Consider a node i that is in a community by itself, and another existing community C with degrees and weights as shown below. Prove that the modularity gain when i merges with $C$ can be given by the following equation

$Q=[\frac{{}_{in}+k_{i,in}}{2m}-(\frac{{}_{\text{t}\text{o}\text{t}}+k_i}{2m}{)}^2]-[\frac{{}_{in}}{2m}-(\frac{{}_{\text{t}\text{o}\text{t}}}{2m}{)}^2-(\frac{k_i}{2m}{)}^2]$

For the network above, We would like to classify nodes into $2$ classes $"+"$ and $"-.$ Labels for node $3,5,8$ and $10$ are given $($red for $"+",$ blue for $"-").$ By using the following relational classifier:

$(P(Y_i=c)=\frac{1}{|N_i|}{\displaystyle \underset{(i)}{\overset{?}{}}},$jinEW$(i,j)P(Y_j=c)$

Assuming all edge weights are equal to $1,$ an unbiased initialization of the unlabeled nodes, and an ordering according to the node ID$,$ show the following $($illustrating your steps$)$:

A$.$ The probability distribution of the positive class over the unlabeled nodes at the second iteration of ICA

B$.$ Assuming at each step a threshold of $0\mathrm{.}5$ is used to determine whether the node is $"+"$ or $"-",$ show the final result of ICA algorithm and the class