What is PageRank?
PageRank is a numeric value that represents how important a page is on the web. Google figures that when one page links to another page, it is effectively casting a vote for the other page. The more votes that are cast for a page, the more important the page must be. Also, the importance of the page that is casting the vote determines how important the vote itself is. Google calculates a page's importance from the votes cast for it. How important each vote is is taken into account when a page's PageRank is calculated.
PageRank is Google's way of deciding a page's importance. It matters because it is one of the factors that determines a page's ranking in the search results. It isn't the only factor that Google uses to rank pages, but it is an important one.
Thursday, September 10, 2009
Sunday, August 30, 2009
Quick Tips To Search Google Like An Expert
If you’re like me, you probably use Google many times a day. But, chances are, unless you are a technology geek, you probably still use Google in its simplest form. If your current use of Google is limited to typing a few words in, and changing your query until you find what you’re looking for, then I’m here to tell you that there’s a better way – and it’s not hard to learn. On the other hand, if you are a technology geek, and can use Google like the best of them already, then I suggest you bookmark this article of Google search tips. You’ll then have the tips on hand when you are ready to pull your hair out in frustration when watching a neophyte repeatedly type in basic queries in a desperate attempt to find something.
Expert Google Search Tips
1. Explicit Phrase:
Lets say you are looking for content about internet marketing. Instead of just typing internet marketing into the Google search box, you will likely be better off searching explicitly for the phrase. To do this, simply enclose the search phrase within double quotes.
Example: "internet marketing"
2. Exclude Words:
Lets say you want to search for content about internet marketing, but you want to exclude any results that contain the term advertising. To do this, simply use the "-" sign in front of the word you want to exclude.
Example Search: internet marketing -advertising
3. Site Specific Search:
Often, you want to search a specific website for content that matches a certain phrase. Even if the site doesn’t support a built-in search feature, you can use Google to search the site for your term. Simply use the "site:somesite.com" modifier.
Example: "internet marketing" site:www.smallbusinesshub.com
4. Similar Words and Synonyms:
Let’s say you are want to include a word in your search, but want to include results that contain similar words or synonyms. To do this, use the "~" in front of the word.
Example: "internet marketing" ~professional
5. Specific Document Types:
If you’re looking to find results that are of a specific type, you can use the modifier "filetype:". For example, you might want to find only PowerPoint presentations related to internet marketing.
Example: "internet marketing" filetype:ppt
6. This OR That:
By default, when you do a search, Google will include all the terms specified in the search. If you are looking for any one of one or more terms to match, then you can use the OR operator. (Note: The OR has to be capitalized).
Example: internet marketing OR advertising
7. Phone Listing:
Let’s say someone calls you on your mobile number and you don’t know how it is. If all you have is a phone number, you can look it up on Google using the phonebook feature.
Example: phonebook:617-555-1212 (note: the provided number does not work – you’ll have to use a real number to get any results).
8. Area Code Lookup:
If all you need to do is to look-up the area code for a phone number, just enter the 3-digit area code and Google will tell you where it’s from.
Example: 617
9. Numeric Ranges:
This is a rarely used, but highly useful tip. Let’s say you want to find results that contain any of a range of numbers. You can do this by using the X..Y modifier (in case this is hard to read, what’s between the X and Y are two periods. This type of search is useful for years (as shown below), prices or anywhere where you want to provide a series of numbers.
Example: president 1940..1950
10. Stock (Ticker Symbol):
Just enter a valid ticker symbol as your search term and Google will give you the current financials and a quick thumb-nail chart for the stock.
Example: GOOG
11. Calculator:
The next time you need to do a quick calculation, instead of bringing up the Calculator applet, you can just type your expression in to Google.
Example: 48512 * 1.02
12. Word Definitions:
If you need to quickly look up the definition of a word or phrase, simply use the "define:" command.
Example: define:plethora
Hope this list of Google search tips proves useful in your future Google searches. If there are any of your favorite Google expert power tips that I’ve missed, please feel free to share them in the comments.
Expert Google Search Tips
1. Explicit Phrase:
Lets say you are looking for content about internet marketing. Instead of just typing internet marketing into the Google search box, you will likely be better off searching explicitly for the phrase. To do this, simply enclose the search phrase within double quotes.
