Personally, I wanted a bit more of the math, so I went back and read the full-length version of “The Anatomy of a Large-Scale Hypertextual Web Search Engine” (a natural first step). This was the paper written by Larry Page and Sergey Brin in 1997. Aka the paper in which they presented Google, published in the Stanford Computer Science Department. (Yes, it is long and I will be working a bit late tonight. All in good fun!)
Now, how much weight does PageRank carry? Like most every other part of the algorithm, it’s questionable. If we listed all the ranking factors, I don’t suspect it would be in the top 5, but it’s important to remember that the key to ranking well is to be the LESS IMPERFECT than your competition. IE: To have more of the right things that send the right signals in the right places so that Google sees you as a better, more relevant, candidate for the top three on page one. If you and your competitor both have optimized (on-page and technically) for the same keyword phrase perfectly, PR could be the deal breaker that pushes your blue link an inch up.
Just do a quick Google search. If you're monitoring to see if a link you built is indexed, or just want to find other areas where you've been mentioned or linked, do a quick search with your company brand name, your web URL or other terms you're following. I've seen plenty of backlinks indexed by the search engine that never showed up in my search console account.
The field is replete with terms that might confuse and perplex the average individual. What is a squeeze page? What's a sales funnel? What's a CPA? What's SEO? How do you setup a good blog to filter the right type of relevant traffic and get your offer in front of eligible users? What's a massive value post (MVP) really mean? Clearly, there are an endless array of terms, some of which you might already know or might not depending on how much you presently know about the field.
In my view there is nothing wrong with saying ‘hey Google, these pages are not important from a search engine perspective, let me not give them so much weight’. Regardless of how Google now views these type of pages from a weight perspective, doing the above as a webmaster should be logical and encouraged. You have said this yourself at least a few times in the past.
I think Matt Grenville’s comment is a very valid one. If your site, for whatever reason, can not attract links naturally and all of your competitors are outranking you by employing tactics that might breach Google’s TOS, what other options do you have? As well as this people will now only link to a few, trusted sites (as this has been clarified in your post as being part of Google’s algorithm) and put a limit on linking out to the smaller guys.

I won’t blame MC. Google, knows what they does. These are things that webmasters need not worry about. Well, it won’t make much difference as far as I think. I don’t use no follow tags specifically – I use WP for blogging purposes and it does rest of the things for me other than writing content which I do. I think it is the content and the external links that sites point to – which should be considered. I mean, if a computer blog owner posts a really fantastic computer article about something related to computer, and also puts some links to external pages (which are really useful for the readers), then that post, should be ranked high in gooogle – And I think google does this well – So, webmasters, just concentrate on yur website/blogs etc and leave rest of the things to Big G.
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.
Thanks for the article (and lead-off links as they were good info too) but I did not quite get – if there was a penalisation by Google for sculpting – from the article or whether it was just bad practice? And also to echo what someone else asked ‘is it WORTH actually undoing this type of work on websites SEO’s have worked on’ or simply change the way we work with new sites?
“So what happens when you have a page with “ten PageRank points” and ten outgoing links, and five of those links are nofollowed? Let’s leave aside the decay factor to focus on the core part of the question. Originally, the five links without nofollow would have flowed two points of PageRank each (in essence, the nofollowed links didn’t count toward the denominator when dividing PageRank by the outdegree of the page). More than a year ago, Google changed how the PageRank flows so that the five links without nofollow would flow one point of PageRank each.”

Because of the size of the actual web, the Google search engine uses an approximative, iterative computation of PageRank values. This means that each page is assigned an initial starting value and the PageRanks of all pages are then calculated in several computation circles based on the equations determined by the PageRank algorithm. The iterative calculation shall again be illustrated by our three-page example, whereby each page is assigned a starting PageRank value of 1.
An omni-channel approach not only benefits consumers but also benefits business bottom line: Research suggests that customers spend more than double when purchasing through an omni-channel retailer as opposed to a single-channel retailer, and are often more loyal. This could be due to the ease of purchase and the wider availability of products.[24]
So enough of these scary stories. Google actually likes backlinks and relies upon them. The whole idea behind them is that they help to tell Google what is good and useful out there. Remember, it is still an algorithm. It doesn’t know that your page describing the best technique for restoring a 1965 Ford Mustang bumper is all that great. But if enough people are talking about how great it is, and thereby referencing that page on other websites, Google will actually know.

