On fitting a Poisson distribution to make sense of data
Unlike other graphs, the degree distribution is a function of N. Specifically, the number of nodes with k=2 is constant (4), the  number of nodes with k=3 grows linearly with the length of one side $4l-4$, while the number of nodes with k=4 grows exponentially with the length of a side, $l^2-4l-4$. In the large $l$ (or $N$) limit, the degree distribution is 4. 

You can get the exact distribution by normalizing this piecewise defined function (divide by $l^2=N$).