# The Google Pagerank Algorithm and How It Works The Google Pagerank Algorithm and How It Works What is Pagerank? In short PageRank is a vote, by all the other pages on the Web, about how important a page is. A link to a page counts as a vote of support PR(A) = (1-d) + d(PR(T1)/C(T1) + +PR(Tn)/C(Tn))

Breaking Down the Equation PR(Tn) - Each page has a notion of its own self-importance. Thats PR(T1) for the first page in the web all the way up to PR(Tn) for the last page C(Tn) - Each page spreads its vote out evenly amongst all of its outgoing links. The count, or number, of outgoing links for

page 1 is C(T1), C(Tn) for page n, and so on for all pages. PR(Tn)/C(Tn) - so if our page (page A) has a backlink from page n the share of the vote page A will get is PR(Tn)/C(Tn) d( - All these fractions of votes are added together but, to stop the other pages having too much influence, this total vote is damped down by multiplying it by 0.85 (the factor d) (1 - d) - The (1 d) bit at the beginning is a bit of probability math magic so the sum of all web pages\ PageRanks will be one: it adds in the bit lost by the d(. It also means that if a page has no links to it (no backlinks) even then it will still get a small PR of 0.15 (i.e. 1 0.85). (Aside: the Google paper says

the sum of all pages but they mean the the normalised sum otherwise known as the average to you and me. How is it Calculated? The PR of each page depends on the PR of the pages pointing to it. But we wont know what PR those pages have until the pages pointing to them have their PR calculated and so on. So what we do is make a guess.

Simple Example Each page has one outgoing link. So that means C(A) = 1 and C(B) = 1. We dont know what their PR should be to begin with, so we will just guess 1 as a safe random number. d (damping factor) = 0.85 PR(A)= (1 d) + d(PR(B)/1)PR(B)= (1 d) +

d(PR(A)/1) i.e. PR(A)= 0.15 + 0.85 * 1 =1 PR(B)= 0.15 + 0.85 * 1 =1 Lets Do It Again with Another Number. Lets try 0 and re-calculate

PR(A)= 0.15 + 0.85 * 0 = 0.15 = 0.15 + 0.85 * PR(B) 0.15 = 0.2775 Now we have calculated a next best guess so we just plug it in the equation again

PR(A)= 0.15 + 0.85 * 0.2775 data = 0.385875 PR(B)= 0.15 + 0.85 * 0.385875 = 0.47799375 And again PR(A)= 0.15 + 0.85 * 0.47799375 = 0.5562946875 PR(B)= 0.15 + 0.85 * 0.5562946875 = 0.622850484375

Principle It doesnt matter where you start your guess, once the PageRank calculations have settled down, the normalized probability distribution (the average PageRank for all pages) will be 1.0 Lets look at a more complicated example

Data Observation: every page has at least a PR of 0.15 to share out. But this may only be in theory there are rumors that Google undergoes a post-spidering phase whereby any pages that have no incoming links at all are completely deleted from the index