I dont know if Google gets its kicks out of keeping Search Engine Marketers and Webmasters jumping through hoops – or if they are in cahoots with the big SEM firms – so that they get this news and these updates before the average guy on the street. Either way, they are seriously getting a bit too big and powerful and the time is RIPE for a new search engine to step in and level the playing field.
By 2004, search engines had incorporated a wide range of undisclosed factors in their ranking algorithms to reduce the impact of link manipulation. In June 2007, The New York Times' Saul Hansell stated Google ranks sites using more than 200 different signals.[25] The leading search engines, Google, Bing, and Yahoo, do not disclose the algorithms they use to rank pages. Some SEO practitioners have studied different approaches to search engine optimization, and have shared their personal opinions.[26] Patents related to search engines can provide information to better understand search engines.[27] In 2005, Google began personalizing search results for each user. Depending on their history of previous searches, Google crafted results for logged in users.[28] 

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The Google algorithm's most important feature is arguably the PageRank system, a patented automated process that determines where each search result appears on Google's search engine return page. Most users tend to concentrate on the first few search results, so getting a spot at the top of the list usually means more user traffic. So how does Google determine search results standings? Many people have taken a stab at figuring out the exact formula, but Google keeps the official algorithm a secret. What we do know is this:
Hemanth Kumar, a good rule of thumb is: if a link on your website is internal (that is, it points back to your website), let it flow PageRank–no need to use nofollow. If a link on your website points to a different website, much of the time it still makes sense for that link to flow PageRank. The time when I would use nofollow are when you can’t or don’t want to vouch for a site, e.g. if a link is added by an outside user that you don’t particularly trust. For example, if an unknown user leaves a link on your guestbook page, that would be a great time to use the nofollow attribute on that link.
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.
Cause if I do that, If I write good content, whilst my 100+ competitors link build, article market, forum comment, social bookmark, release viral videos, buy links, I’ll end up the very bottom of the pile, great content or not and really I am just as well taking my chances pulling off every sneaky trick in the book to get my site top because, everyone does it anyway and if I don’t what do have to lose?”
A variety of methods can increase the prominence of a webpage within the search results. Cross linking between pages of the same website to provide more links to important pages may improve its visibility.[47] Writing content that includes frequently searched keyword phrase, so as to be relevant to a wide variety of search queries will tend to increase traffic.[47] Updating content so as to keep search engines crawling back frequently can give additional weight to a site. Adding relevant keywords to a web page's meta data, including the title tag and meta description, will tend to improve the relevancy of a site's search listings, thus increasing traffic. URL canonicalization of web pages accessible via multiple urls, using the canonical link element[48] or via 301 redirects can help make sure links to different versions of the url all count towards the page's link popularity score.
However, if you're going to understand online marketing, you have to understand the importance of building Google's trust. There are three core components involved here. These three core components are like the pillars of trust that comprise all of Google's 200+ ranking factor rules. Each of those rules can be categorized and cataloged into one of these three pillars of trust. If you want to rank on the first page or in the first spot, you need to focus on all three, and not just one or two out of three.

