A note on network reliability

Let G = (V,E) be a loopless undirected multigraph, with a probability pe, 0 ≤ pe ≤ 1 assigned to every edge e ∈ E. Let Gp be the random subgraph of G obtained by deleting each edge e of G, randomly and independently, with probability qe = 1 − pe. For any nontrivial subset S ⊂ V let (S, 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 S. Define P (S) = ∑ e∈(S,S) pe, and observe that P (S) is simply the expected number of edges of Gp that lie in the cut (S, S). In this note we prove the following.