Example: "internet marketing"
2. Exclude Words:
Lets say you want to search for content about internet marketing, but you want to exclude any results that contain the term advertising. To do this, simply use the "-" sign in front of the word you want to exclude.
Example Search: internet marketing -advertising
3. Site Specific Search:
Often, you want to search a specific website for content that matches a certain phrase. Even if the site doesn’t support a built-in search feature, you can use Google to search the site for your term. Simply use the "site:somesite.com" modifier.
Example: "internet marketing" site:www.smallbusinesshub.com
4. Similar Words and Synonyms:
Let’s say you are want to include a word in your search, but want to include results that contain similar words or synonyms. To do this, use the "~" in front of the word.
Example: "internet marketing" ~professional
5. Specific Document Types:
If you’re looking to find results that are of a specific type, you can use the modifier "filetype:". For example, you might want to find only PowerPoint presentations related to internet marketing.
Example: "internet marketing" filetype:ppt
6. This OR That:
By default, when you do a search, Google will include all the terms specified in the search. If you are looking for any one of one or more terms to match, then you can use the OR operator. (Note: The OR has to be capitalized).
Example: internet marketing OR advertising
7. Phone Listing:
Let’s say someone calls you on your mobile number and you don’t know how it is. If all you have is a phone number, you can look it up on Google using the phonebook feature.
Example: phonebook:617-555-1212 (note: the provided number does not work – you’ll have to use a real number to get any results).
8. Area Code Lookup:
If all you need to do is to look-up the area code for a phone number, just enter the 3-digit area code and Google will tell you where it’s from.
Example: 617
9. Numeric Ranges:
This is a rarely used, but highly useful tip. Let’s say you want to find results that contain any of a range of numbers. You can do this by using the X..Y modifier (in case this is hard to read, what’s between the X and Y are two periods. This type of search is useful for years (as shown below), prices or anywhere where you want to provide a series of numbers.
Example: president 1940..1950
10. Stock (Ticker Symbol):
Just enter a valid ticker symbol as your search term and Google will give you the current financials and a quick thumb-nail chart for the stock.
Example: GOOG
11. Calculator:
The next time you need to do a quick calculation, instead of bringing up the Calculator applet, you can just type your expression in to Google.
Example: 48512 * 1.02
12. Word Definitions:
If you need to quickly look up the definition of a word or phrase, simply use the "define:" command.
Example: define:plethora
Hope this list of Google search tips proves useful in your future Google searches. If there are any of your favorite Google expert power tips that I’ve missed, please feel free to share them in the comments.
Google search basics tricks
Search is simple: just type whatever comes to mind in the search box, hit Enter or click on the Google Search button, and Google will search the web for pages that are relevant to your query.
Most of the time you'll find exactly what you were looking for with just a basic query. However the following tips can help you refine your technique to make the most of your searches. Throughout the article, we'll use square brackets [ ] to signal queries, so [ black and white ] is one query, while [ black ] and [ white ] are two.
Some basic facts
* Every word matters. Generally, all the words you put in the query will be used. There are some exceptions.
* Search is always case insensitive. Searching for [ new york times ] is the same as searching for [ New York Times ].
* With some exceptions, punctuation is ignored (that is, you can't search for @#$%^&*()=+[]\ and other special characters).
Guidelines for better search
* Keep it simple. If you're looking for a particular company, just enter its name, or as much of its name as you can recall. If you're looking for a particular concept, place, or product, start with its name. If you're looking for a pizza restaurant, just enter pizza and the name of your town or your zip code. Most queries do not require advanced operators or unusual syntax. Simple is good.
* Think how the page you are looking for will be written. A search engine is not a human, it is a program that matches the words you give to pages on the web. Use the words that are most likely to appear on the page. For example, instead of saying [ my head hurts ], say [ headache ], because that's the term a medical page will use. The query [ in what country are bats considered an omen of good luck? ] is very clear to a person, but the document that gives the answer may not have those words. Instead, use the query [ bats are considered good luck in ] or even just [ bats good luck ], because that is probably what the right page will say.
