i.e. the PageRank value for a page u is dependent on the PageRank values for each page v contained in the set Bu (the set containing all pages linking to page u), divided by the number L(v) of links from page v. The algorithm involves a damping factor for the calculation of the pagerank. It is like the income tax which the govt extracts from one despite paying him itself.
PageRank as a visible score has been dying a slow death since around 2010, I’d say. Pulling it from the Google Toolbar makes it official, puts the final nail in the visible PageRank score coffin. The few actually viewing it within Internet Explorer, itself a depreciated browser, aren’t many. The real impact in dropping it from the toolbar means that third parties can no longer find ways to pull those scores automatically.
These are paid advertisements via Google AdWords. You can differentiate these from organic (non-paid, earned) results because of the tiny yellow “ad” icon. Normally, you will also see paid ads at the top of the SERP as well. In this book, we are only talking about SEO for organic search results, not advertising through search engines. But it is important to understand the difference as a user, and as a search engine optimizer, as they can both be valuable.
What seems to be happening is that the toolbar looks at the URL of the page the browser is displaying and strips off everything down the last “/” (i.e. it goes to the “parent” page in URL terms). If Google has a Toolbar PR for that parent then it subtracts 1 and shows that as the Toolbar PR for this page. If there’s no PR for the parent it goes to the parent’s parent’s page, but subtracting 2, and so on all the way up to the root of your site.  If it can’t find a Toolbar PR to display in this way, that is if it doesn’t find a page with a real calculated PR, then the bar is greyed out.
Another excellent guide is Google’s “Search Engine Optimization Starter Guide.” This is a free PDF download that covers basic tips that Google provides to its own employees on how to get listed. You’ll find it here. Also well worth checking out is Moz’s “Beginner’s Guide To SEO,” which you’ll find here, and the SEO Success Pyramid from Small Business Search Marketing.
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