Calculating PageRank of a Google Matrix Let $A$ the adjacency matrix of a Web Digraph, with $\{0,1\}$ entries. For sake of clarity, we assume that the matrix is irreducible and without full-zero rows (i.e. no leaf nodes in the graph) . Every page $p_{i}$ has $d(i)$ outlinks towards others pages and its rank $w_{i}$ is distribuited uniformly among them according this rule:
if $p_{i}\rightarrow p_{j}$ then $w_{j}$ (rank of page $p_{j}$) gains a fraction $\tfrac{1}{d(i)}$ of $w_{i}$.
Thus, for each of pages $p_{j}$ we have 
$$w_{j}=\sum_{i\in \textit{inlinks}(j)} \tfrac{1}{d(i)}w_{i}$$
where $\textit{inlinks}(j)=\{i\,|\,p_{i}\rightarrow p_{j}\}$.
It's easy to see that if a page has got autolinks, its rank gains a surplus despite of the case it has not.
For example
$$\text{if  }A=\begin{pmatrix}
1&1&0\\
0&0&1\\
1&0&0
\end{pmatrix}
\Rightarrow \text{a solution is } w_{1}=2 \, , \, w_{2}=w_{3}=1
$$
and 
$$ \text{if }
B=\begin{pmatrix}
0&1&0\\
0&0&1\\
1&0&0
\end{pmatrix}
\Rightarrow \text{a solution is } w_{1}=w_{2}=w_{3}=1
$$
My doubt: since we are considering a social model, where the "importance" of a page is increased by the rank of the others pages linked with, is a good ("democratic") choise not considering autolinks?
 A: 
My doubt: since we are considering a social model, where the "importance" of a page is increased by the rank of the others pages linked with, is a good ("democratic") choise not considering autolinks?

I think that this is not a mathematical question, because this choice mainly depends on the purpose of the model, on that which factors should it reflect, in particular, on the meaning of page rank. 
If we understand democraticity in the sense that the main importance is the public, then number of page autolinks contributes mainly to its self-importance, but not so much to public importance, so it is not so important to page rank. 
In similar estimations of scientific importance, like number of quotations of the paper, there is a problem of artificial surplus of the rank, caused by autoquotations, so, sometimes, autoquotations are excluded from the calculation of the rank. But this doesn’t not exclude a bit more subtle artificial surplus which can be created by cliques quoting themselves. 
As a citizen of Soviet Union I remember well-known words of our famous poet Vladimir Mayakovskiy, that nobody cares about a single person and listens his opinion, but a party (of course, the communist one, we did not have an other) is a hand with millions of fingers, clenched in a united crushing fist. A famous Russian writer Lev Tolstoy was very morally sensitive and said even more, that a human is a fraction where the numerator is what he is and the denominator is what he thinks about himself.