* Describe what you need with as few terms as possible. The goal of each word in a query is to focus it further. Since all words are used, each additional word limits the results. If you limit too much, you will miss a lot of useful information. The main advantage to starting with fewer keywords is that, if you don't get what you need, the results will likely give you a good indication of what additional words are needed to refine your results on the next search. For example, [ weather cancun ] is a simple way to find the weather and it is likely to give better results than the longer [ weather report for cancun mexico ].
* Choose descriptive words. The more unique the word is the more likely you are to get relevant results. Words that are not very descriptive, like 'document,' 'website,' 'company,' or 'info,' are usually not needed. Keep in mind, however, that even if the word has the correct meaning but it is not the one most people use, it may not match the pages you need. For example, [ celebrity ringtones ] is more descriptive and specific than [ celebrity sounds ].
How to read search results
Google's goal is to provide you with results that are clear and easy to read. The diagram below points out four features that are important to understanding the search results page:
search results
1. The title: The first line of any search result is the title of the webpage.
2. The snippet: A description of or an excerpt from the webpage.
3. The URL: The webpage's address.
4. Cached link: A link to an earlier version of this page. Click here if the page you wanted isn't available.
All these features are important in determining whether the page is what you need. The title is what the author of the page designated as the best short description of the page.
The snippet is Google's algorithmic attempt to extract just the part of the page most relevant to your query. The URL tells you about the site in general.
Most of the time you'll find exactly what you were looking for with just a basic query. However the following tips can help you refine your technique to make the most of your searches. Throughout the article, we'll use square brackets [ ] to signal queries, so [ black and white ] is one query, while [ black ] and [ white ] are two.
Some basic facts
* Every word matters. Generally, all the words you put in the query will be used. There are some exceptions.
* Search is always case insensitive. Searching for [ new york times ] is the same as searching for [ New York Times ].
* With some exceptions, punctuation is ignored (that is, you can't search for @#$%^&*()=+[]\ and other special characters).
Guidelines for better search
* Keep it simple. If you're looking for a particular company, just enter its name, or as much of its name as you can recall. If you're looking for a particular concept, place, or product, start with its name. If you're looking for a pizza restaurant, just enter pizza and the name of your town or your zip code. Most queries do not require advanced operators or unusual syntax. Simple is good.
* Think how the page you are looking for will be written. A search engine is not a human, it is a program that matches the words you give to pages on the web. Use the words that are most likely to appear on the page. For example, instead of saying [ my head hurts ], say [ headache ], because that's the term a medical page will use. The query [ in what country are bats considered an omen of good luck? ] is very clear to a person, but the document that gives the answer may not have those words. Instead, use the query [ bats are considered good luck in ] or even just [ bats good luck ], because that is probably what the right page will say.
* Describe what you need with as few terms as possible. The goal of each word in a query is to focus it further. Since all words are used, each additional word limits the results. If you limit too much, you will miss a lot of useful information. The main advantage to starting with fewer keywords is that, if you don't get what you need, the results will likely give you a good indication of what additional words are needed to refine your results on the next search. For example, [ weather cancun ] is a simple way to find the weather and it is likely to give better results than the longer [ weather report for cancun mexico ].
* Choose descriptive words. The more unique the word is the more likely you are to get relevant results. Words that are not very descriptive, like 'document,' 'website,' 'company,' or 'info,' are usually not needed. Keep in mind, however, that even if the word has the correct meaning but it is not the one most people use, it may not match the pages you need. For example, [ celebrity ringtones ] is more descriptive and specific than [ celebrity sounds ].
How to read search results
Google's goal is to provide you with results that are clear and easy to read. The diagram below points out four features that are important to understanding the search results page:
search results
1. The title: The first line of any search result is the title of the webpage.
2. The snippet: A description of or an excerpt from the webpage.
3. The URL: The webpage's address.
4. Cached link: A link to an earlier version of this page. Click here if the page you wanted isn't available.
All these features are important in determining whether the page is what you need. The title is what the author of the page designated as the best short description of the page.
The snippet is Google's algorithmic attempt to extract just the part of the page most relevant to your query. The URL tells you about the site in general.
