There are many articles on the net about how the PageRank algorithm works that all copy from the original paper written by the very founders of Google Larry Page and Sergey Brin. Google itself also has a very good article that explain it with no formulas or numerical explanations. Basically PageRank is like social networks. If you're mentioned by someone important, your importance increases and the people you mention gets upped as well.
We recently had a coursework in discrete mathematics to calculate PageRank values for all web pages in a web matrix. To be able to do this you have to do many simplifications and you're limited in terms of complexity to keep it possible to do "by hand". I wrote a little program that calculates the PageRank for any web with no simplifications. The outcome is that I can quickly calculate the PageRank values for each page.
Here's how to use it:
from PageRank import PageRanker
web = ((0, 1, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1),
(1, 0, 0, 0))
pr = PageRanker(0.85, web)
pr.improve_guess(100)
print pr.getPageRank()
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