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 $i,j$ set $$ a_{ij} = 1/d(i) $$ if the outdegree of vertex $i, d(i)$ is positive and there is an arc from $i$ to $j$ in $D$ Set $$ a_{ij} = 0 $$ if $d(i) >0$ but there is no arc from $i$ to $j$ in $D$

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

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