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.