Nashville Grant, here’s the mental model I’d employ: search engines want to return great content. If you make such a fantastic site that all the web has heard of you, search engines should normally reflect that fact and return your site. A lot of bad SEO happens because people say “I’ll force my way to the top of Google first, and then everyone will find out about my site.” Putting rankings before the creation of a great site is in many ways putting the cart before the horse. Often the search rankings follow from the fact that you’re getting to be well-known on the web completely outside the sphere of search. Think about sites like Twitter and Facebook–they succeed by chasing a vision of what users would want. In chasing after that ideal of user happiness and satisfaction, they became the sort of high-quality sites that search engines want to return, because we also want to return what searches will find useful and love. By chasing a great user experience above search rankings, many sites turn out to be what search engines would want to return anyway.

On-page SEO is the work you do on your own website to get a high rank in search engines. Your goal is obviously that your website will show on the first page and perhaps even among the first three search results. On-page SEO does not carry as much weight as off-page SEO in the rankings, but if you don’t get the basics right… it’s unlikely that your off-page SEO will deliver results, either.
PageRank is a link analysis algorithm and it assigns a numerical weighting to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of "measuring" its relative importance within the set. The algorithm may be applied to any collection of entities with reciprocal quotations and references. The numerical weight that it assigns to any given element E is referred to as the PageRank of E and denoted by {\displaystyle PR(E).} Other factors like Author Rank can contribute to the importance of an entity.
PageRank was developed by Google founders Larry Page and Sergey Brin at Stanford. In fact the name. PageRank is a likely play on Larry Page's name. At the time that Page and Brin met, early search engines typically linked to pages that had the highest keyword density, which meant people could game the system by repeating the same phrase over and over to attract higher search page results. Sometimes web designers would even put hidden text on pages to repeat phrases. 
And my vital question about Amazon affiliate links. I think many people also wonder about it as well. I have several blogs where I solely write unique content reviews about several Amazon products, nothing more. As you know, all these links are full of tags, affiliate IDs whatsoever (bad in SEO terms). Should I nofollow them all or leave as they are?
One thing is certain: interlinking sites doesn't help you from a search engine standpoint. The only reason you may want to interlink your sites in the first place might be to provide your visitors with extra resources to visit. In this case, it would probably be okay to provide visitors with a link to another of your websites, but try to keep many instances of linking to the same IP address to a bare minimum. One or two links on a page here and there probably won't hurt you.
You can confer some of your site's reputation to another site when your site links to it. Sometimes users can take advantage of this by adding links to their own site in your comment sections or message boards. Or sometimes you might mention a site in a negative way and don't want to confer any of your reputation upon it. For example, imagine that you're writing a blog post on the topic of comment spamming and you want to call out a site that recently comment spammed your blog. You want to warn others of the site, so you include the link to it in your content; however, you certainly don't want to give the site some of your reputation from your link. This would be a good time to use nofollow.

Danny, I was on the panel where Matt suggested that and I point blank asked on stage what happened when folks starting abusing the tactic and Google changed their mind if you recall (at the time, I’d seen some of the things being done I knew Google would clarify as abuse and was still a nofollow unenthusiast s a result at that time). And Matt dismissed it. So, I think you can take home two important things from that – 1. SEO tactics can always change regardless of who first endorses them and 2. Not everything Matt says is etched in stone. <3 ya Matt.

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Advertising with Google won't have any effect on your site's presence in our search results. Google never accepts money to include or rank sites in our search results, and it costs nothing to appear in our organic search results. Free resources such as Search Console, the official Webmaster Central blog, and our discussion forum can provide you with a great deal of information about how to optimize your site for organic search.
The amount of link juice passed depends on two things: the number of PageRank points of the webpage housing the link, and the total number of links on the webpage that are passing PageRank. It’s worth noting here that while Google will give every website a public-facing PageRank score that is between 1 and 10, the “points” each page accumulates from the link juice passed by high-value inbound links can — and do — significantly surpass ten. For instance, webpages on the most powerful and significant websites can pass link juice points in the hundreds or thousands. To keep the rating system concise, Google uses a lot of math to correlate very large (and very small) PageRank values with a neat and clean 0 to 10 rating scale.
Site owners are using the toolbar to find “good” sites that they should get links from, regardless of the fact that link context is also important, not to mention many, many other factors that are used by Google to rank a web page. Other site owners, getting a gray PR0 toolbar for their site, immediate assume the worst, that they’ve been blacklisted.
While most search engine companies try to keep their processes a secret, their criteria for high spots on SERPs isn't a complete mystery. Search engines are successful only if they provide a user links to the best Web sites related to the user's search terms. If your site is the best skydiving resource on the Web, it benefits search engines to list the site high up on their SERPs. You just have to find a way to show search engines that your site belongs at the top of the heap. That's where search engine optimization (SEO) comes in -- it's a collection of techniques a webmaster can use to improve his or her site's SERP position.
Optimization techniques are highly tuned to the dominant search engines in the target market. The search engines' market shares vary from market to market, as does competition. In 2003, Danny Sullivan stated that Google represented about 75% of all searches.[63] In markets outside the United States, Google's share is often larger, and Google remains the dominant search engine worldwide as of 2007.[64] As of 2006, Google had an 85–90% market share in Germany.[65] While there were hundreds of SEO firms in the US at that time, there were only about five in Germany.[65] As of June 2008, the market share of Google in the UK was close to 90% according to Hitwise.[66] That market share is achieved in a number of countries.
In this illustration from the “PageRank Citation Ranking” paper, the authors demonstrate how webpages pass value onto other pages. The two pages on the left have a value of 100 and 9, respectively. The page with a value of 100 has two links that point to the pages on the right. That page’s value of 100 is divided between the two links, so that each conveys a value of 50. The other page on the left has three outgoing links, each carrying one-third of the page’s value of 9. One link goes to the top page on the right, which ends up with a total value of 53. The bottom right page has no other backlinks, so its total value is 50.
With focus I mean making sure that your pages focus on the same keyword everywhere, and your site focuses on the same high level keywords and sections in your site focusing on their own high level (but not as high as the keywords for which you want your home page to rank) keywords. Focus few people really understand while the interesting thing is that you do this almost automatically right if you do your site architecture and understanding your customers, right.
But, why do search engines care about backlinks? Well, in the early days of the Internet, search engines were very simple, and relied strictly on keyword matching. It didn’t matter how good the content on a website was, how popular it was, or what the website was for–if a phrase on a page matched a phrase that someone searched for, then that page would likely show up. That meant that if someone had an online journal in which they documented at length how they had to take their car to a “car accident repair shop,” then people searching for a “car accident repair shop” would likely be led to that page. Not terribly useful, right?
(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.
PageRank is one of many, many factors used to produce search rankings. Highlighting PageRank in search results doesn’t help the searcher. That’s because Google uses another system to show the most important pages for a particular search you do. It lists them in order of importance for what you searched on. Adding PageRank scores to search results would just confuse people. They’d wonder why pages with lower scores were outranking higher scored pages.

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