The [Friendship paradox][1] goes *most people have fewer friends than their friends have on average*. The original paper has a simple counter example: [![enter image description here][2]][2]

My question is how exceptional is this example? 

Even in some random garph models where some answer is tractable, the analysis of the [distribution of a neighbour's degree][3] deals with expectation. That is, in some models it is explicitly shown that *the average difference between the mean number of friends of friends and the number of friends is positive*, which is much less likely to get press attention I think.

Given any graph $G=(V,E)$ and $v\in V$, let $N(v) = \{w\in V:\{v,w\} \in E\}$ and $\operatorname{deg}(v) = |N(v)|$. This is the number of friends of $v$. Now consider $\operatorname{ndeg}(v)= \sum_{w \in N(v)}\operatorname{deg}(w)$, the total friends of friends of $v$ and the distribution of
$$
f(v) = \frac{\operatorname{ndeg}(v)}{\operatorname{deg}(v)} - \operatorname{deg}(v).
$$
For a friendship paradox,
$$
g = |\{v:f(v)>0\}|-|\{v:f(v)<0\}|>0.
$$
For the graph pictured above,
$$
\{f(v)\}_{v\in \{A,B,C,D,E,F\}}=\left\{ 1,1,-\frac13-\frac13,-\frac13,-\frac13 \right\}.
$$
Clearly $g = -2$, and no paradox. But, the mean of $f$ is $1/9>0$. 

The mean of $f$ is [shown to be positive][3] in the popular configuration model. And this is usually said to be evidence/proof of the friendship paradox. (Tangential question: is $g>0$ in this model?)

Now the question(s):

 1. Is the mean of $f$ positive for any graph?
 2. What are some equivalent conditions stated in terms of graph properties to $g >0$?

  [1]: https://en.wikipedia.org/wiki/Friendship_paradox
  [2]: https://i.sstatic.net/jmE3Z.png
  [3]: https://en.wikipedia.org/wiki/Degree_distribution#Excess_degree_distribution