A general theory for random walks on transfinite networks whose ranks
are arbitrary natural numbers is established herein. In such networks,
nodes of higher ranks connect together transfinite networks of lower
ranks. The probabilities for transitions through such nodes are obtain
ed as extensions of the Nash-Williams rule for random walks on ordinar
y infinite networks. The analysis is based on the theory of transfinit
e electrical networks, but it requires that the transfinite network ha
ve a structure that generalizes local-finiteness for ordinary infinite
networks. The shorting together of nodes of different ranks are allow
ed; this complicates transitions through such nodes but provides a con
siderably more general theory. It is shown that, with respect to any f
inite set of nodes of any ranks, a transfinite random walk can be repr
esented by an irreducible reversible Makov chain, whose state space is
that set of nodes.