Random walks and the effective resistance of networks (Q756281)

A simple network is a finite electrical network comprised only of unit resistances. Identifying the network with a graph \(G=(V,E)\), a random walk on G is defined as a stochastic process on V in which a transition from x, independently of the past, leads to each of the d(x) (the degree) neighbors of x with probability 1/d(x). A random walk from x to y begins at the former and terminates when it first enters the latter. It is shown that the effective resistance \(R_{xy}\) between nodes x and y is equal to the expected number of traversals along any specific edge (x,z) by a random walk from x to y. Several consequences of this result are discussed, including the reciprocity theorem for networks.