Fitness model for scale free networks
I don't know if my solution is correct, but after some searching, I eventually came up to this conclusion:

The algorithm that is used to build the graph is not compatible with a different choice other than $\theta_i = i^{-\alpha}$. 

This is because the algo is derived from the  Barabási–Albert with some modifies. We start with N nodes, and we add a link between a couple of nodes every time step until we have added $mN$ edges. First we normalized the weights so they will be the probability that we choose the node $i, \;$  $\sum_{i=1}^{N} i^{-\alpha} \sim \frac{1-\alpha}{N^{1-\alpha}}$. We can easily compute the rate at which the node acquires links, since it will be proportional to its degree:
\begin{equation}
\frac{d\,k_i}{dt} = \frac{k_i}{2mN} =  \frac{1-\alpha}{N^{1-\alpha} i^\alpha},
\end{equation}
this means that we will select the node $i$ with probability $p(i) = k_i ^{-\frac{1}{\alpha}}$. This is equal to the probability that a node has a degree $k&gt;k_i$, so it is related to the cumulative probability distribution:
\begin{equation}
P(k&gt;\bar{k}) \sim k^{-\frac{1}{\alpha}} \rightarrow p(k) \sim k^{-\frac{1}{\alpha} -1} \equiv k^{-\gamma},
\end{equation}
where $\gamma = \frac{1+\alpha}{\alpha}$.
Is this a good answer?