Abstract
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 ∈ 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 |→∞, For the special case when each edge-resistance takes the values 1 and ∞ ohms with probabilities p and 1- p respectively, these problems deal with the strength of connectivity of random graphs .
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