Is there any abstract theory of electrical networks? 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.

Many Thanks.
 A: John Baez has written a bit about the mathematics of circuit theory and networks in general, in a categorical framework.
http://ncatlab.org/johnbaez/show/Circuit+theory+I
http://math.ucr.edu/home/baez/networks/
A: The mathematical foundation of electrical networks/electrical circuits is mainly rooted in dynamical systems theory. It may be somewhat of a disappointment, but the models of electrical circuits, etc. are not sufficiently unique that they demand a new, novel field of mathematics. Indeed, the behavior of most systems is describable using the same mathematical techniques of studying spring-mass-damper systems, or the like.
You will not find an axiomatic foundation of any engineering discipline. This is rooted in the nature of physics -- there are no fundamental axioms that we have found in the physical world. There are phenomena that we cannot yet explain, but that is not the same thing as an axiomatic construction.
Of course, any mathematical model is built upon some set of axioms of mathematics (laws of arithmetic, axiom of choice, etc.), but in the advanced study of these models, one very rarely resorts to purely axiomatic arguments. This is because these fields are deliberately constructed to extend the axioms of mathematics to more complex proofs relating the properties and structures of mathematical constructions to other properties/structures in rigorous ways. Therefore, you can study a differential equation without having to fundamentally construct the proof of existence of a solution axiomatically; the Peano existence theorem might be sufficient, instead.
A: The theory of electrical circuits consists of several sub-theories. Predominant mathematical disciplines that arise in the study of electrical circuits are linear algebra, differential equations, functional analysis (Fourier Transform, Laplace Transform) and graph theory. A circuit is a physical system which implements a mathematical function. Thus a circuit can be abstracted from its physics into its mathematical behavior and one can choose to study the latter. Then the mathematical behavior is captured by what is known as a "system", which there are various ways to define mathematically. Some authors begin by defining the "input space" and the "output space" and then a system is a particular kind of "morphism" between those two spaces. Another abstract approach is to define input and output spaces and then take a system to be a subset of the cartesian product of the input and output space, i.e. the set of all input-output signal pairs that can occur in the system. This is known as the behavioral approach. Another approach is the algebraic approach. As an example, Rudolf Kalman, the giant of mathematical systems theory, about 50 years ago, wrote a paper saying that a linear system is actually a module over a principal ideal domain. This opened the road for algebraic systems theory and if you like categories, you will find many interesting things there. But if you want to make a start, on the textbook level, i recommend any good book on Signals and Systems (e.g. Oppenheim's) or on Control Theory e.g. William Brogan's "Modern Control Theory". Warning: the further you go on this road, the less relevant your study will be with actual circuit design and analysis practices used in industry. This is because there is a huge distance to be covered from having a meaningful and useful theory to actually using this theory and modifying it locally in order to obtain something that actually works. Let me give you a simple example: there is only one system model of the BJT transistor (and a couple more equivalent) but there are hundreds of various types of BJT transistors, very different from each other. These differences are not captured in the system theory level, but they are crucial for implementation.
A: There are related approaches to network theory tensors and using (co)homology theory. The more topological approach is outlined in John Baez's Circuits notes already cited by @espen180. There is an excellent exposition of this approach in P. Bamberg and S. Sternberg's A Course of Mathematics for Students of Physics vol 2
The, IMO more powerful, tensor approach originates Gabriel Kron who called it Diakoptics. Kron's work was taken up by Kazuo Kondo's group in Japan (the RAAG) who published much their work in the RAAG Memoirs. Unfortunatelly most of this work predates the internet, and is hard to get hold of electronically or otherwise. The books including the RAAG Memoirs are out of print and rather rare. Papers were often published in obscure now discountinued journals. If you search, some names to look for are Banesh Hoffmann, A Brameller, F H Branin, J P Roth, W J Gibbs, H H Happ, L V Bewley & J W Lynn.
