Nonlinear, resistive, countably infinite electrical networks †

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Abstract
Nonlinear infinite electrical networks can be analysed by using Hilbert-space techniques when the total power in the network is finite. This work attacks the case where the total power is not finite. Graph-theoretic methods are used to show that a unique current flow occurs in a countably infinite, nonlinear, resistive network after the voltage-current pairs in certain specified branches are arbitrarily assigned. The currents and voltages throughout the entire network can be determined by computing them recursively in a sequence of finite subnetworks that partition the network. The main theorem requires that the resistances and conductances in the network satisfy sufficiently strong Lipschitz conditions.

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