You may rely on the reliability polynomial for much more than you might think

The reliability polynomial R-g(p) of a collection g of subsets of a. nite set X- has been extensively studied in the context of network theory. There, X is the edge set of a graph (V, X) and g the collection of the edge sets of certain subgraphs. For example, we may take g to be the collection of edge sets of spanning trees. In that case, R-g(p) is the probability that, when each edge is included with the probability p, the resulting subgraph is connected. The second author defined R-g(p) in an entirely different way enabling one to glean additional information about the collection g from R-g(p). Illustrating the extended information available in the reliability polynomial is the main focus of this article while demonstrating the equivalence of these two definitions is the main theoretical result.