1. PageRank Algorithm
El Habib NFAOUI (elhabib.nfaoui@usmba.ac.ma)
LIIAN Laboratory, Faculty of Sciences Dhar Al Mahraz, Fes
Sidi Mohamed Ben Abdellah University, Fes
2018-2019
3. 1. Introduction
Hyperlinks are a special feature of the Web, which link Web pages to form a huge
network. They have been exploited for many purposes, especially for Web search.
Google’s early success was largely attributed to its hyperlink-based ranking algorithm
called PageRank, which was originated from social network analysis [1].
Two most well known Web hyperlink analysis algorithms: PageRank and HITS
(Hypertext Induced Topic Search).
4. 2. PageRank
PageRank algorithm was first introduced by L. Page, S.Brin (1998), and later became
the skeleton for Google’s Search Engine. Basically, PageRank algorithm calculates
the importance ranking of every web page using the hyperlink structure of the web.
Importance ranking is represented by a global score assigned to every web page.
PageRank is a static ranking of Web pages in the sense that a PageRank value is
computed for each page off-line and it does not depend on search queries. The
PageRank of a node will depend on the link structure of the web graph.
Given a query, a web search engine computes a composite score for each web page
that combines hundreds of features such as cosine similarity and term proximity,
together with the PageRank score. This composite score is used to provide a ranked
list of results for the query.
5. 2.1 PageRank scoring
Consider a random surfer who randomly surfs the web pages:
Start at a random page
At each time step, the surfer go out of the
current page along one of the links
on that page, equiprobably
As the surfer proceeds in this random walk (surf) from node to node, he
visits some nodes more often than others; intuitively, these are nodes with
many links coming in from other frequently visited nodes. The idea behind
PageRank is that pages visited more often in this walk are more
important.
1/3
1/3
1/3
Sec. 21.2
6. 2.2 Teleporting (or teleportation)
What if the current location of the surfer has no out-links?
To address this an additional operation for our random surfer was introduced: the
teleport operation.
In the teleport operation the surfer jumps from a node to any other node in the web
graph. This could happen because he types an address into the URL bar of his browser.
The destination of a teleport operation is modeled as being chosen uniformly at random
from all web pages. In other words, if N is the total number of nodes in the web graph,
the teleport operation takes the surfer to each node with probability 1/N.
How do we model the random surfer process?
7. 3. Markov chains
A Markov chain consists of n states, plus an nn transition probability matrix P.
At each step, we are in one of the states.
For 1 i,j n, the matrix entry Pij tells us the probability of j being the next state,
given we are currently in state i.
i j
Pij
Pii>0
is OK.
Sec. 21.2.1
Clearly, for all i, .1
1
ij
n
j
P
8. 4. Random surfer model
We can view a random surfer on the web graph as a Markov chain (Markov
chains are abstractions of random walks). In this Markov chain model, each
Web page or node in the Web graph is regarded as a state. A hyperlink is a
transition, which leads from one state to another state with a transition
probability. Transition probability represents the probability of moving
from one web page to another. The teleport operation contributes to these
transition probabilities. Thus, this framework models Web surfing as a
stochastic process. It models a Web surfer randomly surfing the Web as a
state transition in the Markov chain.
9. 4. Random surfer model
The adjacency matrix A of the web graph is defined as follows: if there is a hyperlink from
page i to page j, then Aij = 1, otherwise Aij = 0. We can readily derive the transition
probability matrix P for our Markov chain from the N × N matrix A:
with probability, random surfer clicks on one of the hyperlinks. This is known as
transportation. Each hyperlink has an equal probability of being clicked. is a
damping factor usually set to 0.85.
with the complementary probability 1- (=0.15), random surfer jumps to some other
web page (e.g., enters the url into address bar of the browser). This is known as
teleportation. Each web page has an equal probability of being jumped to.
N is the total number of nodes in the web graph.
(Equation 1)
If
Otherwise (if Not)
10. 5. PageRank algorithm
The PageRank of page j is the sum of the PageRank scores of pages i linking to j,
weighted by the probability of going from i to j. In words, the PageRank thesis reads
as follows:
A Web page is important if it is pointed to by other important pages.
