Infinite networks: I--Resistive networks

There are several examples of infinite networks of resistors; it is always assumed that a unique current exists as a consequence of Kirchhoff´s laws. Actually, unlike the situation in finite networks, these laws are insufficient to determine a unique current. A plausible set of network laws are formulated and two main theorems are proved. 1) In an infinite network consisting of nonnegative resistors (with no short circuits) and a finite number of sources, there exists a unique current flow. 2) This current flow is the limit of the unique current flows in finite, subnetworks that approximate the whole network. Methods of algebraic topology and Hilbert space theory are used in the formulations and proofs.