Countably Infinite Networks that Need not be Locally Finite

A network that is not locally finite is an infinite network with at least one node having an infinity of incident branches. Such networks arise naturally when the short circuits in a locally finite infinite network are coalesced into nodes. Although a number of existence and uniqueness theorems for the behavior of a locally finite network have been given in the literature, such is not the case for nonlocally finite networks. This paper provides such a theorem for the branch-voltage drops of a nonlocally finite resistive network having an infinity of current sources satisfying a certain restriction. The conditions under which this result holds are Kirchhoff's loop law, the finiteness of the total power dissipation, and a generalized form of Tellegen's theorem. However, Kirchhoff's node law need not hold at those nodes having an infinity of incident branches. Voltage sources can be taken into account by making appropriate changes of sources.