Presentation on theme: "Google Pagerank: how Google orders your webpages Dan Teague NCSSM."— Presentation transcript:
Google Pagerank: how Google orders your webpages Dan Teague NCSSM
The Problem Imagine a library containing 40 billion documents but with no centralized organization and no librarians. In addition, anyone may add a document at any time without telling anyone. If one of these documents is vitally important to you, how could you find it?
The Basic Idea PageRank is a numeric value that represents how important a page is on the web. Google figures that when one page links to another page, it is effectively casting a vote for the other page. The more votes that are cast for a page, the more important the page must be. Also, the importance of the page that is casting the vote determines how important the vote itself is. http://www.webworkshop.net/pagerank.html
Markov Chain We would like to think of this matrix as a transition matrix (like a Markov chain). If we move around on the graph at random, at which nodes will we spend most of our time? These most important nodes can be found in a Markov chain by considering the powers of H.
Or we can look for solutions to HX = X. This means we want the eigenvector X associated with the eigenvalue of 1. This is why the Pagerank is known as the $25,000,000,000 eigenvector.
What About the Other Problems? Dangling Nodes are easy to find. Cycles and Sub-graph sinks are more difficult and time consuming. Pagerank handles these problems without actually finding them. The Cycle The Sub-graph Sink
Probabilistic Movement Roll a die. If anything but a 6 shows, then follow the web, that is, use our matrix (H + A). However, if you roll a 6, then pick a page at random and go there. This gives us an out when we are trapped either by a cycle or by a sub-graph sink.
How Often Should We Look for an Escape? Would it be better to roll a 20-sided die or flip a coin?
How do you implement the coin flip? Create a matrix all of whose entries are 1. This is the One matrix. If we multiply this matrix by 1/n, where n is the number of nodes in the graph (in our example 11, in reality 40 billion), then we have an equal chance of traveling from any point to any other point. We pretend that the web is a complete graph.
Roll the die We will use the Web-ordered matrix H+A with probability p and the One matrix with probability (1-p). What’s a good value for p?