On the Wikipedia page of Friendship Paradox (https://en.wikipedia.org/wiki/Friendship_paradox), there's a paragraph saying "The average number of friends that a typical friend has can be modeled by choosing a random person (who has at least one friend), and then calculating how many friends their friends have on average. This amounts to choosing, uniformly at random, an edge of the graph (representing a pair of friends) and an endpoint of that edge (one of the friends), and again calculating the degree of the selected endpoint. ". However, I found that choosing a vertex uniformly at random (say it's $v$) then choosing a neighbor of $v$ at random, is not the same as choosing an edge uniformly at random.
See the graph above. If we choose a node uniformly at random and then choose a neighbor of it, the probability of choosing node $B$ is $\frac12$. But if we choose an edge uniformly at random and then choose an endpoint of that edge, then the probability of choosing node $B$ is $\frac38$. Actually, these two different random process give two different expectations (if I did them correctly): $$\mathbb{E}_{(u,v)\in E}\ \mathrm{deg}(v)=2.25 $$ $$\mathbb{E}_{u\in V, v\in N(u)} \mathrm{deg}(v)=\frac{29}{12}\approx2.42$$ where $N(u)$ is the set of neighbors of $u$. These two numbers are both larger than $2$ which is the average number of neighbors of a random node, so it may not affect the conclusion. But I'm still confused about which way is actually the correct way to do?