I have read the docs about the degree distribution of the networks generated by static.power.law.game().It seems that the degree distribution of these networks should be, where lamda is the power,
is a normalization constant and
(but this formula can not show the probability of the degree being zero)
That’s not true; if your graph has N vertices only, then the normalization constant has to be truncated at the N-th term instead of summing all the way to infinity. The degree distribution you wrote is valid for infinite networks only; the reason why this is usually mentioned in the literature is because finite networks cannot be “pure” power-law networks in the strict theoretical sense since there will always be some kind of a finite size effect. For instance, if your graph has 1 million vertices, then the probability of seeing a vertex with 2 million incident edges is zero (unless you allow multiple edges), therefore the degree distribution cannot be a pure power law.
But the analytic line and the real data of in-degree (or out-degree) distribution are not fitted with each other.
They will not fit exactly; all that matters is that the theoretical line is parallel to the plotted empirical distributions, confirming that the *exponents* are the same. The offset you see between the two lines can be derived from the normalization constant. Also, note that the analytic line won’t ever be able to fit the in-degree, the out-degree and the total degree lines exactly because the total degree is always the sum of the in-degree and the out-degree, hence it will always be offset from the origin by a different amount than the in-degree and out-degree lines (but they will be parallel).
So, in a nutshell, I don’t see any problem at all with your figures.
T.