Network/Directed connections/Same weight column stochastic matrix/Example

Let $N$ be a network (a "directed graph“), consisting in a set $K$ of vertices, and a set of directed edges, which can exist between the vertices. For example, $K$ is the set of all web pages, and there exists an arrow from $j\in K$ to $i\in K$, if the web page $j$ has a link to the web page $i$. The linking structure can be expressed by the adjacency matrix

$A=\left(a_{ij}\right),$

where

$a_{ij}:=\{\begin{array}{c}1,\text{there exists a link from }j\text{ to }i,\\ 0,\text{ else},\end{array},$

or by the column stochastic matrix

$B=\left(b_{ij}\right),$

where

$b_{ij}=\frac{a_{ij}}{d_j},$

and $d_j$ is the number of links starting at the vertex $j$. This division ensures that the column sum equals $1$ (we suppose that there is at least one link starting at any vertex).