Google matrix

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 directedarc from vertex $i$ to vertex $j$ exists if and only if page $i$ has a link out topage $j$.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 $v^T\text{𝟏}=1$where 1 is a vector of all ones.Finally, choose a constant $c\in (0,1)$.The Google matrix $G$is

$$G=cA+(1-c)\text{𝟏}{v}^T.$$

Clearly, $G$ is stochastic. For the actual matrix that Google uses $c$ is about .85.