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 $4.$

$($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.