A Comparison of Four Monte Carlo Methods for Estimating the Probability of s-t Connectedness

This paper describes and compares the performance of four alternative Monte Carlo sampling plans for estimating the probability that two nodes, s and t, are connected in an undirected network whose arcs fail randomly and independently. Models of this type are commonly used when computing the reliability of a system of randomly failing components. The first method, dagger sampling, relies for its advantage on inducing negative correlation between the outcomes of the replications in the sample. The second method, sequential destruction/construction exploits permutation properties of arc failures and successes. The third method uses easily determined bounds on the reliability probability to gain an advantage. The fourth approach enumerates all failure sets of the network to achieve its appeal. An example based on a 30-arc 20-node network illustrates the variance reducing features and the time and space needs of each sampling plan. Dagger sampling and sequential construction offer limited benefits when compared to the bounds and failure-sets methods. The failure-sets method performs best on the example for small failure probabilities. However, in general it offers no guarantee of a smaller variance than crude Monte Carlo and requires substantially more memory than the other methods. Moreover, this memory requirement can grow rapidly as the size of the network increases. By contrast, the bounds method guarantees a smaller variance than for crude Monte Carlo sampling and has modest memory requirements.