Probability Distribution I'm thinking about a set of n users on Facebook. Between each of the $\binom{n}{2}$ pairs of distinct friends, lets say an edge (indicating that the two people are friends) is independently present with probability $p$. 
Lets define Friend Count $(FC)$ of a person $i$ (e.g. $FC(i)$) is the number of edges that are connected to person $i$.
I’m trying to determine what the distribution form for $FC(i)$ would be… e.g. geometric, poisson, normal, uniform, binomial etc and after that defining the correlation between $FC(i)$ and $FC(j)$?
Would love any help/thoughts 
 A: There are $n-1$ edges available to person # $i$.   There is an independent probability $p$ of any of these being connected.   This description indicates it is a binomial distribution.
$$\begin{align}
\mathsf E(C_i) & = (n-1)\,p 
\\[2ex]
 \mathsf {Var}(C_i) & = (n-1)\,p\,(1-p)
\end{align}$$
The interdependence of friend counts between persons $i$ and $j$ will be due to the edge between them; all other edges are independent.   We then use the Law of Total Expectation to partition on this eventuality.
$$\begin{align}
\mathsf {Cov}(C_i, C_j) 
& =\mathsf E(C_i C_j) - \mathsf E(C_i)\,\mathsf E(C_j)
\\[1ex]
& = p\,\mathsf E(C_i C_j\mid e_{i,j}=1) + (1-p)\,\mathsf E(C_i C_j\mid e_{i,j}=0) - \mathsf E(C_i)\,\mathsf E(C_j)
\\[1ex] & = p\,((n-2)\,p+1)^2 + (1-p)\,((n-2)\,p)^2 - ((n-1)\,p)^2
\end{align}$$
Simplify, then calculate the correlation.
A: Let's say we have $n$ total users, and we're trying to find $FC(i)$ for a fixed $i$.  For each of the $n-1$ other users---let's say user $j$---there is a probability of $p$ that user $i$ and $j$ are friends.  Since every edge is added independently, for any given $k \in \{1,2,\ldots,n-1\}$, we have $$ \mathbb{P}[FC(i) = k] = \binom{n-1}{k}p^k(1 - p)^{n-k-1} $$
since we can choose the $k$ friends for $i$ to be friends with, and $p^k(1 - p)^{n-k-1}$ is the probability that user $i$ is friends with only these users.  Thus, the distribution of $FC(i) \sim B(n-1,p)$, i.e. $FC(i)$ is binomial.

By the way, what you're considering is a random graph on $n$ nodes, where each edge is added independently with probability $p$.  This is commonly notated as $G(n,p)$.  Your $FC(i)$ term, in graph theory terminology, is called the degree of vertex $i$.
