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Google matrix

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Google's PageRank algorithm uses a particular stochastic matrix called the Google matrix. 
The purpose of the PageRank algorithm is to compute a stationary vector of the Google matrix.
The stationary vector is then used to provide a ranking of the pages on the internet.

A directed graph $D$ is constructed whose vertices correspond to web pages and a directed
arc from vertex $i$ to vertex $j$ exists if and only if page $i$ has a link out to 
page $j$. 
Then a stochastic matrix $A=(a_{ij})$ is constructed from $D$: for each $i,j$
a_{ij} = 1/d(i)
if the outdegree of vertex $i, d(i)$ is positive and there is an arc from $i$ to $j$ in $D$.
a_{ij} = 0
if $d(i) >0$  but there is no arc from $i$ to $j$ in $D$.

a_{ij} = 1/n
if $d(i) = 0$, where $n$ is the order of the matrix. 

Having defined $A$ choose a positive row vector $v^T$ such that $v^T\textbf{1} = 1$
where $\textbf{1}$ is a vector of all ones.
Finally, choose a constant $c \in (0,1)$.
The \emph{Google matrix} $G$
G = cA + (1-c)\textbf{1}v^T .
Clearly, $G$ is stochastic. For the actual matrix that Google uses $c$ is about .85.