GOOGLE PAGE RANKING ALGORITHM
The PageRank Algorithm
The original PageRank algorithm was described by Lawrence Page and Sergey Brin in several publications. It is given by
PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))
where
PR(A) is the PageRank of page A,
PR(Ti) is the PageRank of pages Ti which link to page A,
C(Ti) is the number of outbound links on page Ti and
d is a damping factor which can be set between 0 and 1.
So, first of all, we see that PageRank does not rank web sites as a whole, but is determined for each page individually. Further, the PageRank of page A is recursively defined by the PageRanks of those pages which link to page A.
The PageRank of pages Ti which link to page A does not influence the PageRank of page A uniformly. Within the PageRank algorithm, the PageRank of a page T is always weighted by the number of outbound links C(T) on page T. This means that the more outbound links a page T has, the less will page A benefit from a link to it on page T.
The weighted PageRank of pages Ti is then added up. The outcome of this is that an additional inbound link for page A will always increase page A's PageRank.
Finally, the sum of the weighted PageRanks of all pages Ti is multiplied with a damping factor d which can be set between 0 and 1. Thereby, the extend of PageRank benefit for a page by another page linking to it is reduced.
The Random Surfer Model
In their publications, Lawrence Page and Sergey Brin give a very simple intuitive justification for the PageRank algorithm. They consider PageRank as a model of user behaviour, where a surfer clicks on links at random with no regard towards content.
The random surfer visits a web page with a certain probability which derives from the page's PageRank. The probability that the random surfer clicks on one link is solely given by the number of links on that page. This is why one page's PageRank is not completely passed on to a page it links to, but is devided by the number of links on the page.
So, the probability for the random surfer reaching one page is the sum of probabilities for the random surfer following links to this page. Now, this probability is reduced by the damping factor d. The justification within the Random Surfer Model, therefore, is that the surfer does not click on an infinite number of links, but gets bored sometimes and jumps to another page at random.
The probability for the random surfer not stopping to click on links is given by the damping factor d, which is, depending on the degree of probability therefore, set between 0 and 1. The higher d is, the more likely will the random surfer keep clicking links. Since the surfer jumps to another page at random after he stopped clicking links, the probability therefore is implemented as a constant (1-d) into the algorithm. Regardless of inbound links, the probability for the random surfer jumping to a page is always (1-d), so a page has always a minimum PageRank.
A Different Notation of the PageRank Algorithm
Lawrence Page and Sergey Brin have published two different versions of their PageRank algorithm in different papers. In the second version of the algorithm, the PageRank of page A is given as
PR(A) = (1-d) / N + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))
where N is the total number of all pages on the web. The second version of the algorithm, indeed, does not differ fundamentally from the first one. Regarding the Random Surfer Model, the second version's PageRank of a page is the actual probability for a surfer reaching that page after clicking on many links. The PageRanks then form a probability distribution over web pages, so the sum of all pages' PageRanks will be one.
Contrary, in the first version of the algorithm the probability for the random surfer reaching a page is weighted by the total number of web pages. So, in this version PageRank is an expected value for the random surfer visiting a page, when he restarts this procedure as often as the web has pages. If the web had 100 pages and a page had a PageRank value of 2, the random surfer would reach that page in an average twice if he restarts 100 times.
As mentioned above, the two versions of the algorithm do not differ fundamentally from each other. A PageRank which has been calculated by using the second version of the algorithm has to be multiplied by the total number of web pages to get the according PageRank that would have been caculated by using the first version. Even Page and Brin mixed up the two algorithm versions in their most popular paper "The Anatomy of a Large-Scale Hypertextual Web Search Engine", where they claim the first version of the algorithm to form a probability distribution over web pages with the sum of all pages' PageRanks being one.
In the following, we will use the first version of the algorithm. The reason is that PageRank calculations by means of this algorithm are easier to compute, because we can disregard the total number of web pages.
The Characteristics of PageRank
The characteristics of PageRank shall be illustrated by a small example.
