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.