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Lecture3 - April 5 2010 Franklin Kenter PageRank Spectral...

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April 5, 2010 Franklin Kenter PageRank, Spectral Graph Theory, and the Matrix Tree Theorem Introduction 1 Introduction In this lecture, we will go over the basics of the PageRank algorithm and how it relates to graph theory. Then, we will start our study in spectral graph theory by proving the Matrix Tree Theorem. 2 PageRank There are two predominant ranking algorithms. PageRank and HITS. PageRank was developed by Brin and Page and is the foundation for what is now the Google search engine in 1997. The other, lesser known, is the HITS algorithm which focuses on “hubs” and “authorities” developed by Kleinburg [1] in 1999. Both of these algorithms are important, most notably, because they are able to capture the essence of a graph without doing any global pattern matching. Further, it should be noted that even though Kleinburg “did not receive any Google stock,” mathematically, his algorithm is still very important today. Here, we will focus on the PageRank algorithm, for more information on the HITS algorithm, please see the references below. As developed by Brin and Page, PageRank is a voting system whereby the weight of each vote is linearly propotional to the total value of the votes it receives. [3] This might seem paradoxial. However, this sets up a linear equation! More specifically, let G = ( V, E ) be a directed graph. We seek the vector f R | V | , indexed by the verticies in G, that satisfies the following: f = α 1 /n + (1 - α ) fP f1 * = 1 Where α is a constant between 0 and 1, and P is the probability transition matrix of the directed graph. 1
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In particular, f is an eignvalue of the matrix α 1 * 1 /n +(1 - α ) P with eigenvalue 1. The PerronFrobe- nius Theorem guarentees that there exists an eigenvector with eigenvalue 1, and provided that P is strongly connected, all other eigenvalues have modulus strictly less than 1. Before we continue, we should go over other interpretations of PageRank. Above we gave the vauge voter interpration. However there are two other interprations. First, as described by Brin and Page [3], PageRank also models a bored surfer that surfs the graph (i.e., internet) as follows: at each site, with probability α
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