$($e$)$ Calculate the eigenvector centrality $x_i$ of each node of the network and rank the

nodes, from the most to the least central, according to their eigenvector centrality.

To obtain the eigenvector centrality, start from the initial guess $x^{(0)}=\frac{1}{N}1$ where

$1$ is the $N-$dimensional column vector of elements $1_i=1$AAi$=1,2$dots,$N,$ and use

the following recursive rule

$x^{(n)}=A{x}^{(n-1)},$

where ninN. Finally calculate the eigenvector centrality $x_i$ of each node $i$ of the

network from the limit

$x_i=\underset{n}{lim}\frac{x_i^{(n)}}{{\displaystyle \underset{j=1}{\overset{N}{}}}{x}_j^{(n)}}.$

Can you obtain the same result by directly calculating eigenvalues and

eigenvectors of the adjacency matrix?