A Note on Network Reliability

Let G = (V, E)be a loopless undirected multigraph on n vertices, with a probability p e 0 ⩽ p e ⩽ 1 assigned to every edge e ϵ E. Let G p be the random subgraph of G obtained by deleting each edge e of G, randomly and independently, with probability q e = 1 — p e . For any nontrivial subset S ⊂ V let (S, \(\bar{S}\)) denote, as usual, the cut determined by S, i.e., the set of all edges of G with an end in S and an end in its complement \(\bar{S}\). Define \(P(S) = {{\sum }_{{e \in (S,\bar{S})}}}{{p}_{e}}, \) and observe that P(S) is simply the expected number of edges of Gp that lie in the cut (S,). In this note we prove the following.