Let R be a N-dimensional row vector of PageRank values of all pages, i.e.,
The PageRank vector is then recursively defined as the solution of equation:
(Recursive calculation of the PageRanks. We consider the transportation and the
teleportation operations defined previously)
11. 5. PageRank algorithm
Input:
- The adjacency matrix A of the web graph;
- : damping factor ; // usually set to 0.85
- ε : Pre-specified threshold (desired precision); //used in Stopping condition
Initialization
- Using equation 1, calculate the probability matrix P;
- PageRank vector ;
- ;
Output: PageRank vector
Repeat
Until ε
Simple iterative algorithm for calculating the PageRanks vector R.
The iteration ends when the PageRank
values do not change much or converge.
In this algorithm, the iteration ends after
the L1-norm of the residual vector is less
than the pre-specified threshold. Note
that the L1-norm for a vector is simply
the sum of all the components.
12. 6. Example
Consider the social network given below. PageRank algorithm can find the
importance ranking of the nodes in the network.
: Is the damping factor
13. 6. Example
Transportation:
T matrix gives the pairwise transportation probabilities. Tij gives the probability
that random surfer transports from page i to page j (nodes are numbered in
alphabetic order, i.e., A=1, B=2, ...).
14. 6. Example
Teleportation:
D matrix gives the pairwise teleportation probabilities. Dij gives the probability that
random surfer teleports from page i to page j (nodes are numbered in alphabetic order,
i.e., A=1, B=2, ...). Note that, teleportation probabilities depends only on dangling and
non-dangling property of a node, i.e., node A is dangling, all other nodes are non-
dangling.
Dangling nodes : Nodes with no outgoing edges (links).
15. 6. Example
Random surfing probabilities:
Final probabilities for the random surfer is given by P = T +D.
16. 6. Example
PageRank computation:
As mentioned before, each web page has an initial score, which is 1/11 = 0.0909 (step
0). Using the basic version of PageRank algorithm given previously, we can compute the
PageRank scores of each page. Bellow is the PageRank vectors corresponding to the
given social network:
Converges occurs when L1-norm of PageRank scores is less than 10-6 and it takes 82 steps
to converge. S shows the scores for first 3 steps and last 2 steps. Scores are normalized to
sum to 1. In order to get the percentage of importance, scores can be multiplied by 100. Last
row of S gives the final percentages. (source: Shatlyk Ashyralyyev, CS533 course)
17. 7. Strengths of PageRank
The main advantage of PageRank is its ability to fight spam. A page is important if
the pages pointing to it are important. Since it is not easy for Web page owner to add
in-links into his/her page from other important pages, it is thus not easy to influence
PageRank. Nevertheless, there are reported ways to influence PageRank.
Recognizing and fighting spam is an important issue in Web search.
Another major advantage of PageRank is that it is a global measure and is query
independent. That is, the PageRank values of all the pages on the Web are
computed and saved off-line rather than at the query time. At the query time, only a
lookup is needed to find the value to be integrated with other strategies to rank the
pages. It is thus very efficient at the query time. Both these two advantages
contributed greatly to Google’s success.
We note again that the link-based ranking is not the only strategy used in a search
engine. Many other information retrieval methods, heuristics, and empirical
parameters are also employed. However, their details are not published. Also
PageRank is not the only link-based static and global ranking algorithm. All major
search engines, such as Bing and Yahoo!, have their own algorithms.
18. References
[1] Wasserman, S. and K. Faust. Social Network Analysis. 1994: Cambridge University
Press.
[2] Christopher D. Manning, Prabhakar Raghavan and Hinrich Schütze. An Introduction to
Information Retrieval. 2009, Cambridge University Press
[3] Bing Liu. Web Data Mining. Pub. Date: 2011, Second Edition, pages: 622. ISBN: 978-
3-642-19459-7. Publisher: Springer-Verlag Berlin Heidelberg
[4] Shatlyk Ashyralyyev, CS533 course, Bilkent University
