Summary: [For the entire collection see Zbl 0582.00006.]
An electrical network is a graph \(G=(V,E)\) with two sets I, O of vertices, called input and output vertices, such that each edge e has some electrical resistance R(e) ohms. We suppose that the family \(\{\) R(e): \(e\in E\}\) is a collection of independent, identically distributed random variables, and we are interested in the effective (random) resistance R(G) of the network G between I and O. There are three main cases of interest, when G is a branching tree, or a complete graph or a subsection of some crystalline lattice; for these cases, we discuss the asymptotic properties of R(G) in the limit as \(| V| \to \infty\). For the special case when each edge-resistance takes the values 1 and \(\infty\) ohms with probabilities p and 1-p respectively, these problems deal with the strength of connectivity of random graphs.