Let $G = (V, E)$ be an Erdős-Rényi graph, with $N = |V|$ nodes, $L = |E|$ edges. The distribution of the degree of any particular node is binomial (or Poisson under certain condition). Suppose that the average node degree is $\bar{k}$.
Fix a node $i$ in the graph with degree $k_i$. Then randomly pick another node $j$ which is in the neighborhood of $i$ (randomly = with uniform probability).
What can I say about the distribution of the degree of $j$ knowing that it is in the neighborhood of node $i$ with degree $k_i$?