 # Find the correlation between D_{i} and D_{j} Livia Cardenas 2022-07-23 Answered
Consider a graph having $n=8$
vertices labeled $1,2,...,8$ . Suppose that each edge is independently present with probability p. The degree of vertex i, designated as ${D}_{i}$ , is the number of edges that have vertex i as one of its vertices. Find $Corr\left({D}_{i},{D}_{j}\right)$ , the correlation between ${D}_{i}$ and ${D}_{j}$ .
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Step 1
$\begin{array}{rl}\rho \left({D}_{i},{D}_{j}\right)& =\frac{Cov\left({D}_{i},{D}_{j}\right)}{\sqrt{Var\left({D}_{i}\right)Var\left({D}_{j}\right)}}\\ & =\frac{Var\left({I}_{ij}\right)}{\sqrt{Var\left({D}_{i}\right)Var\left({D}_{j}\right)}}\\ & =\frac{p\left(1-p\right)}{\sqrt{7p\left(1-p\right)\cdot 7p\left(1-p\right)}}\\ & =\frac{1}{7}\end{array}$
The second equality comes from the fact that the covariance between edges is 0 unless the edge is the same. The third: ${I}_{ij}$ is bernoulli with parameter p, and ${D}_{i}$ and ${D}_{j}$ are binomial with parameters 7 and p.
###### Not exactly what you’re looking for? makaunawal5
Step 1
Here's a sketch; if I have time, I'll complete this later.
I denote $I={I}_{ij}$.
Let ${E}_{ij}$ be the edge from vertex i to j. We know that . In fact, we know that there are $7+6+\cdots +1=28$ possible edges, so that