A curious asymmetry:

Say you have an undirected graph with at least one edge {a, b} such that degree(a) != degree(b).

If you pick a random edge, {u, v}, E(degree(u)) = E(degree(v)).

However, if you pick a random node first, u, and *then* pick a random neighbor, v, of u, E(degree(u)) < E(degree(v)).