Exercise $7.$ A graph is constructed in the following way: $2n-$complete cliques are first created

$($each containing $n-$nodes, with every node connected to all $n-1$ nodes in the same clique$).$ We

then consider all possible edges between a single node in clique $1,$ and a single node in clique $2.$

Each such edge is added to the graph with the same probability $p.$

$($a$)$ What is the average degree of a node within this network?

$($b$)$ Calculate the total expected number of edges, $m,$ in the network.

$($c$)$ Calculate the expected modularity score for the network, if the two different cliques are

assigned to two different communities. Recall the formula for modularity is:

$Q=\frac{1}{2m}{\displaystyle \underset{i}{\overset{?}{}}},j(A_{ij}-\frac{k_i{k}_j}{2m}){}_{g_i{g}_j}$