I got to the middle of my course Mathematics of Networks, and definitely a lot of things are much clearer for me now than they were before. I am not sure I will be able to cover some applied aspects of this science, including a detailed discussion of interrelations of networks and epidemics, but the schedule that I planned at the very beginning of the course is definitely works out.
I use a lot of references to prepare for my lectures, the list of useful sources is already lengthy and still growing (see the web page), however in this post I would like to point out one particular source. If I were to choose only one reference, this definitely would be my choice. These are the lecture notes by Remco van der Hofstad, who did an incredible job preparing his mathematically rigorous and very readable text. So, if you are interested in the mathematical development of the random network science, these lecture notes (which hopefully will be a textbook soon) is the choice number one, no doubt about it.