Circuit Theory and Other Discrete Physical Models

Abstract

In this chapter we present linear electric circuit theory as the central physical model for performing calculus on networks. We adopt circuit theory as the central physical model for applying and understanding the concepts of discrete calculus on graphs for three reasons: because much of the progress in graph theory over the last century was created in the context of circuit theory; because of the early connection made between circuit theory and algebraic topology (Branin Jr. in Proc. of Conf. on Neural Networks, pp. 453–491, 1966; Roth in Proceedings of the National Academy of Sciences of the United States of America 41(7):518–521, 1955); and because circuits are physical, realizable systems which need not be seen as discretizations of an underlying continuous domain but rather as a domain unto themselves. Our focus in this chapter will be to cover the main concepts in linear circuit theory from an algebraic standpoint with a focus on operators. This will prepare the reader with notation and concepts that tie naturally to the previous chapter in which the discrete analogues of differential operators were introduced. We begin with definitions of the physical quantities and corresponding quantities in circuit theory, then proceed to define the laws that relate the quantities to each other, treat methods of solving for the unknowns, and end by connecting circuit theory to other discrete processes.