Consider the directed network $G=(V,E)$ with $N=7$ nodes and $L=9$ links, in which

node $1$ points to nodes $2$ and $3$ ; node $2$ points to nodes $1,6$ and $7$ ; node $3$ points to

nodes $4$ and $5$ ; node $4$ points to node $3$ ; node $6$ points to node $2.$

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

directed or undirected?

$($b$)$ How many weakly$-$connected components and how many strongly$-$connected

components are there in the network? List the nodes belonging to each of these

components.

$($c$)$ List the nodes in the out$-$component relative to each strongly$-$connected

component of the network. List the nodes in the in$-$component relative to each

strongly$-$connected component of the network. Does the network have tendrils?

$($d$)$ Consider the undirected network constructed from the network with adjacency

matrix A by neglecting the direction of the links. Calculate the efficiency of node

$1$ and of node $2$ on this undirected network.