Question f and g only please 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 $i.$

$($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 $?$

$($e$)$ Write down the master equation of the model, i$.$e$.$ the equation that describes the evolution of the average number $N_k(t)$ of nodes that at time $t$ have degree $k.$

$($f$)$ Suppose now that you iterate the growing process for a finite number of time steps, until you produce a final network with $N={10}^6$ nodes $($and a minimum degree $m=2).$ Denote as $K$ the natural cutoff in the network. Treating the degree $k$ as a continuous variable, evaluate the natural cutoff $K,$ the normalisation constant in the degree distribution, the average degree $($:$k$:$),$ and $($:$k^2$:$).$

$($g$)$ Calculate the probability of finding a node with $1000$ links in the network obtained in point $($f$).$ Calculate the probability of finding a node with $1000$ links in a Erd$$s$-$R$$nyi random graphs with the same number of nodes and links as in the network obtained in point $($f$).$

$9$