# PageRank Algorithm

Topics: Matrix, Linear algebra, Google Pages: 31 (4244 words) Published: October 21, 2013
PageRank Algorithm

December 9, 2012
Abstract
This paper dicsusses the PageRank algorithm. We carefully go through each step of the algorithm and explain each procedure. We also explain the mathematical setup of the algorithm, including all computations that are used in the PageRank algorithm. Some of the topics that we touch on include the following, but not limited to, are: linear algebra, node analysis, matrix theory, and numerical methods. But primarily this paper concerns itself with the use of the linear algebra involved in the computation of the Google matrix, which results in the Pagerank, which descibribes how important a page is. Importance is placed on the intuition of all related mathematical topics involved in the algorithm and clarity of understanding

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Introduction

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Figure 1 shows our directed graph, a simple fantasy World Wide Web. The arrows represent a hyperlink from 1 page to another page. For example, page A goes to C, E, and D etc. We now set up a table that describes our directed graph. Table 1 shows how we will get our hyperlink matrix. The columns represent the starting page and the rows represtent the ending page. A 1 will be place in this table if a page has a hyperlink to another page. For example, Page A has a hyperlink to Page C and so there is a 1 in the C row and A column.

Now we will “extract” our hyperlink matrix from this. We will denote this matrix by L. Now from Figure 2 we need to do one additional thing. We need to adjust this matrix so that when we add each element from each row, the sum equals 1 (the motivation behind this will be discussed later). We can do this by getting the sum of each row and dividing each element in that row by that sum. Now our revised hyperlink matrix looks like Figure 3 and we will denote it by H. 1

The above process was a technical representation of how to get this H matrix. Now we will present an intuitive way to get an understanding of what will need to be done in calculating the PageRank of each page. While the previous way is useful (it will be useful in developing the code for it), it does not let us see what must be done to calculate a PageRank. The idea Brin and Page had was [3, 13] that each page would transfer a porportion of its PageRank to the page it had a hyperlink to it. So in the above example, Since page A has a hyperlink to pages C, D, and E, page...

References: Princeton University Press, 2006.
[3] Sergey Brin, Lawrence Page, The antaomy of a large-scale hypertextual Web search engine, Computer
Networks and ISDN Systems, 33: 107-17, 1998.
[10] Ron Larson, Elementary Linear Algebra 7th Edition, Brooks Cole, 2012, pp 550-556.
May 2006
[14] Masaaki Kijima, Markov Processes for Stochastic Modeling, CRC Press, 1997, pp 295-297