The following definitions attempt to capture the basic principles for the ranking of Web pages. (The following notation is similar to the original papers, except that we index Web pages by integers rather than consider the Web pages symbollically. This adjustment seems to fit well with the mathematics that follows.)

Let

• W_i be the collection of all accessible pages on the World Wide Web (where i = 1, ..., N with N being the number of these pages).
• R_i is the PageRank (importance) of page W_i.
• N_j is the total number of links from page W_j to other pages.
• A_i,j is 1 / N_j if page W_j has a link to page W_i, and 0 otherwise.
  (Jargon: If page W_j has a link to page W_i, then W_j is said to be a backlink of W_i.)
• c be a positive real number (used for normalization as described later).

[Simplified] PageRank is now defined as satisfying these properties:

•  

  Ri =

  N Σ j=1

  c (Ai,j * Rj)

• and Column vector R (= (R_i) has L₁ norm 1:   ||R|| = 1

The Simplified Vector Equation

Note that (A_i,j) is a modified adjacency matrix. With this interpretation, the PageRank vector R is a unit vector that satisfies the vector equation

Do you recognize this type of equation?

created: 10 February 2007 last revised: 5 March 2007

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