I am not worried by this; I do agree with Danny Sullivan (Great comment Danny, best comment I have read in a long time). I will not be changing much on my site re: linking but it is interesting too see that Google took over a year to tell us regarding the change, but was really happy to tell us about rel=”nofollow” in the first place and advised us all to use it.
By using Internet platforms, businesses can create competitive advantage through various means. To reach the maximum potential of digital marketing, firms use social media as its main tool to create a channel of information. Through this a business can create a system in which they are able to pinpoint behavioral patterns of clients and feedback on their needs.[30] This means of content has shown to have a larger impingement on those who have a long-standing relationship with the firm and with consumers who are relatively active social media users. Relative to this, creating a social media page will further increase relation quality between new consumers and existing consumers as well as consistent brand reinforcement therefore improving brand awareness resulting in a possible rise for consumers up the Brand Awareness Pyramid.[31] Although there may be inconstancy with product images;[32] maintaining a successful social media presence requires a business to be consistent in interactions through creating a two way feed of information; firms consider their content based on the feedback received through this channel, this is a result of the environment being dynamic due to the global nature of the internet.[29] Effective use of digital marketing can result in relatively lowered costs in relation to traditional means of marketing; Lowered external service costs, advertising costs, promotion costs, processing costs, interface design costs and control costs.[32]
The mathematics of PageRank are entirely general and apply to any graph or network in any domain. Thus, PageRank is now regularly used in bibliometrics, social and information network analysis, and for link prediction and recommendation. It's even used for systems analysis of road networks, as well as biology, chemistry, neuroscience, and physics.[45]
In this new world of digital transparency brands have to be very thoughtful in how they engage with current and potential customers. Consumers have an endless amount of data at their fingertips especially through social media channels, rating and review sites, blogs, and more. Unless brands actively engage in these conversations they lose the opportunity for helping guide their brand message and addressing customer concerns.
Of course, it’s possible that the algorithm has some method of discounting internally reflected (and/or directly reciprocal) links (particularly those in identical headers or footers) to such an extent that this isn’t important. Evidence to support this the fact that many boring pages that are linked to by every page in a good site can have very low PR.
Well, it seems that what this article says, is that the purpose of the no-follow link is to take the motivation away from spammers to post spam comments for the purpose of the link and the associated page rank flow; that the purpose of no-follow was never to provide a means to control where a page’s pagerank flow is directed. It doesn’t seem that shocking to me folks.
Backlink is a link one website gets from another website. Backlinks make a huge impact on a website’s prominence in search engine results. This is why they are considered very useful for improving a website’s SEO ranking. Search engines calculate rankings using multiple factors to display search results. No one knows for sure how much weight search engines give to backlinks when listing results, however what we do know for certain is that they are very important.
Backlinks can be time-consuming to earn. New sites or those expanding their keyword footprint may find it difficult to know where to start when it comes to link building. That's where competitive backlink research comes in: By examining the backlink profile (the collection of pages and domains linking to a website) to a competitor that's already ranking well for your target keywords, you can gain insight about the link building that may have helped them. A tool like Link Explorer can help uncover these links so you can and target those domains in your own link building campaigns.
A: I wouldn’t recommend it, because it isn’t the most effective way to utilize your PageRank. In general, I would let PageRank flow freely within your site. The notion of “PageRank sculpting” has always been a second- or third-order recommendation for us. I would recommend the first-order things to pay attention to are 1) making great content that will attract links in the first place, and 2) choosing a site architecture that makes your site usable/crawlable for humans and search engines alike.
Another way to get sites to link back to something valuable on your site is by offering a free tool. A free tool could be a basic tool (like an auto loan calculator) or a scaled down version of a paid tool (like Alexa’s Site Overview and Audience Overlap tools). If the tools are valuable enough, others will link to them in their content. Plus, on free versions of paid tools, you can add call-to-actions to sign up for the full product/service which drives acquisition in addition to awareness.
Digital marketing methods such as search engine optimization (SEO), search engine marketing (SEM), content marketing, influencer marketing, content automation, campaign marketing, data-driven marketing,[6] e-commerce marketing, social media marketing, social media optimization, e-mail direct marketing, display advertising, e–books, and optical disks and games are becoming more common in our advancing technology. In fact, digital marketing now extends to non-Internet channels that provide digital media, such as mobile phones (SMS and MMS), callback, and on-hold mobile ring tones.[7] In essence, this extension to non-Internet channels helps to differentiate digital marketing from online marketing, another catch-all term for the marketing methods mentioned above, which strictly occur online.
Internet Marketing Inc. provides integrated online marketing strategies that help companies grow. We think of ourselves as a business development consulting firm that uses interactive marketing as a tool to increase revenue and profits. Our management team has decades of combined experience in online marketing as well as graduate level education and experience in business and finance. That is why we focus on creating integrated online marketing campaigns designed to maximize your return on investment.
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