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Link to and from sites that are complimentary to what your site is about. Create your marketing/linking plan with visitors in mind, not the robots.
The formula is trivial, for anyone who's passed a semester of linear algebra, and incomprehensible otherwise. But if you're thinking in terms of modeling it in terms of bean-counting, then you should either find some simpler concept to model, or learn to like padded walls.
The principles are simple:
1) Incoming links to your pages are good -- the better the site that links to them, the better. If you have a choice of two incoming links, both are better than either one. But be warned that Google knows some sites are VERY bad, and links from them won't help you at all.
2) Clear navigation within the site is good: a casual human visitor or googlebot should be able to get anywhere in the site within 5 clicks or so. Fancy dynamic menus are probably not good: daunting to both humans and bots.
3) Link only to sites that are worth something to real visitors. Linking to sites Google has already identified as "GRAW" (generally recognized as worthless) will hurt your site.
4) Don't depend on link exchanges for page rank boost. A lot of people cheat on them; and even if they don't cheat, they still may lure you into linking to GRAW sites. And even if they weren't GRAW before, exchanging links with a site (like yours) desperate for links may drag both of you into the graw pits.
5) Patience. If this has to be accomplished by next month or you go crazy -- better start filling out insurance papers and checking out the asylums.
6) Don't try to second-guess Google engineers. They are the sort of people who tell rocket scientists how to deal with the difficult problems; and you are not. Instead, read the Google recommendations, and follow them. (You'd be surprised how many idiots build Flash sites and then obsess over their page rank. And even in these forums, the number of proposals for bean-counting schemes, involving intricate sorts on strain, color, number and size of spots, etc., is incredible.
The PageRank Citation Ranking: Bringing Order the the Web [citeseer.nj.nec.com] by Sergey Brin and Lawrence Page
As noted earlier in this thread, these two papers describe the original page rank system. It may well have changed over the years.
What are C() & d?
d is the damping factor, i.e. the fraction of PR which is transferred from one page to other pages. In the first paper d=1 was used, later they changed to d=0.85 (e.g. to avoid problems coming from leaf nodes). Currently Google is using a different value. (1-d) is some kind of self contribution of a page.
PR(x) denotes the PR of page x linking to you; C(x) is the number of links on that page.
The PR formula leads to an eigen value problem for d=1 and a system of linear equations for d<1, respectively. For large systems the solution is derived by iteration schemes.
The real PR is related to the toolbarPR by a logarithmic scale.