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in S that is larger than some case that a graph has a go o d larger than the secondlargest if it quickly time logarithmic in the size of the graph converges to a limiting distribution on the set of no des in the graph It is also the case that a random walk is rapidlymixing on a graph if and only if the graph is an expander or has an eigenvalue separation 100000000 10000000 1000000 100000 10000 1000 100 10 factor times jS j here is called the expansion factor It is the expansion factor if and only if the largest eigenvalue is suciently eigenvalue A random walk on a graph is said to b e rapidlymixing To relate all this to the PageRank computation note that it is essentially the determination of the limiting distribution of a random walk on the Web graph The imp ortance ranking of a no de is essentially the limiting probability that the random walk will b e at that no de after a suciently large time The fact that the PageRank computation terminates in logarithmic time is equivalent to saying factor future in has a go o d expansion that the random walk is rapidly mixing or that the underlying graph Expander graphs have many desirable prop erties that we may b e able to exploit in the computations involving the Web graph Convergence of PageRank Computation 0 7.5 15 22.5 30 37.5 45 52.5 Figure Rates of Convergence for Full Size and Half Size Link Databases Searching with PageRank A ma jor application of PageRank is searching We have implemented two search engines which use PageRank The rst one we will discuss is a simple titlebased search engine The second search engine is a full text search engine called Go ogle BP Go ogle utilizes a numb er of factors to rank search results including standard IR measures proximity anchor text text of links p ointing to web pages and PageRank While a comprehensive user study of the b enets of PageRank is b eyond the scop e of this pap er we have p erformed some comparative exp eriments and provide some sample results in this pap er Number of Iterations 322 Million Links 161 Million Links Total Difference from Previous IterationPDF Image | PageRank Citation Ranking Bringing Order to the Web
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