MULTI-TERMINAL NETWORK CONNECTEDNESS on SERIES-PARALLEL NETWORKS

Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.