Universal Reliability Bounds for Sparse Networks

Consider a graph with perfect nodes and edges subject to independent random failures with identical probability. The all-terminal reliability is the probability that the resulting subgraph is connected. First, we fully characterize uniformly least reliable graphs (ULRG) whose co-rank is not greater than four. Universal reliability bounds are here introduced for those graphs. It is formally proved that ULRG is invariant under bridge-contractions, and maximize the number of bridges among all connected simple graphs with a prescribed number of nodes and edges. A closed-form for the maximum number of bridges is also given, which has an intrinsic interest from a graph-theoretic point of view. Finally, the cost-reliability tradeoff is discussed, comparing the number of edges required to reduce the reliability gaps between the least and most reliable graphs. A remarkable conclusion is that the network design is critical under rare event failures, where the reliability-gap between least and most-reliable networks is monotonically increasing with the number of terminals.