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

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

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).