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

The reliability polynomial R_φ(p) of a collection φ of subsets of a finite set X has been extensively studied in the context of network theory. There, X is the edge set of a graph (V, X) and φ the collection of the edge sets of certain subgraphs. For example, we may take φ to be the collection of edge sets of spanning trees. In that case, R_φ(p) is the probability that, when each edge is included with the probability p, the resulting subgraph is connected. The second author defined R_φ(p) in an entirely different way enabling one to glean additional information about the collection φ from R_φ(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.