A Fast and Simple Unbiased Estimator for Network (Un)reliability

The following procedure yields an unbiased estimator for the disconnection probability of an n-vertex graph with minimum cut c if every edge fails independently with probability p: (i) contract every edge independently with probability 1- n
^-2/c
, then (ii) recursively compute the disconnection probability of the resulting tiny graph if each edge fails with probability n
^2/c
p. We give a short, simple, self-contained proof that this estimator can be computed in linear time and has relative variance O(n
²
). Combining these two facts with a standard sparsification argument yields an O(n
³
log n)-time algorithm for estimating the (un)reliability of a network. We also show how the technique can be used to create unbiased samples of disconnected networks.