Designing electrical networks is among the highly mathematical engineering disciplines, which uses a vast scope of techniques from Fourier analysis and complex function theory, to logic, combinatorics and topology. But, at least to me, with my minor knowledge of electrical engineering, it seems that at the end of the day, what is physically built out of these theories--I mean in a manufacturer laboratory-- is always a finite graphs with nodes labeled with "simple functions", in a way that the whole thing is again a function, with desired characteristics. But, from a mathematical point of view, it is customary to investigate such structured functions, in a categorical context and exploit the language and power of category theory, much like what programmers do.
Here, I do not dare to further these vague ideas and pose my question:
What is the right and fruitful mathematical foundation for the theory of electrical networks? Is there any purely axiomatic approach to the subject, accessible to a mathematics enthusiast with minor background in electrical engineering.