

5. Definitions Adjacency matrix A Transition matrix P 1 2 3 1 2 3
1/2 1 2 3
Any edge weigthing is possible: Any proposals?

6. What is a random walk
t=0 1
1/2

7. What is a random walk 1
1/2
t=1
t=0 1
1/2

8. What is a random walk 1
1/2
t=0 1
1/2
t=1
t=2 1
1/2

9. What is a random walk 1
1/2
t=1
t=0 1
1/2
t=2 1
1/2
t=3 1
1/2

10. Probability Distributions
xt(i) = probability that surfer is at node i at time t
xt+1(i) = ∑j (Probability of being at node j)*Pr(j->i) = ∑j xt(j)*P(j,i) = xt * P
t=2 1 2 3
1/2
Transition matrix P
1/2
xt+1 xt =

11. Probability Distributions
xt(i) = probability that surfer is at node i at time t
xt+1(i) = ∑j (Probability of being at node j)*Pr(j->i) = ∑j xt(j)*P(j,i) = xt P
xt+1 = xt P = xt-1* P * P = xt-2* P * P * P = …= x0 Pt+1
What happens when the surfer keeps walking for a long time? So called Stationary distribution

12. Stationary Distribution
The stationary distribution at a node is related to the amount/proportion of time a random walker spends visiting that node.
It is when the distribution does not change anymore: i.e. xT+1 = xT  xT P = 1* xT
(left eigenvector of eigenvalue 1)
For “well-behaved” graphs this does not depend on the start distribution: xt+1 = x0 P t+1

13. Interesting questions
Does a stationary distribution always exist? Is it unique?
Yes, if the graph is “well-behaved”, namely the markov chain is irreducible and aperiodic.
How fast will the random surfer approach this stationary distribution?
Mixing Time!

14. Well behaved graphs
Irreducible: There is a path from every node to every other node ( it is an SCC).
Irreducible
Not irreducible

15. Well behaved graphs
Aperiodic: The GCD of all cycle lengths is 1. The GCD is also called period.
Aperiodic
Periodicity is 3

16. About undirected graphs
A connected undirected graph is irreducible
A connected non-bipartite undirected graph has a stationary distribution proportional to the degree distribution!
Makes sense, since larger the degree of the node more likely a random walk is to come back to it.

17. Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
PageRanks and HITS
Paolo Ferragina
Dipartimento di Informatica
Università di Pisa

18. Query-independent ordering
Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
Query-independent ordering
First generation: using link counts as simple measures of popularity.
Undirected popularity:
Each page gets a score given by the number of in-links plus the number of out-links (es. 3+2=5).
Directed popularity:
Score of a page = number of its in-links (es. 3).
Easy to SPAM

19. Second generation: PageRank
Deploy the graph structure, and each link has its own importance !
PageRank is
independent of the query
many interpretations:
Linear algebra – eigenvectors, eigenvalues
Markov chains – steady state probability distribution
Social interpretation – a sort of voting scheme
Paolo Ferragina, Web Algorithmics

20. Basic Intuition… d 1-d Random jump to any node Random jump
to neighbors d
Paolo Ferragina, Web Algorithmics

21. Google’s Pagerank Random jump B(i) : set of pages linking to i.
Principal
eigenvector
B(i) : set of pages linking to i.
#out(j) : number of outgoing links from j.
e : vector of components 1/sqrt{N}.
Paolo Ferragina, Web Algorithmics

22. Pagerank: use in Search Engines
Preprocessing:
Given graph of links, build matrix P
Compute its principal eigenvector r
r[i] is the pagerank of page i
We are interested in the relative order
At query time:
Retrieve pages containing query terms
Rank them by their Pagerank
The final order is query-independent
Paolo Ferragina, Web Algorithmics

23. Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
How to compute it
Fast (approximate) computation:
Given the Web graph, build the matrix P
Compute r = e * Pt for t = 0, 1, …
r[i] is the pagerank of page i

24. (Personalized) Pagerank
Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
(Personalized) Pagerank
Bias the random jump:
Substitute e = [1 1 ….1] with a preference vector which jumps to preferred pages (topics)
If e = ei , then r[j] = relatedness between node j and node i

25. CoSim Rank to «compare» nodes
Personalized PageRank, p(0)(i) varies with node i
The adjacency matrix A has normalized rows
>>> Set d=1, so no teleport step
You can normalize it multiplying by (1-c)

26. HITS: Hypertext Induced Topic Search
Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
HITS: Hypertext Induced Topic Search

27. Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
Calculating HITS
It is query-dependent
Produces two scores per page:
Authority score: a good authority page for a topic is pointed to by many good hubs for that topic.
Hub score: A good hub page for a topic points to many authoritative pages for that topic.

28. Authority and Hub scores
Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
Authority and Hub scores 2 5 3 1 1 6 4 7
a(1) = h(2) + h(3) + h(4)
h(1) = a(5) + a(6) + a(7)

29. HITS: Link Analysis Computation
Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
HITS: Link Analysis Computation
Where
a: Vector of Authority’s scores
h: Vector of Hub’s scores.
A: Adjacency matrix in which ai,j = 1 if ij
Symmetric
matrices
Thus, h is an eigenvector of AAt
a is an eigenvector of AtA

30. Prof. Paolo Ferragina, Algoritmi per "Information Retrieval"
Weighting links
Weight more if the query occurs in the neighborhood of the link (e.g. anchor text).