Transfinite graphs and electrical networks

Abstract

All prior theories of infinite electrical networks assume that such networks are finitely connected, that is, between any two nodes of the network there is a finite path. This work establishes a theory for transfinite electrical networks wherein some nodes are not connected by finite paths but are connected by transfinite paths. Moreover, the voltages at those nodes may influence each other. The main difficulty to surmount for this extension is the construction of an appropriate generalization of the concept of connectedness. This is accomplished by extending the idea of a node to encompass infinite extremities of a graph. The construction appears to be novel and leads to a hierarchy of transfinite graphs indexed by the finite and infinite ordinals. Two equivalent existence and uniqueness theorems are established for transfinite resistive electrical networks based upon Tellegen’s equation, one using currents and the other using voltages as the fundamental quantities. Kirchhoff’s laws do not suffice for this purpose and indeed need not hold everywhere in infinite networks. Although transfinite countable electrical networks have in general an uncountable infinity of extremities, called "tips," the number of different tip voltages may be radically constrained by both the graph of the network and its resistance values. Conditions are established herein under which various tip voltages are compelled to be the same. Furthermore, a theorem of Shannon-Hagelbarger on the concavity of resistance functions is extended to the driving-point resistance between any two extremities of arbitrary ranks. This is based upon an extension of Thomson’s least power principle to transfinite networks.