Consider the network with $N=4$ nodes and adjacency matrix A given by

$A=([0,1,1,1],[1,0,1,1],[0,0,0,0],[0,0,0,0])$

$($a$)$ Calculate the number of links in the network and draw the network. Is the

network directed or undirected?

$($b$)$ Is the network weakly connected? Is the network strongly connected? List the

nodes belonging to each of the strongly$-$connected components in the network.

$($c$)$ Calculate the in$-$ and the out$-$degree centrality $x_i$ of each node $i=1,2,$dots,$N$ in

the network and rank the nodes, from the most to the least central, according to

their in$-$degree centrality.

$($d$)$ Calculate the eigenvector centrality $x_i$ of each node $i=1,2,$dots,$N$ of the network

and rank the nodes, from the most to the least central, according to their

eigenvector centrality. To this end 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,$N.$ Consider

the iteration

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

for ninN.

Finally, calculate the eigenvector centrality $x_i$ of each node $i$ of the network by

finding the limit

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

$($e$)$ State the definition of the PageRank centrality. Calculate the PageRank

centrality $x_i$ of each node $i=1,2,$dots,$N$ of the network and rank the nodes, from

the most to the least central, according to their PageRank centrality.