We regard a small web consisting of three pages A, B and C, whereby page A links to the pages B and C, page B links to page C and page C links to page A. According to Page and Brin, the damping factor d is usually set to 0.85, but to keep the calculation simple we set it to 0.5. The exact value of the damping factor d admittedly has effects on PageRank, but it does not influence the fundamental principles of PageRank. So, we get the following equations for the PageRank calculation:
PR(A) = 0.5 + 0.5 PR(C)
PR(B) = 0.5 + 0.5 (PR(A) / 2)
PR(C) = 0.5 + 0.5 (PR(A) / 2 + PR(B))
These equations can easily be solved. We get the following PageRank values for the single pages:
PR(A) = 14/13 = 1.07692308
PR(B) = 10/13 = 0.76923077
PR(C) = 15/13 = 1.15384615
It is obvious that the sum of all pages' PageRanks is 3 and thus equals the total number of web pages. As shown above this is not a specific result for our simple example.
For our simple three-page example it is easy to solve the according equation system to determine PageRank values. In practice, the web consists of billions of documents and it is not possible to find a solution by inspection.
The Iterative Computation of PageRank
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.
Iteration PR(A) PR(B) PR(C)
0 1 1 1
1 1 0.75 1.125
2 1.0625 0.765625 1.1484375
3 1.07421875 0.76855469 1.15283203
4 1.07641602 0.76910400 1.15365601
5 1.07682800 0.76920700 1.15381050
6 1.07690525 0.76922631 1.15383947
7 1.07691973 0.76922993 1.15384490
8 1.07692245 0.76923061 1.15384592
9 1.07692296 0.76923074 1.15384611
10 1.07692305 0.76923076 1.15384615
11 1.07692307 0.76923077 1.15384615
12 1.07692308 0.76923077 1.15384615
We see that we get a good approximation of the real PageRank values after only a few iterations. According to publications of Lawrence Page and Sergey Brin, about 100 iterations are necessary to get a good approximation of the PageRank values of the whole web.
Also, by means of the iterative calculation, the sum of all pages' PageRanks still converges to the total number of web pages. So the average PageRank of a web page is 1. The minimum PageRank of a page is given by (1-d). Therefore, there is a maximum PageRank for a page which is given by dN+(1-d), where N is total number of web pages. This maximum can theoretically occur, if all web pages solely link to one page, and this page also solely links to itself.
The Implementation of PageRank
The original PageRank algorithm was described by Lawrence Page and Sergey Brin in several publications. It is given by
PR(A) = (1-d) + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))
where
PR(A) is the PageRank of page A,
PR(Ti) is the PageRank of pages Ti which link to page A,
C(Ti) is the number of outbound links on page Ti and
d is a damping factor which can be set between 0 and 1.
So, first of all, we see that PageRank does not rank web sites as a whole, but is determined for each page individually. Further, the PageRank of page A is recursively defined by the PageRanks of those pages which link to page A.
The PageRank of pages Ti which link to page A does not influence the PageRank of page A uniformly. Within the PageRank algorithm, the PageRank of a page T is always weighted by the number of outbound links C(T) on page T. This means that the more outbound links a page T has, the less will page A benefit from a link to it on page T.
The weighted PageRank of pages Ti is then added up. The outcome of this is that an additional inbound link for page A will always increase page A's PageRank.
Finally, the sum of the weighted PageRanks of all pages Ti is multiplied with a damping factor d which can be set between 0 and 1. Thereby, the extend of PageRank benefit for a page by another page linking to it is reduced.
The Random Surfer Model
In their publications, Lawrence Page and Sergey Brin give a very simple intuitive justification for the PageRank algorithm. They consider PageRank as a model of user behaviour, where a surfer clicks on links at random with no regard towards content.
The random surfer visits a web page with a certain probability which derives from the page's PageRank. The probability that the random surfer clicks on one link is solely given by the number of links on that page. This is why one page's PageRank is not completely passed on to a page it links to, but is devided by the number of links on the page.
So, the probability for the random surfer reaching one page is the sum of probabilities for the random surfer following links to this page. Now, this probability is reduced by the damping factor d. The justification within the Random Surfer Model, therefore, is that the surfer does not click on an infinite number of links, but gets bored sometimes and jumps to another page at random.
The probability for the random surfer not stopping to click on links is given by the damping factor d, which is, depending on the degree of probability therefore, set between 0 and 1. The higher d is, the more likely will the random surfer keep clicking links. Since the surfer jumps to another page at random after he stopped clicking links, the probability therefore is implemented as a constant (1-d) into the algorithm. Regardless of inbound links, the probability for the random surfer jumping to a page is always (1-d), so a page has always a minimum PageRank.
