Ah. Zemanian, A MAXIMUM PRINCIPLE FOR NODE VOLTAGES IN FINITELY STRUCTURED, TRANSFINITE, ELECTRICAL NETWORKS, Potential analysis, 5(1), 1996, pp. 73-101
Transfinite electrical networks have unique finite-powered voltage-cur
rent regimes given in terms of branch voltages and branch currents, bu
t they do not in general possess unique node voltages. However, if the
ir structures are sufficiently restricted, those node voltages will ex
ist and will satisfy a maximum principle much like that which holds fo
r ordinary infinite electrical networks. The structure that is imposed
in order to establish these results generalizes the idea of local-fin
iteness, Other properties that do not hold in general for transfinite
networks but do hold under the imposed structure are Kirchhoff's curre
nt laws for nodes of any ranks and the permissibility of connecting pu
re voltage sources to such nodes. This work lays the foundation for a
theory of transfinite random walks, which will be the subject of a sub
sequent work.