Consider the following model to grow simple networks. At time $t=1$ we start with a

complete network with $n_0=6$ nodes. At each time step $t>1$ a new node is added to

the network. The node arrives together with $m=2$ new links, which are connected to

$m=2$ different nodes already present in the network. The probability ${}_i$ that a new

link is connected to node $i$ is:

${}_i=\frac{k_i-1}{Z},$ with $Z={\displaystyle \underset{j=1}{\overset{N(t-1)}{}}}(k_j-1)$

where $k_i$ is the degree of node $i,$ and $N(t-1)$ is the number of nodes in the network at

time $t-1.$

$($d$)$ Derive the degree distribution $P(k)$ of the network for $t1$ in the mean$-$field

approximation. Does the model produce scale$-$free networks? If so$,$ what is the

value of the degree exponent $?$