A Different Notation of the PageRank Algorithm
Lawrence Page and Sergey Brin have published two different versions of their PageRank algorithm in different papers. In the second version of the algorithm, the PageRank of page A is given as
PR(A) = (1-d) / N + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))
where N is the total number of all pages on the web. The second version of the algorithm, indeed, does not differ fundamentally from the first one. Regarding the Random Surfer Model, the second version's PageRank of a page is the actual probability for a surfer reaching that page after clicking on many links. The PageRanks then form a probability distribution over web pages, so the sum of all pages' PageRanks will be one.
Contrary, in the first version of the algorithm the probability for the random surfer reaching a page is weighted by the total number of web pages. So, in this version PageRank is an expected value for the random surfer visiting a page, when he restarts this procedure as often as the web has pages. If the web had 100 pages and a page had a PageRank value of 2, the random surfer would reach that page in an average twice if he restarts 100 times.
As mentioned above, the two versions of the algorithm do not differ fundamentally from each other. A PageRank which has been calculated by using the second version of the algorithm has to be multiplied by the total number of web pages to get the according PageRank that would have been caculated by using the first version. Even Page and Brin mixed up the two algorithm versions in their most popular paper "The Anatomy of a Large-Scale Hypertextual Web Search Engine", where they claim the first version of the algorithm to form a probability distribution over web pages with the sum of all pages' PageRanks being one.
In the following, we will use the first version of the algorithm. The reason is that PageRank calculations by means of this algorithm are easier to compute, because we can disregard the total number of web pages.
The Characteristics of PageRank
The characteristics of PageRank shall be illustrated by a small example.
We regard a small web consisting of three pages A, B and C, whereby page A links to the pages B and C, page B links to page C and page C links to page A. According to Page and Brin, the damping factor d is usually set to 0.85, but to keep the calculation simple we set it to 0.5. The exact value of the damping factor d admittedly has effects on PageRank, but it does not influence the fundamental principles of PageRank. So, we get the following equations for the PageRank calculation:
PR(A) = 0.5 + 0.5 PR(C)
PR(B) = 0.5 + 0.5 (PR(A) / 2)
PR(C) = 0.5 + 0.5 (PR(A) / 2 + PR(B))
These equations can easily be solved. We get the following PageRank values for the single pages:
PR(A) = 14/13 = 1.07692308
PR(B) = 10/13 = 0.76923077
PR(C) = 15/13 = 1.15384615
It is obvious that the sum of all pages' PageRanks is 3 and thus equals the total number of web pages. As shown above this is not a specific result for our simple example.
For our simple three-page example it is easy to solve the according equation system to determine PageRank values. In practice, the web consists of billions of documents and it is not possible to find a solution by inspection.
The Iterative Computation of PageRank
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.
Iteration PR(A) PR(B) PR(C)
0 1 1 1
1 1 0.75 1.125
2 1.0625 0.765625 1.1484375
3 1.07421875 0.76855469 1.15283203
4 1.07641602 0.76910400 1.15365601
5 1.07682800 0.76920700 1.15381050
6 1.07690525 0.76922631 1.15383947
7 1.07691973 0.76922993 1.15384490
8 1.07692245 0.76923061 1.15384592
9 1.07692296 0.76923074 1.15384611
10 1.07692305 0.76923076 1.15384615
11 1.07692307 0.76923077 1.15384615
12 1.07692308 0.76923077 1.15384615
We see that we get a good approximation of the real PageRank values after only a few iterations. According to publications of Lawrence Page and Sergey Brin, about 100 iterations are necessary to get a good approximation of the PageRank values of the whole web.
Also, by means of the iterative calculation, the sum of all pages' PageRanks still converges to the total number of web pages. So the average PageRank of a web page is 1. The minimum PageRank of a page is given by (1-d). Therefore, there is a maximum PageRank for a page which is given by dN+(1-d), where N is total number of web pages. This maximum can theoretically occur, if all web pages solely link to one page, and this page also solely links to itself.
The Implementation of PageRank
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