A Note on Network Reliability

Abstract

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.

This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the NSF and by the Sloan Foundation, Grant No. 93-6-6.