I have an empirical undirected network. I assume, that a degree distribution is $ F(k) = 1 - e^{1 - \frac{k}{m}} $. and would like to estimate $m$.
The only method I'm aware of for such task is MLE.
If I write a likelihood function $ L = \prod_{i=1}^{N}{p(x_i)} $ (where $ p(k) = \frac{1}{m}e^{1 - \frac{k}{m}} $ and ${x_i}$ is a degree of node $i$), then I can estimate $ m = \bar{x} $, but as I see the result is twice as much as it should be.
I believe this is because I should not write the likelihood function as I did, because degrees of two connected vertices are not independent, so I should take this into account, but I don't quite undestand how can I do this.
Could you confirm or reject this, and if I am right, please could you help me to build the correct likelihood function?
The problem here is that the support of the variable is not $(0;\infty)$ but it is $k \in [m;\infty)$, I suppose (you didn't say where the variable is defined).
So your density is this (I prefer to indicate $x$ the variable instead of $k$)
$f(x,m)=\frac{1}{m}e^{1-\frac{x}{m}}\mathbb{1}_{[m;\infty)}(x)$
and, I suppose, $m >0$
Now, as you can see, the support depends on the parameter. Are you able to continue by yourself?
EDIT: anyway...here is the solution
The likelyhood is the following
$L(m)=\frac{1}{m^n}e^{n-\frac{\sum_{i}x_i}{m}}\mathbb{1}_{(0;x_{(1)}]}(m)$
so the MLE estimator is
$\hat{m}=min[x_{(1)};\bar{x}]=x_{(1)}$
where $x_{(1)}=min(x)$