Linear algebra and sequential importance sampling for network reliability

The reliability polynomial of a graph gives the probability that a graph is connected as a function of the probability that each edge is connected. The coefficients of the reliability polynomial count the number of connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph's reliability polynomial. We develop a new bottom-up SIS algorithm for estimating the reliability polynomial by choosing a spanning tree and adding edges. This algorithm improves on existing bottom-up algorithms in that it has lower complexity ≈ O(E
²
) as opposed to O(EV
³
), and it uses importance sampling to reduce variance.