Question c only complex networksConsider 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 $t-1.$

$($a$)$ Find an expression for the number of nodes, $N(t),$ and the number of links, $L(t),$ in the network as a function of time $t.$ Find an expression for the value of $Z$ as a function of time $t.$

$($b$)$ What is the average node degree $($:$k$:$)$ at time $t?$ What is the average node degree in the limit $t?$

$($c$)$ Write down the differential equation governing the time evolution of the degree $k_i$ of node i for $t1$ in the mean$-$field approximation. Solve this equation with the initial condition $k_i(t_i)=m,$ where $t_i$ is the time of arrival of node $i.$

$7$