The model: We consider the Barabási–Albert model
The result I want to show: for that model, show that the degree distribution follows a power law with exponent $=3$.
I read this proof (page 2) and this one from the original paper which are the same. But I don't understand:
- Why can we consider $t$ as continuous? (how to show that it's legit).
- Why can you use the derivative and why $\dfrac{d}{dt}(d_i(t))= \dfrac{d_i(t)}{\sum d_i(t)}$?
The short answer is, well, because, all the "proofs" you refer to are not rigorous mathematical proofs. Moreover even the statement of the model is ambiguous: they start with a network of $m$ nodes without even specifying what exactly this network is.
So here are two ways to proceed: Accept this kind of reasoning and train your intuition (very often supported by numerical evidence) to become a physicist. Or read the careful mathematical proof, which can be found, e.g., in Random Graphs and Complex Networks: Volume 1 by Remco van der Hofstad (amazon but pdf should be easy to find). Remember, however, that the actual proof takes a little more than two pages.