Consider the directed network $G=(V,E)$ with $N=5$ nodes and $L=8$ links, in which

node $1$ points to nodes $2$ and $3,$ node $2$ points to node $4,$ node $3$ points to nodes $2$ and

$4,$ node $4$ points to node $2,$ and node $5$ points to nodes $3$ and $4.$

$($a$)$ Draw the network and write down its adjacency matrix $A.$

$($b$)$ How many weakly$-$connected components and how many non$-$trivial $($i$.$e$.$ with

more than one node$)$ strongly$-$connected components are there in the network?

List all the nodes belonging to each one of these components. List all the nodes

belonging, respectively, to the in$-$component and the out$-$component of each of the

non$-$trivial strongly$-$connected components.

$($c$)$ Determine the in$-$degree sequence $\{k_1^{in},k_2^{in},k_3^{in},k_4^{in},k_5^{in}\}$ and the out$-$degree

sequence $\{k_1^{\text{o}\text{u}\text{t}},k_2^{\text{o}\text{u}\text{t}},k_3^{\text{o}\text{u}\text{t}},k_4^{\text{o}\text{u}\text{t}},k_5^{\text{o}\text{u}\text{t}}\}$ of the network. Write down the average node

in$-$degree, the average node out$-$degree, the node in$-$degree distribution $P^{in}(k)$ and

the node out$-$degree distribution $P^{\text{o}\text{u}\text{t}}(k).$

$($d$)$ Calculate the normalised in$-$degree centrality $x_i$ of each node of the network and

rank the nodes, from the most to the least central, according to their in$-$degree

centrality.

$($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?

$($f$)$ State the definition of the Katz centrality. Calculate the Katz centrality $x_i$ of each

node of the network and rank the nodes, from the most to the least central,

according to their Katz centrality. Do you get a different ranking from that

obtained based on the eigenvector centrality? Explain why.

$($g$)$ Calculate the $NN$ matrix $d$ whose element $d_{ij}$ is the length of the shortest

paths from node $i$ to node $j.$ Calculate the efficiency centrality $x_i$ of each node of

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

their efficiency centrality.